New STPM Mathematics (T) Chapter Past Year Question

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NEW STPM Mathematics (T) Topical Past Year Questions Compiled by: KK LEE (Lee Kian Keong) August 8, 2016

Abstract

This documents contains all the questions from STPM Past Year Papers and i sorted all the questions according to the chapters (New STPM syllabus). Download the file from scribd is not allowed. If you willing to download the file, please contact me by facebook me or download directly from my website http://kkleemaths.com. I used more than 5 years to make this file. Please appreciate my hard work.

Contents

1 Functions

2 Sequences and Series 3 Matrices

4 Complex Numbers 5 Analytic Geometry 6 Vectors

7 Limits and Continuity 8 Differentiation 9 Integration

10 Differential Equations 11 Maclaurin Series

12 Numerical Methods 13 Data Description 14 Probability

15 Probability Distributions 16 Sampling and Estimation 17 Hypothesis Testing 18 Chi-squared Tests

2 15 24 33 36 38 44 48 64 79 89 96 99

120

131 166 175 178

1

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1: Functions

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Functions

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1

STPM MATHEMATICS (T)

1. [STPM ] Find the value of x, with 0◦ < x < 360◦ , which satisfies equation sec x + tan x = 4. Give your answers correct to the nearest 0.1◦ . [6 marks] [Answer : 62◦ ]

2. [STPM ] √ Sketch the graph of y = |1 − 2x|, x ∈ R and the graph of y = x, x ≥ 0 on the same coordinate system. Solve the inequality |1 − 2x| >

√

x.

3. [STPM ] Function f is defined as

[3 marks] [4 marks]

1 4

[Answer : {x : 0 ≤ x < , x > 1}]

( x(x − π), 0 ≤ x < 2π; f (x) = 2 π sin(x − π), 2π ≤ x ≤ 3π.

(a) Sketch the graph of f . (b) Find the range of f .

[4 marks] [3 marks]

(c) Determine whether f is a one-to-one function. Give reasons for your answer.

[Answer : (b) {y : π 2 ≤ y < 2π 2 } ; (c) f is not one-to-one function.]

4. [STPM ] Solve the equation

2 logx 3 − log3

5. [STPM ] The function f is defined as follows:

√

3 x= . 2

[6 marks]

[Answer : x = 3,

f : x → 4 + (x − 1)2 , x ∈ R.

(a) Sketch the graph of f . (b) State the range of f . (c) Determine if f

[2 marks]

−1

exist.

1 ] 81

[2 marks] [1 marks] [2 marks]

[Answer : (b) {y : y ≥ 4} ; (c) No]

6. [STPM ] Given that x + 2 is a factor of f (x) = x3 + (a + 2b)x2 + (a − 3b)x + 8. Find a in terms of b, and find q(x) so that f (x) = (x + 2)q(x) holds for all values of b. [5 marks] Determine the values of b so that f (x) = 0 has at least two distinct real roots. 6 2 Sketch on different diagram, the graph of y = f (x) when b = − and b = . 5 5 2

[6 marks]

[4 marks]

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1: Functions 6 5

2 5

Function f is defined by f (x) = f (x) > f (x − 1).

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7. [STPM ]

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[Answer : a = −7b ; q(x) = x2 − (5b + 2)x + 4 ; {b : b < − , b ≥ }]

1 with x ∈ R and x 6= 0. Determine the set of values of x so that x [5 marks]

[Answer : {x : 0 < x < 1}]

8. [STPM ] Given that x3 + mx2 + nx − 6 is divisible by x − 3 and x + 2. Find the values of m and n.

[5 marks]

[Answer : m = 0, n = −7]

9. [STPM ] Given that f (x) = log2 (15 − 2x − x2 ). Find the range of x so that f (x) is defined. 2

[3 marks]

Find the maximum value of 15 − 2x − x and hence deduce the maximum value of f (x).

[4 marks]

[Answer : {x : −5 < x < 3} ; 16, 4]

10. [STPM ] Express sin x − 3 cos x in the form r sin(x − α), with r > 0 and 0◦ < α < 90◦ , giving the value of α correct to the nearest 0.1◦ . Sketch the curve y = sin x − 3 cos x for 0◦ ≤ x ≤ 360◦ . [8 marks] Find the range of values of x between 0◦ and 360◦ which satisfies the inequality sin x − 3 cos x ≥ 2. Find the largest and the smallest value for [Answer :

11. [STPM ] Solve the equation

√

1 . sin x − 3 cos x + 5

[4 marks] [3 marks]

10 sin(x − 71.6◦ ) ; {x : 110.3◦ < x < 212.9◦ } ;

s

4x √ = 3. 1− x

5−

1 √

10

,

5+

1 √

10

]

[5 marks]

[Answer : x =

9 ] 16

12. [STPM ] Determine the values of k so that the quadratic equation x2 + 2kx + 4k − 3 = 0 has two distinct real roots. [4 marks]

13. [STPM ] The function f is defined as follows:

f :x→

5x + 2 , x 6= 5 x−5

(a) Find f 2 and hence deduce f −1 .

[Answer : {k : k < 1, k > 3}]

[3 marks]

3

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[3 marks]

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(b) Find f 13 (2). [Answer : (a) f 2 (x) = x , f −1 (x) =

14. [STPM ] Show that the roots of x2 + bx + c = 0, a 6= 0 are given by √ −b ± b2 − 4ac x= . 2a

5x + 2 ; (b) −4] x−5

Deduce that if m + ni, with m, n ∈ R, is a root of this equation, then m − ni is another root. [5 marks] (a) Show that 2+i is a root of f (x) = 0 where f (x) = 2x3 − 5x2 − 2x + 15, and find its other roots. [5 marks]

(b) Find a polynomial g(x) so that f (x) − xg(x) = 15 − 7x. Express g(x) in the form p(x − q)2 + r, 1 . [5 marks] with p, q, r ∈ R, find the maximum of g(x) 3 2

[Answer : (a) 2-i , − ; (b) g(x) = 2 x −

15. [STPM ] The function f is defined by

( 2 − |x − 1|, f (x) = x2 − 9x + 18,

(a) Sketch the graph of f .

x < 3, x ≥ 3.

5 4

2

+

15 8 , ] 8 15

[5 marks]

x (b) Determine the set of x so that f (x) > 1 − . 6

[5 marks]

[Answer : (b) {x : 0 < x <

12 , x > 6}] 5

16. [STPM ] Express 9 sin θ − 6 cos θ in the form r sin(θ − α), with r > 0 and 0◦ < α < 90◦ . Hence, find the smallest and the largest value for 9 sin θ − 6 cos θ − 1. [6 marks] √

√

√

[Answer : 3 13 sin(θ − 33.7◦ ) , −3 13 − 1 , 3 13 − 1]

17. [STPM ] Given that f (x) = x3 + px2 + 7x + q where p, q are constants. When x = −1, f 0 (x) = 0. When f (x) is divided by (x + 1), the remainder is −16. Find the values of p and q. [4 marks] (a) Show that f (x) = 0 only has one real roots. Find the set of values of x such that f (x) > 0. x+9 (b) Express in partial fraction. f (x)

[Answer : p = 5, q = −13 ; (a) {x : x > 1} ; (b)

4

[6 marks]

[5 marks]

x+5 1 − ] 2(x − 1) 2(x2 + 6x + 13)

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1: Functions

19. [STPM ]

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1 − 2x as partial fractions. x2 (1 + 2x2 )

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Express

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18. [STPM ] [5 marks]

2 x

[Answer : − +

1 4x − 2 + ] 2 x 1 + 2x2

1 1 Express the function f : x → | x − 1| + | x + 1|, x ∈ R, in the form that does not involve the modulus 2 2 sign. Sketch the graph of f and determine its range. [7 marks]   −x, [Answer : f : x → 2,   x,

x < −2 −2 ≤ x < 2 , x≥2 R={y : y ≥ 2}]

20. [STPM ] Function f is defined by f (x) = x2n − (p + 1)x2 + p, where n and p are positive integers. Show that x − 1 is a factor of f (x) for all values of p.

[3 marks]

When p = 4, x − 2 is a factor of f (x). Find the value of n and factorise f (x) completely.

[5 marks]

2

With the value of n you have obtained, find the set of values of p such that f (x) + 2x − 2 = 0 has roots which are distinct and real. [7 marks] [Answer : n = 2 , (x − 2)(x + 2)(x − 1)(x + 1) ; {p : p > 2, p 6= 3}]

21. [STPM ] Solve the simultaneous equations

1 log4 (xy) = , (log2 x)(log2 y) = −2. 2

22. [STPM ] The functions f and g are defined by

[6 marks]

f : x → 2x, x ∈ R;

g : x → cos x − | cos x|, −π ≤ x ≤ π.

(a) Find the composite function f ◦ g and state its domain and range. (b) Show, by definition, that f ◦ g is an even function. (c) Sketch the graph of f ◦ g.

1 2

[4 marks]

[2 marks] [2 marks]

[Answer : (a) f ◦ g : x → 2(cos x − | cos x|), D = {x : −π ≤ x ≤ π}, R = {y : −4 ≤ y ≤ 0}]

23. [STPM ]

√

1 3x + 1, x ∈ R, x ≥ − . 3 and state its domain and range.

The function f is defined by f : x → Find f −1

1 2

[Answer : x = , y = 4, x = 4, y = ]

5

[4 marks]


x2 − 1 1 , Df −1 = {x : x ≥ 0}, Rf −1 = {x : x ≥ − }] 3 3

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[Answer : f −1 : x →

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24. [STPM ]q √ √ √ Express 59 − 24 6 as p 2 + q 3 where p and q are integers.

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[7 marks]

√

√

[Answer : 4 2 − 3 3]

25. [STPM ] Show that polynomial 2x3 − 9x2 + 3x + 4 has x − 1 as factor. Hence,

[2 marks]

(a) find all the real roots of 2x6 − 9x4 + 3x2 + 4 = 0.

[5 marks]

(b) determine the set of values of x so that 2x3 − 9x2 + 3x + 4 < 12 − 12x.

[6 marks]

[Answer : x = 1, x = −1, x = 2, x = −2 ; x < 1]

26. [STPM ] √ π Express cos x + 3 sin x in the form r cos(x − α), with r > 0 and 0 < α < . [4 marks] 2 √ Hence, find the set of values of x with 0 ≤ x ≤ 2π, which satisfies the inequality 0 < cos x+ 3 sin x < 1. [5 marks]

[Answer : 2 cos x −

5π 11π π 2π
27. [STPM ] Show that −1 is the only one real root of the equation x3 + 3x2 + 5x + 3 = 0. 28. [STPM ] Find the set of values of x such that −2 < x3 − 2x2 + x − 2 < 0.

29. [STPM ]

[5 marks]

[7 marks]

[Answer : {x : 0 < x < 1, 1 < x < 2}]

1 Sketch, on the same coordinate axes, the graphs of y = 2 − x and y = 2 + . x 1 Hence, solve the inequality 2 − x > 2 + . x

[4 marks]

[4 marks]

√

[Answer : {x : x < 2 − 5}]

30. [STPM ] Express cos θ + 3 sin θ in the form r cos(θ − α), where r > 0 and 0◦ < α < 90◦ .

[4 marks]

[Answer :

√

10 cos(θ − 71.6◦ )]

31. [STPM ] Find all values of x, where 0◦ < x < 360◦ , which satisfy the equation tan x + 4 cot x = 4 sec x. [5 marks]

6

[Answer : 41.8◦ , 138.2◦ ]

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1: Functions

33. [STPM ] The functions f and g are given by f (x) =

1 where x 6= 0. x

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Find the solution set of inequality |x − 2| <

ex − e−x ex + e−x

(a) State the domains of f and g,

[7 marks]

√

[Answer : {x : 0 < x < 1 + 2, x 6= 1}]

and g(x) =

ex

2 . + e−x

[1 marks]

(b) Without using differentiation, find the range of f , 2

2

[4 marks]

(c) Show that f (x) + g(x) = 1. Hence, find the range of g.

(x2

[6 marks]

[Answer : (a) {x : x ∈ R}, {x : x ∈ R} ; (b) {y : −1 < y < 1} ; (c) {y : 0 < y ≤ 1}]

34. [STPM ] Express

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32. [STPM ]

Ax + B C 2x + 1 in the form 2 + where A, B and C are constants. + 1)(2 − x) x +1 2−x

35. [STPM ] Functions f , g and h are defined by f :x→

x ; x+1

(a) State the domains of f and g.

g:x→

x+2 ; x

[Answer :

h:x→3+

(b) Find the composite function g ◦ f and state its domain and range. (c) State the domain and range of h.

(d) State whether h = g ◦ f . Give a reason for your answer.

2 . x

[3 marks]

1 x + ] x2 + 1 2 − x

[2 marks]

[5 marks]

[2 marks]

[2 marks]

[Answer : (a) {x : x ∈ R, x 6= −1}, {x : x ∈ R, x 6= 0} ; 2 (b) g ◦ f (x) = 3 + , D={x : x ∈ R, x 6= 0, x 6= −1}, R={y : y ∈ R, y 6= 1, y 6= 3} ; x (c) D={x : x ∈ R, x 6= 0}, R={y : y ∈ R, y 6= 3} ; (d) h 6= g ◦ f ]

36. [STPM ] The polynomial p(x) = x4 + ax3 − 7x2 − 4ax + b has a factor x + 3 and when divided by x − 3, has remainder 60. Find the values of a and b and factorise p(x) completely. [9 marks] 1 Using the substitution y = , solve the equation 12y 4 − 8y 3 − 7y 2 + 2y + 1 = 0. [3 marks] x 1 3

1 1 2 2

[Answer : a = 2, b = 12, (x + 3)(x − 1)(x + 2)(x − 2) , y = − , 1, − , ]

37. [STPM ] Express 4 sin θ − 3 cos θ in the form R sin(θ − α), where R > 0 and 0◦ < α < 90◦ . Hence, solve the equation 4 sin θ − 3 cos θ = 3 for 0◦ < α < 360◦ . [6 marks] 7

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38. [STPM ] Find the domain and the range of the function f defined by

Sketch the graph of f .

f (x) = sin−1

2(x − 1) . x+1

[4 marks] [3 marks]

[Answer : D={x :

π π 1 ≤ x ≤ 3} , R={y : − ≤ y ≤ }] 3 2 2

39. [STPM] x If loga 2 = 3 loga 2 − loga (x − 2a), express x in terms of a. a

40. [STPM ] Simplify √ √ ( 7 − 3)2 √ , (a) √ 2( 7 + 3)

[6 marks]

[Answer : x = 4a]

[3 marks]

√

√

[Answer : 2 7 − 3 3]

41. [STPM ] Find the constants A, B, C and D such that

42. [STPM ]

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[Answer : 5 sin(θ − 36.9◦ ), θ = 73.7◦ , 180.0◦ ]

A B C D 3x2 + 5x = + + + . 2 2 2 (1 − x )(1 + x) 1 − x 1 + x (1 + x) (1 + x)3

[8 marks]

[Answer : A = 1, B = 1, C = −1, D = −1]

1 4 1 Using the substitution y = x + , express f (x) = x3 − 4x − 6 − + 3 as a polynomial in y. [3 marks] x x x Hence, find all the real roots of the equation f (x) = 0. [10 marks] √ 3± 5 ] [Answer : y − 7y − 6 ; x = −1, 2 3

43. [STPM ] Find, in terms of π, all the values of x between 0 and π which satisfies the equation tan x + cot x = 8 cos 2x.

[4 marks]

[Answer :

8

1 5 12 17 π, π, π, π] 24 24 24 24

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44. [STPM ] The function f and g are defined by

Find f ◦ g and its domain.

45. [STPM ]

f :x→

1: Functions

1 , x ∈ R \ {0}; x

g : x → 2x − 1, x ∈ R.

[4 marks]

[Answer : f ◦ g(x) =

1 1 , D={x : x ∈ R, x 6= }] 2x − 1 2

1 The polynomial p(x) = 2x3 + 4x2 + x − k has factor (x + 1). 2 (a) Find the value of k. (b) Factorise p(x) completely.

46. [STPM ]

[2 marks] [4 marks]

3 2

1 2

[Answer : (a) k = − ; (b) (x + 1)(2x + 3)(2x − 1)]

4 > 3 − 3. Find the solution set of the inequality x − 1 x

47. [STPM ]

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[10 marks]

[Answer : {x : 0 < x < 1, 1 < x < 3}]

θ 2t 1 − t2 If t = tan , show that sin θ = and cos θ = . [4 marks] 2 1 + t2 1 + t2 Hence, find the values of θ between 0◦ and 360◦ that satisfy the equation 10 sin θ − 5 cos θ = 2.[3 marks]

48. [STPM ]

Determine the set of values of x satisfying the inequality

49. [STPM ]

Given that loga (3x − 4a) + loga 3x =

x 1 ≥ . x+1 x+1

[Answer : θ = 36.9◦ , 196.3◦ ]

[4 marks]

[Answer : {x : x < −1, x ≥ 1}]

2 1 + loga (1 − 2a), where 0 < a < , find x. log2 a 2

50. [STPM ] Find the values of x if y = |3 − x| and 4y − (x2 − 9) = −24.

[7 marks]

[Answer :

2 ] 3

[9 marks]

[Answer : x = 7, x = −9]

51. [STPM ] The polynomial p(x) = 6x4 − ax3 − bx2 + 28x + 12, where a and b are real constants, has factors (x + 2) and (x − 2). 9

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1: Functions

(a) Find the values of a and b, and hence, factorise p(x) completely. 3

[7 marks]

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(b) Give that p(x) = (2x − 3)[q(x) − 41 + 3x ], find q(x), and determine its range when x ∈ [−2, 10]. [8 marks]

[Answer : (a) a = 7, b = 27 , (x + 2)(x − 2)(2x − 3)(3x + 1) ; (b) q(x) = (x − 6)2 + 1 , R=[1,65]]

52. [STPM ] Find the values of x, where 0 ≤ x ≤ π, which satisfy the equation sin3 x sec x = 2 tan x.

[Answer : x = 0, π]

53. [STPM ] Solve the following simultaneous equations:

log3 (xy) = 5

54. [STPM ] The graph of a function f is as follows:

(a) State the domain and range of f .

and

log9

[4 marks]

x2 y

= 2.

[4 marks]

[Answer : x = 27, y = 9]

[2 marks]

(b) State whether f is a one-to-one function or not. Give a reason for your answer.

[2 marks]

[Answer : (a) D={x : −3 ≤ x < −1 − 1 < x ≤ 2} , R={y : −1 < y < 2} ; (b) f is not one to one function.]

55. [STPM ] The polynomial p(x) = 2x4 − 7x3 + 5x2 + ax + b, where a and b are real constants, is divisible by 2x2 + x − 1. (a) Find a and b.

(b) For these values of a and b, determine the set of values of x such that p(x) ≤ 0.

[4 marks] [4 marks]

1 2

[Answer : (a) a = 9, b = −5 ; (b) {x : −1 ≤ x ≤ }] 10

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1: Functions

1 . 5 sin θ + 12 cos θ + 15

57. [STPM ] Solve the equation ln x + ln(x + 2) = 1.

[7 marks]

[Answer : 13 sin(θ + 67.4◦ ) ,

f :x→

√

[Answer : −1 + 1 + e]

[6 marks]

[Answer : {x : x ≤ 2}]

1 x for x 6= ; 2x − 1 2

g : x → ax2 + bx + c, where a, b and c are constants.

(a) Find f ◦ f , and hence, determine the inverse function of f . −3x2 + 4x − 1 (b) Find the values of a, b and c if g ◦ f (x) = . (2x − 1)2 x2 − 2 in terms of f and p. (c) Given that p(x) = x2 − 2, express h(x) = 2 2x − 5 [Answer : (a) f ◦ f (x) = x , f −1 (x) =

1 1 , ] 2 28

[4 marks]

58. [STPM ] Find the set of values of x satisfying the inequality 2x − 1 ≤ |x + 1|.

59. [STPM ] Functions f and g are defined by

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56. [STPM ] Express 5 sin θ + 12 cos θ in the form r sin(θ + α), where r > 0 and 0◦ < α < 90◦ . Hence, find the maximum and minimum values of the expression

[4 marks]

[4 marks]

[2 marks]

x ; (b) a = 1, b = 0, c = −1 ; (c) h = f ◦ p] 2x − 1

60. [STPM ] The polynomial p(x) = ax3 + bx2 − 4x + 3, where a and b are constants, has a factor (x + 1). When p(x) is divided by (x − 2), it leaves a remainder of −9. (a) Find the values of a and b, and hence, factorise p(x) completely. [6 marks] p(x) (b) Find the set of values of x which satisfies ≥ 0. [4 marks] x−3 p(x) (c) By completing square, find the minimum value of , x 6= 3, and the value of x at which it x−3 occurs. [4 marks] [Answer : (a) a = 2, b = −5, (x − 3)(2x − 1)(x + 1) ; (b) {x : x ≤ −1,

1 ≤ x < 3, x > 3} ; (c) Minimum 2 9 1 value=− , x = − ] 8 4

61. [STPM ] √ The expression cos x − 3 sin x may be written in the form r cos(x + α) for all values of x, where r is positive and α is a acute. 11


(a) Determine the values of r and α.

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(b) State the minimum and maximum values of cos x − values of x in the interval 0 ≤ x ≤ 2π. √ (c) Sketch the curve y = cos x − 3 sin x for 0 ≤ x ≤ 2π.

√

1: Functions [3 marks]

3 sin x, and determine the corresponding

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[3 marks] [3 marks]

By drawing an appropriate line on the graph, determine the number of roots of the equation √ 3 x cos x − 3 sin x = 4π in the interval 0 ≤ x ≤ 2π. (d) Solve the equation cos x − [Answer : (a) r = 2, α =

√

[3 marks]

3 sin x = −1 for 0 ≤ x ≤ 2π.

[3 marks]

π 2π 5π π ; (b) minimum=-2 when x = , maximum=2 when x = ; (c) 3 roots ; (d) , π] 3 3 3 3

62. [STPM ] Given that 2 − x − x2 is a factor of p(x) = ax3 − x2 + bx − 2. Find the values of a and b. Hence, find the set of values of x for which p(x) is negative. [6 marks] 1 2

[Answer : a = −2, b = 5 , {x : −2 < x < , x > 1}]

63. [STPM ] √ Functions f and g ◦ f are defined by f (x) = ex+2 and (g ◦ f )(x) = x, for all x ≥ 0. (a) Find the function g, and state its domain. 3

(b) Determine the value of (f ◦ g)(e ).

64. [STPM ]

[5 marks]

[2 marks]

[Answer : (a) g(x) =

√

ln x − 2 , D={x : x ≥ e2 } ; (b) e3 ]

3 x Solve the simultaneous equations log9 = and (log3 x)(log3 y) = 1. y 4


(a) Find f −1 , and state its domain.

[8 marks]

[Answer : x = 9, y =

1 f : x → x2 − x, for x ≥ . 2

[Answer : (a) f

−1

66. [STPM ] Sketch a graph of y = cos 2θ in the range of 0 ≤ θ ≤ π.

√

3 or x =

−1

1 (x) = + 2

−1

.

.

[3 marks]

[3 marks]

r x+

1 1 , D={x : x ≥ − } ; (b) (2,2)] 4 4

Hence, find the set of values of θ, where 0 ≤ θ ≤ π, satisfying the inequality 4 sin2 θ ≥ 2 − 12

1 3 ,y = ] 3 9

[4 marks]

(b) Find the coordinates of the point of intersection of graph f and f (c) Sketch, on the same coordinates axes, the graph of f and f

√

√

[2 marks]

3.[5 marks]

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1: Functions

f : x 7→ e2x , x ∈ R;

g : x 7→ (ln x)2 , x > 0.

π 11π ≤x≤ }] 12 12

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67. [STPM ] The functions f and g are defined by

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[Answer : {x :

(a) Find f −1 and state its domain. 1 = g(2), and state, with a reason, whether g has an inverse. (b) Show that g 2 [Answer : (a) f −1 (x) =

68. [STPM ]

[3 marks]

[4 marks]

1 ln x , D={x : x > 0}] 2

1 Express cos x + sin x in the form r cos(x − α), where r > 0 and 0 < α < π. Hence, find the minimum 2 and maximum values of cos x + sin x and the corresponding values of x in the interval 0 ≤ x ≤ 2π. [7 marks]

(a) Sketch the graph of y = cos x + sin x for 0 ≤ x ≤ 2π.

[3 marks]

(b) By drawing appropriate lines on your graph, determine the number of roots in the interval 0 ≤ x ≤ 2π of each of the following equations. 1 [1 marks] i. cos x + sin x = − , 2 ii. cos x + sin x = 2, [1 marks] (c) Find the set of values of x in the interval 0 ≤ x ≤ 2π for which | cos x + sin x| > 1. [Answer :

√

2 cos(x −

√ √ π 5π π ) , minimum value=− 2 when x = , maximum value= 2 when x = ; (b) (i) two 4 4 4 π 3π roots , (ii) no roots ; (c) {x : 0 < x < , π < x < }] 2 2

69. [STPM ] The function f is defined by f (x) = ln(1 − 2x), x < 0. (a) Find f −1 , and state its domain.

[3 marks]

(b) Sketch, on the same axes, the graphs of f and f −1 .

(c) Determine whether there is any value of x for which f (x) = f

70. [STPM ]

[3 marks]

[4 marks]

−1

(x). 1 2

[3 marks]

[Answer : (a) f −1 (x) = (1 − ex ) ; D={x : x > 0} ; (c) No]

1 Sketch the graph of y = sin 2x in the range 0 ≤ x ≤ π. Hence, solve the inequality | sin 2x| < , where 2 0 ≤ x ≤ π. [6 marks] [Answer : {x : 0 ≤ x <

13

π 5π 7π 11π ,
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1: Functions

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(a) Determine the values of h and k.

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71. [STPM ] The polynomial p(x) = hx4 + kx3 + 2x − 1, where h and k are constants, leaves a remainder of 4 when divided by x − 1, and a remainder of −2 when divided by x + 1. [3 marks]

2

(b) Express the polynomial p(x) in the form (x − 1)q(x) + r(x), where q(x) is quadratic and r(x) is linear. [4 marks] (c) Express q(x) in a completed square form a(x + b)2 + c.

[2 marks]

i. Deduce that q(x) is always positive for all real values of x. ii. Deduce the minimum value of q(x) and the corresponding value of x.

(d) Determine the set of values of x for which p(x) > 3x + 1.

[1 marks]

[2 marks]

[3 marks]

[Answer : (a) h = 2, k = 1 ; (b) (x2 − 1)(2x2 + x + 2) + (3x + 1) ;

(c) q(x) = 2 x +

1 4

2

72. [STPM ]

The function f is defined as f (x) = (a) Show that f has an inverse.

+

15 1 15 (ii) minimum value= , when x = − ; (d) {x : x < −1, x > 1}] 8 8 4

1 x e − e−x , where x ∈ R. 2

[3 marks]

(b) Find the inverse function of f , and state its domain.

[7 marks]

[Answer : (b) f −1 (x) = ln(x +

p

x2 + 1), D={x : x ∈ R}]

73. [STPM ] Sketch, on the same axes, the graphs of y = |2x + 1| and y = 1 − x2 . Hence, solve the inequality |2x + 1| ≥ 1 − x2 . [8 marks]

74. [STPM ]

Determine the set of values of x satisfying the inequality x + 4 ≤

75. [STPM ] Functions f and g are defined by

√

[Answer : {x : x ≤ 1 − 3, x ≥ 0}]

3 . x

[6 marks]

√

√

[Answer : {x : x ≤ −2 − 7, 0 < x ≤ −2 + 7}]

f (x) = x2 + 4x + 2, x ∈ R, g(x) =

3 , x 6= −3, x ∈ R x+3

(a) Sketch the graph of f , and find its range.

(b) Sketch the graph of g, and show that g is a one-to-one function. (c) Give a reason why g −1 exists. Find g −1 , and state its domain. (d) Give a reason why g ◦ f exists. Find g ◦ f , and state its domain.

14

[4 marks]

[3 marks]

[4 marks] [4 marks]

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1: Functions

3 − 3 , D={x : x ∈ R, x 6= 0} ; x 2 (d) g ◦ f (x) = 2 , D={x : x ∈ R}] x + 4x + 5

76. [STPM ] Find the value of x such that (3 − log3 x) log3x 3 = 1.

77. [STPM ] π Express 12 cos θ − 5 sin θ in the form r cos(θ + α), where r > 0 and 0 < α < . 2 Hence,

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[Answer : (a) R={y : y ≥ −2} ; (c) g −1 (x) =

[4 marks]

[Answer : x = 3]

[4 marks]

(a) state the minimum and maximum values of 12 cos θ − 5 sin θ for real values of θ,

[1 marks]

(b) solve the equation 12 cos θ − 5 sin θ = 0, 0 ≤ θ ≤ 2π,

[3 marks]

(c) sketch the graph of y = 12 cos θ − 5 sin θ for 0 ≤ θ ≤ 2π and determine the range of values of θ in this interval satisfying the inequality −5 ≤ 12 cos θ − 5 sin θ ≤ 0. [7 marks] [Answer : 13 cos(θ + 0.395) ; (a) 13, −13 ; (b) 1.176, 4.317 ; (c) {θ : 1.176 ≤ θ ≤

π 5π , ≤ θ ≤ 4.318}] 2 4

78. [STPM ] The polynomial p(x) = ax4 + x3 + bx2 − 10x − 4, where a and b are constants, has a factor (2x + 1). When p(x) is divided by (x − 1), the remainder is −15. (a) Determine the values of a and b. (b) Factorise p(x) completely.

(c) Find the set of values of x which satisfies the inequality p(x) < 0

[4 marks] [3 marks]

[4 marks]

1 2

[Answer : (a) a = 2 b = −4 ; (b) (2x + 1)(x − 2)(x2 + 2x + 2) ; (c) {x : − < x < 2}]

79. [STPM ] Solve the equation cos x − 2 sin x = 2 for 0◦ ≤ x ≤ 360◦ .

[7 marks]

[Answer : 270◦ , 323.1◦ ]

80. [STPM ] √ Functions f and g are defined by f (x) = ln(x − 1), where x > 1 and g(x) = x − 2, where x ≥ 2. (a) Sketch, on separate diagrams, the graphs of f and g. (b)

−1

i. Explain why f exists. ii. Hence, determine f −1 and state its domain.

(c) Find the composite function f ◦ g and state its domain and range. p (d) Express ln(x − 1) − 2 as a composition of functions which involves f and g.

[3 marks]

[6 marks]

[4 marks] [2 marks]

[Answer : (b)(ii) f −1 (x) = ex + 1 , Domain={x : x ∈ R} ;

(c) f ◦ g(x) = ln(

p

(x − 2) − 1) , Domain={x : x > 3} , Range={y : y ∈ R} ; (d) g ◦ f (x)]

15

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1: Functions

f (x) = 2x2 + ax + 7, x ∈ R.

31 Without using differentiation, find the values of a if the range of f is ,∞ . 8

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[5 marks]

[Answer : a = ±5]

82. [STPM ] The polynomial p(x) = x4 + mx3 + nx2 + 2x + 2, where m and n are real constants, has a quadratic factor x2 − 1. (a) Find the values of m and n.

[4 marks]

p(x) ≥ −3 for all x. [4 marks] x2 − 1 (c) By using long division, obtain the remainder when p(x) is divided by x2 − x − 6. Hence, deduce the remainder when p(x) is divided by (x + 2). [5 marks]

(b) Find the other quadratic factor of p(x) and hence, show that

(d) Find g(x) in terms of p(x) such that (x − 2) is a factor of g(x).

[2 marks]

[Answer : (a) n = −3, m = −2 ; (b) x2 − 2x − 2 ; (c)14 − 2x , 18 ; (d) g(x) = (x − 2)p(x)]

16

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2: Sequences and Series

Evaluate

∞ X 1 . 102r r=1

[2 marks]

¯ as a rational number in its lowest form. Express 0.18

2. [STPM ]

Find the expansion of

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1. [STPM ]

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Sequences and Series

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2


[2 marks]

[Answer :

(1 + x2 )p in ascending powers of x until the term in x3 . (1 − x)q

2 1 , ] 99 11

[5 marks]

r 13 1 to be estimated using the above (a) If p = q = , suggest a suitable value of x that enables 2 10 r 13 correct to four decimal places. [7 marks] expansion. Hence, estimate 10 1 [3 marks] (b) If p = − and q lies in the interval [0,9], and the largest possible coefficient of x3 . 3

[Answer : 1 + qx + p +

q(q + 1) 2

q(q + 1)(q + 2) 1 x2 + pq + x3 + . . . ; (a) x = , 1.1395 ; (b) 162] 6 5

3. [STPM ] √ √ Express ( 2 − 1)5 in the form a 2 + b, where a, b are integers.

[3 marks]

√

[Answer : 29 2 − 41]

4. [STPM ] The sum of the first 2n terms of a series P is 20n − 4n2 . Find in terms of n, the sum of the first n terms of this series. Show that this series is an arithmetic series. [4 marks] Series Q is an arithmetic series such that the sum of its first n even terms is more than the sum of its first n odd terms by 4n. Find the common difference of the series Q. [5 marks] If the first term of series Q is 1, determine the minimum value of n such that the difference between the sum of the first n terms of series P and the sum of the first n terms of series Q is more than 980.

5. [STPM ] √ √ Simplify (1 + 2 3)5 − (1 − 2 3)5 .

6. [STPM ]

[6 marks]

[Answer : Sn = 10n − n2 ; d = 4 ; n = 21]

[4 marks]

√

[Answer : 1076 3]

1 − . . . and S∞ denotes the 3 sum to infinity of this series. Find the smallest n such that |S∞ − Sn | < 0.0001. [7 marks]

(a) Sn denotes the sum of the first n terms of a geometric series 3 − 1 +

17

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(a) the n-th term of the series is log pq 2n ,

8. [STPM ]

1

[2 marks]

[3 marks]

Expand (1+x) 5 in ascending power of x until the term in x3 . By taking x = 1

3 2

[Answer : (a) 10 ; (b) a = 12 , d = 6 , r = ]

7. [STPM ] Given that the sum of the first n terms of a series is n log pq n+1 . Show that

(b) this is an arithmetic series.

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(b) The first and the second term of an arithmetic series equal respectively the first and the second term of a geometric series. The third term of the geometric series exceeds the third term of the arithmetic series by 3. The arithmetic series has a positive common difference and the sum of its first three terms equals 54. Find the first term of both series. Find also the common difference of the arithmetic series and the common ratio of the geometric series. [8 marks]

1 , find the approximation 40

for 32.8 5 correct to four decimal places. [7 marks] 1 1 + ax and (1 + x) 5 are the same until the term in x2 , find the values of a and b. If the expansion of 1 + bx 203 Hence, show that 32.8 ≈ . [8 marks] 101 1 5

[Answer : 1 + x −

9. [STPM ]

2 2 6 3 3 2 x + x + . . . , 2.0101 ; a = , b = ] 25 125 5 5

√ 1 1 Expand (1 + 8x) 2 in the ascending power of x until the term in x3 . By taking x = , find 3 correct 100 to five decimal places. [4 marks]

10. [STPM ]

[Answer : 1 + 4x − 8x2 + 32x3 + . . . ; 1.73205]

a(1 − rn ) . 1−r Give the condition on r such that lim Sn exists, and express this limit in terms of a and r. [5 marks] Given that Sn = a + ar + ar2 + . . . + arn−1 , with a 6= 0. Show that Sn = n→∞

(a) Determine the smallest integer n such that

(b) Find the sum to infinity

4 1+ + 3

2 n 4 4 + ... + > 21. 3 3

32 (1 − x)2 + 33 (1 − x)3 + . . . + 3r (1 − x)r + . . .

and determine the set of x such that this sum exists. [Answer : |r| < 1,

18

[5 marks]

[5 marks]

a 9(1 − x)2 2 4 ; (a) 7 ; (b) , {x : < x < }] 1−r 3x − 2 3 3

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11. [STPM ] The sum and product of three consecutive terms of an arithmetic progression are −3 and 24 respectively. Determine the three possible terms of the arithmetic progression. [5 marks] [Answer : 4,-1,-6 or -6,-1,4]

12. [STPM ] x n Expand 1 − where n is a positive integer in ascending powers of x until the term in x3 . If the n 1 coefficient of x3 is − , find n. [6 marks] 27 1 x n (1 − x) 2 in ascending powers of x until the With this value of n, obtain the expansion of 1 − n term in x3 . [5 marks] √ 1 [4 marks] Hence, by taking x = − , find the approximation of 10 accurate to 3 decimal places. 10 [Answer : 1 − x +

√ n − 1 2 (n − 1)(n − 2) 3 3 17 61 3 x − x + . . . , n = 3 ; 1 − x + x2 − x . . . ; 10 ≈ 3.162] 2 2n 6n 2 24 432

13. [STPM ] Express

1 in partial fractions. Hence show that (4r − 3)(4r + 1) n X r=1

1 1 = (4r − 3)(4r + 1) 4

14. [STPM ]

Given that y = √

1−

1 4n + 1

.

[6 marks]

[Answer :

1 1 − ] 4(4r − 3) 4(4r + 1)

1 1 √ , where x > − , show that, provided x 6= 0, 2 1 + 2x + 1 + x y=

√ 1 √ ( 1 + 2x − 1 + x). x

[3 marks]

Using the second form for y, express y as a series of ascending powers of x as far as the term in x2 . Hence, by putting x =

1 , show that 100 √

[6 marks]

79407 10 √ ≈ . 160000 102 + 101

[3 marks]

[Answer : y =

7 1 3 − x + x2 + . . .] 2 8 16

15. [STPM ] Determine the set of x such that the geometric series 1 + ex + e2x + . . . converges. Find the exact value of x so that the series converges to 2. [6 marks] 19

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Express

1 as partial fraction. −1

4k 2

Hence, find a simple expression for Sn =

n X k=1

[4 marks]

1 and find lim Sn . n→∞ −1

[4 marks]

4k 2

1 1 1 − ; Sn = [Answer : 2(2k − 1) 2(2k + 1) 2

17. [STPM ] 1 1+x 2 Express as a series of ascending powers of x up to the term in x3 . 1 + 2x √ 1 By taking x = , find 62 correct to four decimal places. 30

18. [STPM ]

1 2

7 8

[Answer : 1 − x + x2 −

r2

1−

1 2n + 1

;

1 ] 2

[6 marks]

[3 marks]

√ 25 3 x + . . . ; 62 = 7.8740] 16

2 in partial fractions. + 2r Using the result obtained, Express ur =

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16. [STPM ]

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[Answer : {x : x < 0} ; x = − ln 2]

[3 marks]

1 1 1 1 (a) show that u2r = − + 2 + + , [2 marks] r r r + 2 (r + 2)2 n ∞ ∞ X X X 1 1 3 1 (b) show that − and determine the values of ur = − ur and ur+1 + r . 2 n+1 n+2 3 r=1

r=1

r=1

[9 marks]

[Answer :

1 3 4 1 − ; (b) , ] r r+2 2 3

19. [STPM ] √ 1 Expand (1 − x) 2 in ascending powers of x up to the term in x3 . Hence, find the value of 7 correct to five decimal places. [5 marks] 1 2

1 8

[Answer : 1 − x − x2 −

20. [STPM ] Prove that the sum of the first n terms of a geometric series a + ar + ar2 + . . . is a(1 − rn ) . 1−r

1 3 √ x ; 7 ≈ 2.64609] 16

[3 marks]

(a) The sum of the first five terms of a geometric series is 33 and the sum of the first ten terms of the geometric series is -1023. Find the common ratio and the first term of the geometric series. 3 − . . . are 2 respectively. Determine the smallest value of n such that |Sn − S∞ | < 0.001. [7 marks]

(b) The sum of the first n terms and the sum to infinity of the geometric series 6 − 3 + Sn and S∞

[5 marks]

20

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[Answer : (a) r = −2, a = 3 ; (b) n = 12] 21. [STPM ] For the geometric series 7 + 3.5 + 1.75 + 0.875 + ..., find the smallest value of n for which the different between the sum of the first n terms and the sum to infinity is less than 0.01. [6 marks]

22. [STPM ]

Express f (x) =

x2 − x − 1 in partial fractions. (x + 2)(x − 3)

1 1 up to the term in 3 . x x Determine the set of values of x for which this expansion is valid. Hence, obtain an expansion of f (x) in ascending powers of

[Answer : 1 +

[5 marks]

[6 marks] [2 marks]

1 5 5 1 − , 1 + 2 + 3 + . . . , {x : x < −3, x > 3}] x−3 x+2 x x

23. [STPM ] If x is so small that x2 and higher powers of x may be neglected, show that x 10 ≈ 29 (2 − 7x). (1 − x)6 2 + 2

24. [STPM ]

[Answer : 11]

[4 marks]

10−Tn 5 The nth term of an arithmetic progression is Tn , show that Un = (−2)2( 17 ) is the nth term of a 2 geometric progression. [4 marks] ∞ X 1 Un . [4 marks] If Tn = (17n − 14), evaluate 2

n=1

[Answer : −

10 ] 3

25. [STPM ] Express the infinite recurring decimal 0.72˙ 5˙ (= 0.7252525 . . . ) as a fraction in its lowest terms.[4 marks] [Answer :

359 ] 495

26. [STPM ] At the beginning of this year, Mr. Liu and Miss Dora deposited RM10 000 and RM2000 respectively in a bank. They receive an interest of 4% per annum. Mr Liu does not make any additional deposit nor withdrawal, whereas, Miss Dora continues to deposit RM2000 at the beginning of each of the subsequent years without any withdrawal. (a) Calculate the total savings of Mr. Liu at the end of n-th year. (b) Calculate the total savings of Miss Dora at the end of n-th year.

[3 marks]

[7 marks]

(c) Determine in which year the total savings of Miss Dora exceeds the total savings of Mr. Liu. [5 marks]

[Answer : (a) 10000(1.04)n ; (b) 52000[1.04n − 1]; (c) 6] 21

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27. [STPM ]

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3 + . . ., obtain the smallest value of n if the difference between the sum 2 45 . [6 marks] of the first n + 4 terms and the sum of first n terms is less than 64 For geometric series 6 + 3 +

[Answer : 5]

28. [STPM ] Determine the set of values of x such that the geometric series e−x + e−2x + e−3x + . . . converges. Find the exact value of x if the sum to infinity of the series is 3.

29. [STPM ]

Given f (x) =

x3 − 3x − 4 , (x − 1)(x2 + 1)

[6 marks]

4 3

[Answer : {x : x > 0} ; x = ln ]

Cx + D B + 2 , [5 marks] x−1 x +1 (b) when x is sufficiently small such that x4 and higher powers can be neglected, show that f (x) ≈ 4 + 7x + 3x2 − x3 . [4 marks] (a) find the constants A, B, C and D such that f (x) = A +

30. [STPM ]

Show that

n X r2 + r − 1 r=1

r2 + r

=

n2 . n+1

[Answer : (a) A = 1, B = −3, C = 4, D = 0]

[4 marks]

31. [STPM ] The sum of the first n terms of a progression 3n2 . Determine the n-th term of the progression, and hence, deduce the type of progression. [4 marks]

32. [STPM ] Express in partial fractions

Show that

and hence, find

[Answer : 6n − 3, Arithmetic Progression]

3 . (3r − 1)(3r + 2)

n X r=1

1 1 3 = − , (3r − 1)(3r + 2) 2 (3n + 2)

∞ X r=1

1 . (3r − 1)(3r + 2)

[4 marks]

[2 marks]

[2 marks]

[Answer : 22

1 1 1 − , ] 3r − 1 3r + 2 6

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A sequence is defined by ur = e−(r−1) − e−r for all integers r ≥ 1. Find deduce the value of

∞ X

ur .

r=1

34. [STPM ] The sequence u1 , u2 , u3 , . . . is defined by un+1 = 3un , u1 = 2. (a) Write down the first five terms of the sequence. (b) Suggest an explicit formula for ur .

35. [STPM ]

n X r=1

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33. [STPM ] ur in terms of n, and [5 marks]

[Answer : 1 − e−n ; 1]

[2 marks]

[2 marks]

[Answer : 2, 6, 18, 54, 162 , ur = 2(3)r−1 ]

1 + ax , where |b| < 1, in ascending powers of x up to the term in x3 . Determine 1 + bx the set of values of x for which both the expansions are valid. [7 marks] 2

Expand (1 + x) 3 and

If the two expansions are identical up to the term in x2 ,

(a) determine the values of a and b, [3 marks] √ 212 1 3 . [3 marks] (b) use x = to obtain the approximation 81 ≈ 8 49 (c) find, correct to five decimal places, the difference between the terms in x3 for the two expansions 1 with x = . [2 marks] 8 2 3

1 9

[Answer : 1 + x − x2 +

4 3 5 1 x + . . . , 1 + (a − b)x + b(b − a)x2 + b2 (a − b)x3 + . . ., |x| < 1 ; (a) a = , b = ; 81 6 6 (c) 0.00006]

36. [STPM ] A sequence a1 , a2 , a3 , . . . is defined by an = 3n2 . The difference between successive terms of the sequence forms a new sequence b1 , b2 , b3 , . . .. (a) Express bn in terms of n.

[2 marks]

(b) Show that b1 , b2 , b3 , . . . is an arithmetic sequence, and state its first term and common difference. [3 marks]

(c) Find the sum of the first n terms of the sequence b1 , b2 , b3 , . . . in terms of an and bn .

[2 marks]

[Answer : (a) 6n + 3 , (b) 9, 6 ; (c) an + bn − 3]

37. [STPM ] ˙ Write the infinite recurring decimal 0.131˙ 8(= 0.13181818 . . .) as the sum of a constant and a geometric series. Hence, express the recurring decimal as a fraction in its lowest terms. [4 marks] [Answer :

23

29 ] 220

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1 in partial fractions, and deduce that (r2 − 1) 1 1 1 1 . ≡ − r(r2 − 1) 2 r(r − 1) r(r + 1)

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(a) Express

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38. [STPM ]

[4 marks]

Hence, use the method of differences to find the sum of the first (n − 1) terms, Sn−1 , of the series 1 1 1 1 + + + ... + + ..., 2 2 × 3 3 × 8 4 × 15 r(r − 1)

and deduce Sn .

[6 marks]

1 (b) Explain why the series converges to , and determine the smallest value of n such that 4 1 − Sn < 0.0025. 4

1 1 1 − ; (b) Sn−1 = [Answer : (a) 2(r − 1) 2(r + 1) 2

[5 marks]

1 1 1 1 1 − − , Sn = ; (c) 13] 2 n(n + 1) 2 2 (n + 1)(n + 2)

39. [STPM ] √ √ √ √ 6 6 6 3 + 2) and ( 3 − 2) to evaluate ( 3 + 2) + ( 3 − 2)6 . Hence, Use the binomial expansions of ( √ 6 show that 2701 < ( 3 + 2) < 2702. [7 marks]

40. [STPM ]

[Answer : 2702]

(a) Show that for a fixed number x 6= 1, 3x2 + 3x3 + . . . + 3xn is a geometric series, and find its sum in terms of x and n. [4 marks] (b) The series Tn (x) is given by

Tn (x) = x + 4x2 + 7x3 + . . . + (3n − 2)xn , for x 6= 1.

By considering Tn (x) − xTn (x) and using the result from (a), show that

Hence, find the value of

Tn (x) =

20 X r=1

x + 2x2 − (3n + 1)xn+1 + (3n − 2)xn+2 . (1 − x)2

[5 marks]

2r (3r − 2), and deduce the value of

[Answer : (a) sum=

24

19 X r=0

2r+2 (3r + 1).

[6 marks]

3x2 (xn−1 − 1) ; (b) 115343370 , 230686740] x−1

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41. [STPM ]

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1 8

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1 − 5x in ascending powers of x are 1 − 3x the same. Determine the range of values of x for which both expansions are valid. [6 marks] 1 1 − 5x [3 marks] Use the result (1 − 8x)4 ≈ to obtain an approximation of (0.84) 4 as a fraction. 1 − 3x Show that the first three terms in the expansions of (1 − 8x)4 and

1 8

1

[Answer : {x : x < − , x > } , 0.84 4 ≈

42. [STPM ] The r-th term, ur , of an infinite series is given by

2r+1 2r−1 1 1 + , ur = 3 3

45 ] 47

A , where A is a constant. [2 marks] 32r+1 (b) Find the sum of the first n terms of the series, and deduce the sum of the infinite series. [6 marks] (a) Express ur in the form

[Answer : (a) ur =

10 32r+1

; (b)

n 1 5 5 ] 1− , 12 9 12

43. [STPM ] A recursive formula for the general term of a sequence is given by ur+1 = ur + 2r + 3, where u0 = 1. (a) Write down the first four terms of the sequence.

[2 marks]

(b) Suggest an explicit formula for the general term and verify your answer.

44. [STPM ]

[Answer : (a) 1, 4, 9, 16 ; (b) ur = (r + 1)2 ]

1 A convergent sequence is defined by ur+1 = 1 + ur and u1 = 1. 3 (a) Write down each of the terms u2 , u3 and u4 in the form r 3 1 formula for ur is given by ur = 1− . 2 3 (b) Determine the limit of ur , as r tends to infinity.

45. [STPM ]

Show that

∞ k k−2 X 1 1 k=1

2

3

r−1 k X 1 k=0

3

, and show that an explicit

is a convergent series. Give a reason for your answer.

Hence, determine the sum of the convergent series.

[3 marks]

[5 marks]

[2 marks]

[Answer : (b)

[3 marks] [2 marks]

[Answer :

25

3 ] 2

9 ] 5

Work Smart, Not Hard

Do not share this past year paper.

Refer the full solution at kkleemaths.com.

26


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3: Matrices



 2 3 1 (a) Given M = −1 0 4. 1 −1 1  1 2  (b) Given matrices A = 2 3 3 1 neous equation

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1. [STPM ]

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Matrices

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3


Show that M3 − 3M2 + 8M − 24I3 = 0. Deduce M−1 .

[7 marks]

   3 −5 1 7 1, B =  1 7 −5. Find AB, and hence solve the simulta2 7 −5 1 −5x + y + 7z = 8, x + 7y − 5z = −16, 7x − 5y + z = 14.

[Answer : (a) A−1

 4 1  5 = 24 1

2. [STPM  ]

−4 1 5

[8 marks]

  12 18 0 −9 ; (b) AB =  0 18 3 0 0

 1 −2 −6 9 , find A2 and A3 . Hence, find A100 . If A = −3 2 2 0 −3  −5 [Answer : A2 =  9 −4

3. [STPM ]

 1 1 2 Given P = 1 2 1. 2 1 1 

(a) Find R so that R = P2 − 4P − I3 . (b) Show that PR + 4I3 = 0.

4. [STPM ] The matrices A and B are given by 

−6 10 −4

  −6 1 9  , A3 = −3 −3 2

[4 marks]

−2 2 0

  −6 −5 9  , A100 =  9 −3 −4

−6 10 −4

 −6 9 ] −3

[3 marks] [2 marks]



1 [Answer : (a) R =  1 −3

   5 0 0 −2 0 0 A = 1 8 0 , B =  − −5 0  . 1 3 5 −1 −3 −2

(a) Determine whether A and B commute.

 0 0  , x = 1, y = −1, z = 2] 18

1 −3 1

 −3 1 ] 1

[3 marks]

(b) Show that there exist numbers m and n such that A = mB + nI, where I is the 3 x 3 identity matrix, and find the values of m and n. [6 marks] 27

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3: Matrices

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5. [STPM ]

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[Answer : (a) Commute. (b) m = −1, n = 3]  2b − 1 a2 b2 Determine the values of a, b, c so that the matrix  2a − 1 a bc  is a symmetric matrix. b b + c 2c − 1 

6. [STPM ]



[5 marks]

[Answer : a = 1, b = 0, c = 0]

   −10 4 9 2 3 4 Matrix M and N is given by M =  15 −4 −14 , and N =  4 3 1 . −5 1 6 1 2 4 −1 Find MN and deduce N . [4 marks] Product X, Y , Z are assembled from three components A, B, C according to different proportions. Each product X consists of two components of A, four components of B, and one component of C; each product of Y consists of three components of A, three components of B, and two components of C; each product of Z consists of four components of A, one component of B, and four components of C. A total of 750 components of A, 1000 components of B, and 500 components of C are used. With X, Y , Z representing the number of products of X, Y , and Z assembled, obtain a matrix equation representing the information given. [4 marks] Hence, find the number of products of X, Y , and Z assembled. [4 marks]

7. [STPM ]

[Answer : x=200, y=50, z=50.]

 1 2 −3 The matrix A is given by A =  3 1 1 . 0 1 −2 

(a) Find the matrix B such that B = A2 − 10I, where I is the 3× identity matrix. (b) Find (A + I)B, and hence find (A + I)21 B.  −3 [Answer : (a)  6 3

8. [STPM ]



3 3 Matrix A is given by A =  5 4 1 2 Find the adjoint of A. Hence, find

 4 1 . 3 A−1 .

1 −2 −1

  5 −3 −10 ; (b)  6 −5 3



10 −1 [Answer : −14 5 6 −3

1 −2 −1

[3 marks] [6 marks]

  5 −3 −10 ,  6 −5 3

1 −2 −1

 5 −10] −5

[6 marks]

 

5/6 −13 17  ; −7/6 1/2 −3

9. [STPM ] The matrices P and Q, where PQ = QP, are given by     2 −2 0 −1 1 0 0 −1  P =  0 0 2  and Q =  0 a b c 0 −2 2

−1/12 5/12 −1/4

 −13/12 17/12 ] −1/4

Determine the values of a, b and c. [5 marks] Find the real numbers m and n for which P = mQ + nI, where I is the 3 × 3 identity matrix.[5 marks] 28

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3: Matrices

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[Answer : a = 0, b = 4, c = −4 ; m = −2, n = 0] 10. [STPM ] A, B, C are square matrices such that BA = B−1 and ABC = (AB)−1 . Show that A−1 = B2 = C. 

 1 2 0 If B =  0 −1 0 , find C and A. 1 0 1

11. [STPM ] The matrix A is given by

[3 marks] [7 marks]



1 [Answer : A =  0 −2



 k 1 5 A =  2 k 8 . 8 −3 2

0 1 −2

  0 1 0 , C = 0 1 2

0 1 2

 0 0] 1

Determine all values of k for which the equation AX = B, where B is a 3 × 1 matrix, does not have a unique solution. [3 marks] For each of these values of k, find the solution, if any, of the equation   1 AX = −2 . 4

[7 marks]

[Answer : k = 3, k = 5 ; For k = 3, no solution. For k = 5, x =

12. [STPM ]

 k 1 3 Determine the values of k such that the determinant of the matrix  2k + 1 −3 2  is 0. [4 marks] 0 k 2

13. [STPM ]



17 12 30 7 − t, y = − − t, z = t] 23 23 23 23

1 4

[Answer : k = − , k = 2]

   5 2 3 a 1 −18 −1 12  and PQ = 2I, where I is the 3 × 3 identity matrix, If P =  1 −4 3 , Q =  b 3 1 2 −13 −1 c determine the values of a, b and c. Hence find P−1 . [8 marks] Two groups of workers have their drinks at a stall. The first group comprising ten workers have five cups of tea, two cups of coffee and three glasses of fruit juice at a total cost of RM11.80. The second group of six workers have three cups of tea, a cup of coffee and two glasses of fruit juice at a total cost of RM7.10. The cost of a cup of tea and three glasses of fruit juice is the same as the cost of four cups of coffee. If the costs of a cup of tea, a cup of coffee and a glass of fruit juice are RM x, RM y and RM z respectively, obtain a matrix equation to represent the above information. Hence determine the cost of each drink. [6 marks] 

11/2 1/2 [Answer : a = 11, b = −7, c = 22 ;  −7/2 −1/2 −13/2 −1/2

29

 −9 6  ; x=RM 1, y=RM 1.30, z=RM 1.40] 11

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14. [STPM ] Consider the system of equations

x + y + pz = q, 3x − y − 2z = 1, 6x + 2y + z = 4,

for the two cases: p = 2, q = 1 and p = 1, q = 2.

3: Matrices

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For each case, find the unique solution if it exists or determine the consistency of the system if there is no unique solution. [7 marks] [Answer : For p = 2, q = 1, unique solution, x = 1/2, y = 1/2, z = 0. For p = 1, q = 2, no unique solution. The system is not consistent.]

15. [STPM ] The matrices A and B are given by     −35 19 18 −1 2 1 A = −3 1 4 , B = −27 −13 45 . −3 12 5 0 1 2 Find the matrix A2 B and deduce the inverse of A. Hence, solve the system of linear equations

[5 marks]

x − 2y − z = −8, 3x − y − 4z = −15, y + 2z = 4.

 121 0 [Answer :  0 121 0 0


[5 marks]

   0 −2/11 −3/11 7/11 0  ,  6/11 −2/11 1/11 ; x = −3, y = 2, z = 1] 121 −3/11 1/11 5/11

λx + y + z = 1, x + λy + z = λ, x + y + λz = λ2 ,

where λ is a constant.

(a) Determine the values of λ for which this system has a unique solution, infinitely many solutions and no solution. [5 marks] (b) Find the unique solution in terms of λ.

[5 marks]

[Answer : (a) For unique solution, λ 6= 1, −2. For infinitely many solution, λ = 1.

17. [STPM ]



 1 0 0 Matrix A is given by A =  1 −1 0 . 1 −2 1

30

For no solution, λ = −2. λ+1 λ+1 (λ + 1)2 (b) x = − ,y = − ,z = ] λ+2 λ+2 λ+2

STPM MATHEMATICS (T) matrix, and deduce A−1 .  4 3 2 1 . 0 2

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(a) Show that A2 = I, where I is the 3 × 3 identity  1 (b) Find the matrix B which satisfies BA =  0 −1

3: Matrices

 1 [Answer : (a) 1 1

18. [STPM ]

(a) The matrix P, Q and R  1 5  2 −2 P= 1 −3

6x + 10y + 8z = 4500 x − 2y + z = 0 x + 2y + 3z = 1080

  32 72 4 , 0 12 0

19. [STPM ] The matrix A is given by

0 72 0

  0 1/18 7/72 0  ,  1/72 −5/72 72 −1/36 1/72

 −13/72 −1/72  ; (b) x = 220, y = 190, z = 160 11/72 ]

 1 1 c A = 1 2 3  1 c 1

(a) Find the values of c for which the equation AX = B does not have a unique solution. (b) For each value of c, find the solutions, if any, of the equation   1 AX =  −3  . −11

20. [STPM ]

 3 1] 2

[5 marks]



and B is a 3 × 1 matrix.

−10 −4 −4

[5 marks]

(b) Using the result in (a), solve the system of linear equations

24 40 [Answer : (a)  4 −8 4 8

  8 0 0 −1 0 ; (b) 3 1 −2 1

[4 marks]

are given by      6 −13 −50 −33 4 7 −13 4  , Q =  −1 −6 −5  , R =  1 −5 −1  2 7 20 15 −2 1 11

Find matrices PQ and PQR and hence, deduce (PQ)−1 .



[4 marks]

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[3 marks]

[5 marks]

[Answer : (a) c 6= 1, 4 ; (b) For c = 1, no solution. For c = 4, x = 5 − 5t, y = −4 + t, z = t]

  1 2 1 3 . Matrix P is given by P = 2 1 2 −1 −1

31

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3: Matrices

(a) Find the determinant and adjoint of P. Hence, find P−1 .

[6 marks]

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(b) A factory assembles three types of toys Q, R and S. The total time taken to assemble one unit of R and one unit of S exceeds the time taken to assemble two units of Q by 8 minutes. One unit of Q, two units of R and one unit of S take 31 minutes to be assembled. The time taken to assemble two units of Q, one unit of R and three units of S is 48 minutes. If x, y and z represent the time, in minutes, taken to assemble each unit of toys Q, R and S respectively, i. write a system of linear equations to represent the above information, ii. using the results in (a), determine the time taken to assemble each type of toy. 

2 [Answer : (a) 14,  8 −4

21. [STPM ] A and B are two matrices such that   −4 −3 6 A = −2 −2 4  2 2 −3

1 −3 5

(b) Using A

obtained in (a), find B.

 −2 6 0 and A2 B =  2 0 4 . 0 4 2

22. [STPM ] Matrix A is given by



−2 [Answer : (a) 2 ,  2 0



[6 marks]

[4 marks]

  3 0 −1 0 4 ,  1 2 2 0

3/2 0 1



 1 x 1 A = −1 −1 0 1 0 0

and A2 = A−1 . Determine the value of x.


[5 marks]

 1/14 5/14 −3/14 −1/14 ; (b)(ii) 5, 8, 10 ] 5/14 −3/14

  1/7 5 −1,  4/7 −2/7 −3

(a) Find the determinant and adjoint of A. Hence, determine A−1 . −1

[2 marks]

  0 −8 2 ; (b)  9 1 0

27 2 18

 0 18] 10

[7 marks]

[Answer : 2]

x + 3y + 2z = −2, 3x + ay + 2z = 2a − 1, 2x + 6y + az = b.

(a) If a 6= 9 and a 6= 4, show that the system has a unique solution. (b) If a = 5 and b = 6, find the unique solution.

[5 marks] [3 marks]

(c) If a = 4, show that the system does not have a solution unless b = −4.

[3 marks]

(d) If a = 9, determine whether there are any values of b for which the system has a solution.[3 marks] [Answer : (b) x = 32

77 55 131 , y = − , z = 10 ; (d) b = − ] 4 4 4


3: Matrices

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24. [STPM ] The matrices P and Q, where PQ = QP, are given by     2 −2 0 −1 1 0 0 −1  P =  0 0 2  and Q =  0 a b c 0 −2 2

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Determine the values of a, b and c. [5 marks] Find the real numbers m and n for which P = mQ + nI, where I is the 3 × 3 identity matrix.[5 marks] [Answer : a = 0, b = 4, c = −4 ; m = −2, n = 0]

25. [STPM ] Using an augmented matrix and elementary row operations, find the solution of the system of equations 3x − 2y − 5z = −5, x + 3y − 2z = −6, 5x − 4y + z = 11.

26. [STPM ] A system of linear equations is given by

x + y + z = k, x − y + z = 0, 4x + 2y + λz = 3.

[9 marks]

[Answer : x = 1, y = −1, z = 2]

where λ and k are real numbers. Show that the augmented matrix for the system may be reduced to   k 1 1 1  0 −2 −k  . 0 0 0 λ − 4 3 − 3k [5 marks]

Hence, determine the values of λ and k so that the system of linear equations has (a) a unique solution, (b) infinitely many solutions, (c) no solution.

[1 marks]

[1 marks]

[1 marks]

[Answer : (a) λ 6= 4 ; (b) λ = 4, k = 1 ; (c) λ = 4, k 6= 1]

27. [STPM ]



 5 4 −2 5 −2. Given that matrix M =  4 −2 −2 2

Show that there exist non-zero constants a and b such that M2 = aM + bI, where I is the 3 × 3 identity matrix. [6 marks] Hence, find the inverse of the matrix M.

[3 marks]

33

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3: Matrices

28. [STPM ] A matrix P is given by



 −5 0 2 P =  0 2 −1 . −1 4 −2

By using elementary row operations, find the inverse of P.


x − 3y = 2, px + qz = −1, py + z = −1.

−0.4 0.6 0.2

 0.2 0.2] 0.9

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0.6 [Answer : a = 11, b = −10 ; −0.4 0.2

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

[5 marks]

 0 2 −1 [Answer : 1/4 3 −5/4] 1/2 5 −5/2 

(a) Write the augmented matrix for the system of linear equations, and show that it may be reduced to   1 −3 0 2  0 p 1 −1  . 0 0 3 − q 2p − 2 [5 marks]

(b) Determine the values of p and q such that the system has infinitely many solutions, and find the general solution. [4 marks] [Answer : (b) p = 1 an q = 3 ,x = −1 − 3t, y = −t − 1, z = t]

30. [STPM ] The matrices M and N are given by     1 b ca 1 a bc M = 1 b ca , N = 1 a bc  . 3 3c 3ab 1 c ab Show that det M = (a − b)(b − c)(c − a). Deduce det N.

31. [STPM ] The variables x, y and z satisfy the system of linear equations

where k is a real constant.

2x + y + 2z = 1, 4x + 2y + z = k, 8x + 4y + 7z = k 2 ,

(a) Write a matrix equation for the system of linear equations. 34

[4 marks]

[2 marks]

[1 marks]

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3: Matrices

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(b) Reduce the augmented matrix to row-echelon form, and show that the system of linear equations does not have a unique solution. [6 marks] (c) Determine all the values of k for which the system of linear equations has infinitely many solutions, and find the solutions in the case when k is positive. [6 marks] (d) Find the set of values of k for which the system of linear equations is inconsistent. 5 3

[Answer : (c) k = , 2 , x =

1−t 5 , y = t, z = 0 ; (d) {k : k ∈ R, k 6= − , k 6= 2}] 2 3

32. [STPM ] Using Gaussian elimination, solve the system of linear equations x + y − z = 0 2x − y − 2z = 4 . 5x − y + z = 2

33. [STPM ]



[8 marks]

2

(b) Show that P(P − 6P + 11I) = 6I, where I is 3 x 3 identity matrix, and deduce P


x + y + z = 1, 2x + 3y + 2z = 3, 2x + 3y + mz = m2 − 1.

(a) Use Gaussian elimination to reduce the augmented matrix for the system above. (b) Determine the value of m for which the system of linear equations i. has a unique solutions, ii. has infinitely many solutions, iii. has no solution.

35

4 3

1 3

[Answer : x = , y = − , z = −1]

 1 1 2 A matrix P is given by P =  0 2 2. −1 1 3 (a) Find P2 − 6P + 11I.

[2 marks]

[3 marks]

−1

.

[5 marks]

[Answer : ]

[4 marks]

[2 marks] [3 marks]

[3 marks]

[Answer : ]

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4: Complex Numbers

√ 2 + ai √ is a real number and find this real number. Determine the value of a if 1 + 2i

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1. [STPM ]

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Complex Numbers

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4


[4 marks]

[Answer : a = 2,

2. [STPM ] If (x + iy)2 = i, find all the real values of x and y.

√

2]

[6 marks]

1 2

1 2

1 2

1 2

[Answer : x = √ , y = √ ; x = − √ , y = − √ ]

3. [STPM ] √ The complex numbers z1 and z2 satisfy the equation z 2 = 2 − 2 3i. (a) Express z1 and z2 in the form a + bi, where a and b are real numbers. (b) Represent z1 and z2 in an Argand diagram.

(c) For each of z1 and z2 , find the modulus, and the argument in radians. [Answer : (a) z1 =

√


√ 5π π 3 − i, z2 = − 3 + i ; (c) |z1 | = 2, |z2 | = 2 , arg(z1 )=− , arg(z2 )= ] 6 6

4. [STPM ] √ Let zl = 1, z2 = x + iy, z3 = y + ix, where x, y ∈ R, x > 0 and i = −1. If z1 , z2 , . . ., zn is a geometric progression, (a) find x and y,

(b) express z2 and z3 in the polar form,

(c) find the smallest positive integer n such that z1 + z2 + . . . + zn = 0, (d) find the product z1 z2 z3 . . . zn , for the value of n in (c). √

[Answer : (a) x =

√ 2(1 + 3i) , where i = −1. 2 (1 − 3i)

6. [STPM ]

If z = cos θ + i sin θ, show that

[2 marks]

[5 marks]

[3 marks]

π π 1 π 3 π , y = ; (b) z2 = cos + i sin , z3 = cos + i sin ; (c) 12 ; (d) -1] 2 2 6 6 3 3

5. [STPM ] Simplify (a)

[3 marks]

[3 marks]

[Answer : (a) −

13 9 − i] 25 25

1 1 1 = (1 − i tan θ) and express in a similar form. [4 marks] 1 + z2 2 1 − z2

7. [STPM ] Find the roots of the equation (z − iα)3 = i3 , where α is a real constant. 36

[Answer :

1 (1 + i cot θ)] 2

[3 marks]

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4: Complex Numbers

[2 marks]

3

(b) Solve the equation [z − (1 + i)] = (2i) . 2

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3

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(a) Show that the points representing the roots of the above equation form an equilateral triangle. [5 marks]

(c) If ω is a root of the equation ax + bx + c = 0, where a, b, c ∈ R and a 6= 0, show that its conjugate ω ∗ is also a root of this equation. Hence, obtain a polynomial equation of degree six with three of its roots also the roots of the equation (z − i)3 = i3 . [5 marks] [Answer : (1 + α)i , −

√

√ √ √ 3 3 1 1 + (α − )i , + (α − )i ; (b) 1+3i , 1 − 3 1 + 3 ; (c) x6 + 3x4 − 3x2 + 4 = 0] 2 2 2 2

8. [STPM ]

1 . 1−z

If z is a complex number such that |z| = 1, find the real part of

[6 marks]

[Answer :

1 ] 2

9. [STPM ] The equation z 4 − 2z 3 + kz 2 − 18z + 45 = 0 has an imaginary root. Obtain all the roots of the equation and the value of the real constant k. [8 marks]

10. [STPM ]

[Answer : Roots=1 − 2i, 1 + 2i, 3i, −3i , k = 14]

(a) Find the roots of ω 4 = −16i, and sketch the roots on an Argand diagram.

11. [STPM ]

[Answer : (a) 2 cos

4k − 1 8

π + i sin

4k − 1 8

(a) Find the fifth roots of unity in the form cos θ + i sin θ, where −π < θ ≤ π. [Answer : (a) cos

2kπ 5

+ i sin

[5 marks]

2kπ 5

π , k = 0, 1, 2, 3]

[4 marks]

, k = −2, −1, 0, 1, 2]

12. [STPM ] √ The complex number z is such that z − 2z ∗ = 3 − 3i, where z ∗ denotes the conjugate of z. (a) Express z in the form a + bi, where a and b are real numbers. (b) Find the modulus and argument of z.

[3 marks]

(c) Represent z and its conjugate in an Argand diagram.

13. [STPM ]

Show that

1 + cos 2θ + i sin 2θ = i cot θ. 1 − cos 2θ − i sin 2θ

[3 marks]

[3 marks]

√

π 3

[Answer : (a) z = − 3 − i ; (b) modulus=2, argument=− ]

π Hence, show that the roots of the equation (z + i)5 = (z − i)5 are ± cot and ± cot 5 2 π 2 2π 2 π 2 2π Deduce the values of cot + cot and cot cot . 5 5 5 5 37

[2 marks]

2π .[6 marks] 5 [4 marks]

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4: Complex Numbers

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2i . (1 + 3i)2

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Given that z 2 =

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14. [STPM ]

(a) Find the real and imaginary parts of z 2 . Hence, obtain z1 and z2 which satisfy the above equation. [10 marks]

2

(b) Given that z1 and z3 are roots of 5x + ax + b = 0, where a and b are integers. i. Find the values of a and b. ii. Determine z3 and deduce the relationship between z1 and z3 .

[Answer : (a) real part=

[3 marks]

[3 marks]

4 2 1 2 1 3 , imaginary part=− , z1 = − i, z 2 = − + i 25 25 5 5 5 5 2 z3 = + 5

; (b)(i) a = −4, b = 1 (ii) 1 i. z3 is conjugate of z1 .] 5

15. [STPM ] In an Argand diagram the points R and S represent the complex numbers w = u + iv and z = x + iy z−i respectively which are related by w = . 1 − iz (a) Express u and v in terms of x and y.

[Answer : u =

[3 marks]

x2 + y 2 − 1 2x , v = ] x2 + y 2 + 2y + 1 x2 + y 2 + 2y + 1

16. [STPM ] √ Express the complex number z = 1 − 3i in polar form. 1 1 Hence, find z 5 + 5 and z 5 − 5 . z z

[4 marks]

[4 marks]

h π π i 1025 1023√3 1023 1025√3 + i, + i] [Answer : 2 cos − +i − , 3 3 64 64 64 64

17. [STPM ] √ The complex number z is given by z = 1 + 3i. (a) Find |z| and arg z.

[3 marks]

√ (b) Using de Moivre’s theorem, show that z = 16 − 16 3i. (c) Express

z4

z∗

5

[3 marks]

in the form x + yi, where z ∗ is the conjugate of z and x, y ∈ R.

[Answer : (a) 2,

[3 marks]

√ π ; (c) 4 − 4 3i] 3

18. [STPM ] √ √ √ √ Express the complex number 6−i 2 in polar form. Hence, solve the equation z 3 = 6−i 2.[9 marks]

19. [STPM ]

√

π π 6 6 √ 11π 11π √ 13π 13π √ π π z = 2[cos( ) + i sin( )], 2[cos(− ) + i sin(− )], 2[cos(− ) + i sin(− )]] 18 18 18 18 18 18

The complex numbers z and w are given by z = −1 + i and w = 38

[Answer : 2 2[cos(− ) + i sin(− )] ;

i+z . 1 − iz

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4: Complex Numbers

[3 marks]

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(b) Express w in polar form.

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[3 marks]

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(a) Find w in the form x + yi, where x, y ∈ R. State the real and imaginary parts of w.

(c) Using de Moivre’s theorem, determine the cube roots of w. Give your answer in cartesian form. [5 marks]

√ √ 3 1 3 1 π π [Answer : (a) i ; (b) w = 1(cos + i sin ) ; (c) + i,− + i , -i] 2 2 2 2 2 2

39

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5: Analytic Geometry

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Analytic Geometry

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5


1. [STPM ] Show that the mid-points of the parallel chords of the parabola y 2 = 4ax with gradient 2 lie on a straight line parallel to the x-axis. [7 marks] 2. [STPM ] The sum of the distance of the point P from the point (4,0) and the distance of P from the origin is (x − 2)2 y2 8 units. Show that the locus of P is the ellipse + = 1 and sketch the ellipse. [7 marks] 16 12 3. [STPM ] Find the perpendicular distance from the centre of the circle x2 + y 2 − 8x + 2y + 8 = 0 to the straight line 3x + 4y = 28. Hence, find the shortest distance between the circle and the straight line. [7 marks] [Answer : 4, 1]

4. [STPM ] Show that x2 + y 2 − 2ax − 2by + c = 0 is the equation of the circle with centre (a, b) and radius p a2 + b2 − c. [3 marks] C3

C1

C2

The above figure shows three circles C1 , C2 and C3 touching one another, where their centres lie on a straight line. If C1 and C2 have equations x2 + y 2 − 10x − 4y + 28 = 0 and x2 + y 2 − 16x + 4y + 52 = 0 respectively. Find the equation of C3 . [7 marks] [Answer : 5x2 + 5y 2 − 74x + 12y + 156 = 0]

5. [STPM ] The equation of a hyperbola is 4x2 − 9y 2 − 24x − 18y − 9 = 0. (a) Obtain the standard form for the equation of the hyperbola. (b) Find the vertices and the equations of the asymptotes of the hyperbola. [Answer : (a)

[3 marks]

[6 marks]

(x − 3)2 (y + 1)2 2 − = 1 ; (b) Vertices are (0,-1) and (6,-1). Asymptotes are y = x − 3 and 32 22 3 2 y = − x + 1] 3

6. [STPM ] c Show that the parametric equations x = ct and y = , where c is a constant, define a point on the t rectangular hyperbola xy = c2 . [2 marks] The points P , Q, R and S, with parameters p, q, r and s respectively, lie on the rectangular hyperbola xy = c2 . 40

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5: Analytic Geometry

(a) Show that pqrs = −1 if the chords P Q and RS are perpendicular.

[4 marks]

[Answer : (b) y = −

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(b) Find the equation of the line passing through the points P and Q. Deduce the equation of the tangent to the rectangular hyperbola at the point P . [4 marks] c c 1 2c 1 x + + , y = − 2x + ] pq p q p p

7. [STPM ] The equation of an ellipse is 3x2 + y 2 + 30x + 10y + 79 = 0. (a) Obtain the standard form for the equation of the ellipse.

[3 marks]

(b) Find the coordinates of the centre C, the focus F1 , and the focus F2 of the ellipse.

[4 marks]

(c) Sketch the ellipse, and indicate the points C, F1 and F2 on the ellipse.

[2 marks]

[Answer : (a)

√ √ (x + 5)2 (y + 5)2 + = 1 ; (b) C(−5, −5) , F1 (−5, −5 + 14) , F2 (−5, −5 − 14)] 7 21

8. [STPM ] The parametric equations of a conic are x = a cos θ − 3 and y = b sin θ + 4, where a and b are positive constants and 0 ≤ θ ≤ 2π. (a) Find the standard form of the equation of the conic, and identify the type of conic.

[3 marks]

(b) If a = b = 5, determine and sketch the conic.

[3 marks]

[Answer : (a)

41

(x + 3)2 (y − 4)2 + = 1 ; (b) Circle] 2 a b2

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6: Vectors

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Vectors

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6


1. [STPM ] Find a vector r that is perpendicular to the vectors p = i + j and q = 2j − k. Express vector w = 2j + k in terms of p, q, and r. [8 marks] 2 3

1 3

2 3

[Answer : r = −i + j + 2k ; w = p + q + r]

2. [STPM ] Equation of plane π is given by r = j + 2k + λ(i − 3j − 2k) + µ(2i + j + k). Find the equation of the plane that contains the point (0, 3, −3) and parallel to π. [6 marks] [Answer : x + 5y − 7z = 36]

3. [STPM ] The position vector of points A and B respect to O are a = 2i − 2j − 9k and b = −2i + 4j + 15k respectively. Find the vector equation for the line passes through the midpoint of AB and perpendicular to the plane OAB. [3 marks] [Answer : r = j + 3k + λ(3i − 6j + 2k)]

4. [STPM ] Given that origin, O and position vectors of P , Q, R, and S are 4i + 3j + 4k, 6i + j + 2k, 9j − 6k and −i + j + k respectively. Find the equation of the plane OP Q. [2 marks] Show that the point S lies on the plane OP Q.

Show that the line RS are perpendicular to the plane OP Q. Find the acute angle between the line P R and the plane OP Q.


[Answer : x + 8y − 7z = 0 ; 60◦ ]

5. [STPM ] Forces (4i+3j) N, (3i+7j) N, and (−5i−6j) N act at a point. Calculate the magnitude of the resultant force and the cosine of the angle between the resultant force and the unit vector i. [5 marks] √ [Answer : 2 5 N,

√

5 ] 5

6. [STPM ] Position vectors of the points P and Q relative to the origin O are 2i and 3i + 4j respectively. Find −−→ −−→ the angle between vector OP and vector OQ [Answer : 53.1◦ ]

7. [STPM ] Given that points O, P , and Q non-colinear, R lies on the line P Q. Position vector of P , Q, and R respect to O are p, q, and r respectively. Show that r = µp + (1 − µ)q, where µ is a real number. [2 marks]

8. [STPM ] Prove that the planes ax + by + cz = d and a0 x + b0 y + c0 z = d0 are parallel if and only if a : b : c = a0 : b0 : c0 . Find the equation of the plane π that is parallel to the plane 3x + 2y − 5z = 2 and contains the point (-1, 1, 3). [5 marks] Find the perpendicular distance between the plane π and the plane 3x + 2y − 5z = 2. 42

[2 marks]

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6: Vectors 9√ 38] 19

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[Answer : 3x + 2y − 5z = −16 ;

9. [STPM ] Let u = cos φ i + sin φ j and v = cos θ i + sin θ j, where i and j are perpendicular unit vectors. Show that 1 1 |u − v| = sin (φ − θ). 2 2 [5 marks]

10. [STPM ] The vector equations of two intersecting lines are given by r = 2i + j + λ(i + j + 2k) and r = 2i + 2j − k + µ(i + 2j + k). (a) Determine the coordinates of the point of intersection of the two lines. (b) Find the acute angle between the two lines.

[3 marks]

[4 marks]

[Answer : (a) (1, 0, −2) ; (b) 33.6◦ ]

11. [STPM ] The line l has equation r = 2i + j + λ(2i + k) and the plane π has equation r = i + 3j − k + µ(2i + k) + v(−i + 4j). (a) The points L and M have coordinates (0, 1, −1) and (1, −5, −2) respectively. Show that L lies on l and M lies on π. [3 marks] Determine the sine of the acute angle between the line LM and the plane π and the shortest distance from L to π. [6 marks]

12. [STPM ]

[Answer :

4 1 If the angle between the vectors a = and b = is 135◦ , find the value of p. 8 p

2 2 √ , ] 3 38 3

[6 marks]

[Answer : −3]

13. [STPM ] The planes π1 and π2 with equations x − y + 2z = 1 and 2x + y − z = 0 respectively intersect in the line l. The point A has coordinates (1,0, 1). (a) Calculate the acute angle between π1 and π2 . [2 marks]     1 2 (b) Explain why the vector −1 ×  1  is in the direction of l. Hence, show that the equation 2 −1 of l is     0 −1 r = 1 + t  5  . 1 3 where t is a parameter.

[5 marks]

(c) Find the equation of the plane passing through A and containing l.

[3 marks]

(d) Find the equation of the plane passing through A and perpendicular to l.

[2 marks]

43

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6: Vectors

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[3 marks]

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(e) Determine the distance from A to l.

r

◦

[Answer : (a) 80.4 ; (c) 3x + 3y − 4z + 1 = 0 ; (d) −x + 5y + 3z = 2 ; (e)

34 ] 35

14. [STPM ] The position vectors of the points A, B, C and D, relative to an origin, are i + 3j, −5i − 3j, (x − 3)i − 6j and (x + 3)i respectively. (a) Show that, for any value of x, ABCD is a parallelogram. (b) Determine the value of x for which ABCD is a rectangle.

[3 marks]

[4 marks]

[Answer : (b) x = 1]

15. [STPM ] The diagram below shows non-collinear points O, A and B, with P on the line OA such that OP : P A = 2 : 1 and Q on the line AB such that AQ : QB = 2 : 3. The lines P Q and OB produced meet −→ −−→ at the point R. If OA = a and OB = b, R

Q

B

A

P

O

−−→ 1 2 (a) show that P Q = − a + b, 15 5 (b) find the position vector of R, relative to O, in terms of b.

[5 marks]

[5 marks]

[Answer : (b) 4b]

16. [STPM ] Two straight lines l1 and l2 have equations −2x + 4 = 2y − 4 = z − 4 and 2x = y + 1 = −z + 3 respectively. Determine whether l1 and l2 intersect. [7 marks] [Answer : No intersecton.]

17. [STPM ] The position vectors of the points A, B and C, with respect to the origin O, are a, b and c respectively. The points L, M , P and Q are the midpoints of OA, BC, OB, and AC respectively. 1 1 (a) Show that the position vector of any point on the line LM is a + λ(b + c − a) for some scalar 2 2 λ, and express the position vector of any point on the line P Q in terms of a, b and c. [6 marks] (b) Find the position vector of the point of intersection of the line LM and the line P Q.

44

[4 marks]

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6: Vectors 1 2

1 2

1 4

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[Answer : (a) b + µ(a + c − b) ; (b) (a + b + c)] 18. [STPM ] Find the equation of the plane which is parallel to the plane 3x + 2y − 6z − 24 = 0 and passes through the point (1, 0, 0). Hence, determine the distance between these two planes. [6 marks]

19. [STPM ]

[Answer : 3x + 2y − 6z − 3 = 0 , 3]

    0 1 The points A and B lie on the line r = 3 + λ −1, and the distance of each point is three units 6 −4 from the origin O. (a) Determine the coordinates of A and B. (b) Find the area of the triangle OAB.

20. [STPM ]

[6 marks]

[3 marks]

[Answer : (a) A(1, 2, 2), B(2, 1, −2) ; (b) 4.5]

(a) Find the equation of line l1 , passing through points A and B, where the position vectors of points A and B are a and b respectively. [1 marks] (b) R is a point on the line l1 in (a). If point C has position vector c, −→ i. find CR in terms of vectors a, b and c. −→ −−→ ii. prove that CR × AB = a × b + b × c + c × a,

21. [STPM ]

22. [STPM ]

The line l has the equation

[3 marks]

[Answer : (a) r = a + λ(b − a) ; (b)(i) (1 − λ)a + λb − c]

Find the coordinates of the point P on the line Q(9, 4, −3).

[1 marks]

y−1 z−3 x = = which is closest to the point 5 1 −2 [6 marks]

[Answer : P (10, 3, −1)]

x+7 y−4 z−5 = = and the plane π has the equation 4x − 2y − 5z = 8. 1 −3 2

(a) Determine whether the line l is parallel to the plane π.

[5 marks]

(b) Find the equation of the plane that is perpendicular to the plane π and contains the points Q(−2, 0, 3) and R(2, 1, 7). [6 marks] [Answer : (a) parallel. (b) x + 12y − 4z = −14]

23. [STPM ] The position vectors of three non-collinear points B, C and D relative to the origin O are b, c and d respectively. If b · (c − d) = 0 and c · (d − b) = 0, show that BC is perpendicular to OD. [4 marks] 45

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6: Vectors

24. [STPM ]

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(a) Determine the values of p and q.

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          1 3 5 1 0          The line r = 2 + λ p is perpendicular to the plane r = 0 + s −1 + t 1, where p and 3 1 1 q 2 q are constants. [5 marks]

(b) Using the values of p and q in (a), find the position vector of the point of intersection of the line and the plane. [5 marks]

25. [STPM ]

[Answer : (a) p = −2, q = −5 ; (b) 4i + 4k]

Find the equation of the plane which contains the straight line x − 3 = dicular to the plane 3x + 2y − z = 3.

z+1 y−4 = and is perpen3 2 [6 marks]

[Answer : x − y + z = −2]

26. [STPM ] The position vectors a, b and c of three points A, B and C respectively are given by a = i + j + k,

b = i + 2j + 3k, c = i − 3j + 2k.

(a) Find a unit vector parallel to a + b + c.

[3 marks]

(b) Calculate the acute angle between a and a + b + c.

[3 marks]

(c) Find the vector of the form i + λj + µk perpendicular to both a and b.

[2 marks]

(d) Determine the position vector of the point D which is such that ABCD is a parallelogram having BD as a diagonal. [3 marks] (e) Calculate the area of the parallelogram ABCD.

[4 marks]

1 5

[Answer : (a) √ (i + 2k) ; (b) 39.2◦ ; (c) i − 2j + k ; (d) i − 4j ; (e) 9]

27. [STPM ] Show that the point A(2, 0, 0) lies on both planes 2x − y + 4z = 4 and x − 3y − 2z = 2. Hence, find the vector equation of the line of intersection of both planes. [5 marks] [Answer : r = 2i + λ(14i + 8j − 5k)]

28. [STPM ] −→ −−→ A tetrahedron OABC has a base OAB and a vertex C, with OA = 2i + j + k, OB = 4i − j + 3k and −−→ OC = 2i − j − 3k. −−→ −→ −−→ (a) Show that OC is perpendicular to both OA and OB. ◦

[3 marks]

(b) Calculate, to the nearest 0.1 , the angle between the edge AC and base OAB of the tetrahedron. [5 marks]

(c) Calculate the area of the base OAB and the volume of the tetrahedron. 46

[7 marks]

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6: Vectors √

14,

14 ] 3

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[Answer : (b) 56.8◦ ; (c)

29. [STPM ] Three vectors a = pi + qj, b = −5i + j and c = 4i + 7j are such that a and b are perpendicular and the scalar product of a and c is 78. (a) Determine the values of p and q. (b) Find the angle between a and c.

[4 marks]

[3 marks]

[Answer : (a) p = 2, q = 10 ; (b) 18.4◦ ]

30. [STPM ] A parallelepiped for which OABC, DEF G, ABF E and OCGD are rectangles is shown in the diagram below.

−→ −−→ The unit vectors i and j are parallel to OA and OC respectively, and the unit vector k is perpendicular −→ −−→ −−→ to the plane OABC, where O is the origin. The vectors OA, OB and OD are 4i, 4i + 3j and i + 5k respectively. √ 13 35 (a) Show that cos ∠BEG = . [6 marks] 175 (b) Calculate the area of the triangle AEG. [6 marks] (c) Find the equation of the plane AEG.

[Answer : (b)

47

[3 marks]

1√ 634 ; (c) 15x + 20y − 3z = 60] 2

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7: Limits and Continuity

( (x − 1)2 , x ≤ 1, f (x) = a 1− , x > 1. x If f is continuous at x = 1, determine the value of a and sketch the graph of f .


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1. [STPM ] Function f is defined by

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Limits and Continuity

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7


[5 marks]

[Answer : a = 1]

 x  x<1 1 + e , f (x) = 3, x=1   2 + e − x, x > 1

(a) Find lim f (x) and lim f (x). Hence, determine whether f is continuous at x = 1.

[4 marks]

(b) Sketch the graph of f .

[3 marks]

x→1−

x→1+


 x − 1, f (x) = x + 2 ax2 − 1,

[Answer : (a) 1 + e , 1 + e ; not continuous ]

0≤x<2

x≥2

where a ∈ R. Find the value of a if lim f (x) exists. With this value of a, determine whether f is x→2 continuous at x = 2. [6 marks]

4. [STPM ]

[Answer : a =

5 ; f continuous at x = 2] 16

1 1 The x-coordinate of the point of intersection of the curves y = x2 + and y = 2 , where x > 0, is p. x x Show that 0.5 < p < 1. 5. [STPM ] The function f is defined by

( x2 − 1, x ≤ 1, f (x) = k(x + 1), x > 1.

(a) If f is continuous, find the value of k.


(√

f (x) =

x + 1, |x| − 1, 48

−1 ≤ x < 1, otherwise.

[2 marks]

[Answer : (a) k = 0]

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x→1−

√

[Answer : (a) 0 , 0 ,

7. [STPM ]

√

2 , 0 ; (b) continuous at x = −1 , discontinuous at x = 1]

f (x) − f (x + h) . h→0 h

x, find lim

8. [STPM ] The graph of a function f is as follows:

(a) State the domain and range of f .

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x→1+

(b) Determine whether f is continuous at x = −1 and x = 1.

Given x > 0 and f (x) =


lim f (x), lim f (x) and lim f (x).

x→−1+

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lim f (x),

x→−1−

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(a) Find


1

[Answer : − √ ] 2 x

[2 marks]

(b) State whether f is a one-to-one function or not. Give a reason for your answer.

[2 marks]

(c) Determine whether f is continuous or not at x = −1. Give a reason for your answer.

[3 marks]


(a) Find lim f (x). x→−1

[Answer : (a) D={x : −3 ≤ x < −1, −1 < x ≤ 2} , R={y : −1 < y < 2} ; (b) f is not one to one function ; (c) f discontinuous at x = −1.]

(√

f (x) =

x + 1, |x| − 1,

x ≥ −1; otherwise.

(b) Determine whether f is continuous at x = −1.


 4  √ , x < 0,   √ 4 − x f (x) = 2, x = 0,   x  √ , x > 0. 1+x−1

(a) Show that lim f (x) exists.

[3 marks]

[2 marks]

[Answer : (a) 0 ; (b) Yes.]

[5 marks]

x→0

49

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[2 marks]

[XXAnswer : (a) 2 ; (b) f is not continuous at x = 0.]

11. [STPM ] Evaluate

√

2x2 + 25 − 5 , x→0 x2 √ 9x2 + 1 (b) lim . x→∞ 3x − 1 (a) lim

12. [STPM ]

Given that f (x) =

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(b) Determine whether f is continuous at x = 0.

[3 marks] [3 marks]

[Answer : (a)

3x2 + x . Find lim f (x) and lim f (x). x→∞ 3x2 − 8x − 3 x→− 13


1 ; (b) 1] 5

[4 marks]

[XXAnswer :

 x  1 − e , x < 0, f (x) = 1, x = 0,   x e − 1, x > 0.

(a) Determine the existence of the limit of f (x) as x approaches 0.

1 , 1] 10

[4 marks]

(b) State, with a reason, whether f is continuous at x = 0. Hence, give the interval(s) on which f is continuous. [3 marks] (c) Sketch the graph of f .

14. [STPM ] Function f is defined by

[3 marks]

[XXAnswer : (a) 0, (b) (−∞, 0) ∪ (0, ∞)]

 2  x − 4 , x 6= 2, f (x) = |x − 2|  4, x = 2.

Determine whether f is continuous at x = 2.

15. [STPM ] A continuous function f is defined by

 x3 − 1, −1 ≤ x < 2, f (x) = 1 − (x − 3)2 + c, 2 ≤ x ≤ 8, 2

where c is a constant. 50

[5 marks]

[XAnswer : No]

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(a) Determine the value of c. (c) Sketch the graph of f .

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(b) Find the values of x such that f (x) = 0.

(d) Find the maximum and minimum values of f .

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[3 marks]

(e) State whether f is a one-to-one function or not. Give a reason for your answer. [XXAnswer : (a) c =

16. [STPM ]

Evaluate

6(x − 2) , x3 − 8 x−8 , (b) lim √ √ x→8 6− x−2 (a) lim

x→2

17. [STPM ]

[2 marks]

√ 15 15 ; (b) x = 1, 3 + 15 ; (d) max= , min=−5 ; (e) one-to-one] 2 2

( x2 + 2x + 5, The function g is defined by g((x) = 3ex + k

x < 0, x ≥ 0.

[2 marks]

[3 marks]

[Answer : (a)

√ 1 ; (b) − 6] 2

(a) Find lim g(x) and lim g(x), and determine the value of k such that the function g is continuous x→0−

x→0+

at x = 0.

(b) Describe the continuity of the function g for x = 0, x < 0 and x > 0. 18. [STPM ] Function f is defined by

f (x) =

Determine whether lim f (x) exists. x→3

 1   ,   2x

x ≤ 3,

    x−3 , x2 − 9

x > 3.

51

[5 marks]

[3 marks]

[5 marks]

[Answer : Yes]

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8: Differentiation

1. [STPM ] Differentiate with respect to x and simplify your answer as far as possible: x2 − x + 1 , x2 + x − 1 (b) e−2x [2 cos(3x) − 3 sin(3x)]. (a)

[Answer : (a)

2x(x − 2) ; (b) −13e−2x cos(3x)] (x2 + x − 1)2

2. [STPM ] Differentiate with respect to x and simplify your answer as far as possible: cos x + sin x , cos x − sin x (b) xn loge x. (a)

[Answer : (a)

3. [STPM ]

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Differentiation

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8


2 ; (b) xn−1 (1 + n ln x)] 1 − sin 2x

dy where dx i. y = (2x − 1)3 (3x + 2)4 and express your result in the form of its factors, ii. y = e−3x (2 cos 2x + 3 sin 2x). 1 1 (b) If x = t − and y = 2t + , where t is a non-zero parameter, prove that t t (a) Find

Deduce that −1 <

3 dy =2− 2 . dx t +1

dy < 2. dx

4. [STPM ]

[Answer : (a) (i) 42x(2x − 1)2 (3x + 2)3 ; (ii) −13e−2x sin 2x]

ln x dy Find the x-coordinate of the point on the curve y = 2 (x > 0) such that = 0, and determine if x dx it is a maximum or minimum point. Sketch the curve for x > 0. You can assume that y → 0 when x → ∞.

5. [STPM ]

[Answer : x =

√

e]

1 tan3 x with respect to x, and express your answer in terms of tan x. 3 d2 y dy (b) Given y = ae−mx cos px, prove that + 2m + (m2 + p2 )y = 0. 2 dx dx (a) Differentiate x − tan x +

52


8: Differentiation

6. [STPM ]

Given that y =

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1 dy (c) Given y = ln(1 + x) − x + x2 , show that ≥ 0 for all values of x > −1. 2 dx

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[Answer : (a) tan4 x]

d2 y cos x − sin x dy , show that = 0. + 2y cos x + sin x dx2 dx

7. [STPM ] A curve with the equation y = ax4 + bx3 + cx2 + dx + e, where a, b, c, d, e are constants, has the following characteristics: (a) It is symmetrical about the y-axis,

(b) It passes through the point (2, −18) and has gradient zero at this point, (c) y = 0 when x = 1.

Show that b = d = 0 and find the values of a, c and e. Sketch the curve and give the coordinates of its turning points.

8. [STPM ]

(a) Given that y = (x + 2)2 (3x − 1)3 , find (b) If y =

[Answer : a = 2, c = −16, e = 14]

dy as a product of its factors. dx

e−x dy , show that (1 + x2 ) + (1 + x)2 y = 0. 1 + x2 dx

9. [STPM ] The parametric equation of a curve are

[Answer : (a) (x + 2)(15x + 16)(3x − 1)2 ]

x = a cos3 θ, y = a sin3 θ

where a is a positive constant and 0 ≤ θ < 2π.

Find the equation of the tangent at the point with the parameter θ. This tangent meets the axes at L and M . Prove that the length of LM is independent of θ. [Answer : y cos θ + x sin θ = a sin θ cos θ , LM = a]

10. [STPM ] A spherical balloon is being inflated at a constant rate of 500 m3 s−1 . Find the rate of increase in the total surface area of the balloon when its radius is 20 m.

11. [STPM ] Given that y = αe−2x sin(x + β), where α and β are constants, verify d2 y dy +4 + 5y = 0. 2 dx dx 53

[Answer : 50]

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8: Differentiation

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12. [STPM ] (a) For the curve y = sin x cos3 x, where 0 ≤ x ≤ π, find the x and y coordinates of the points where dy = 0. Sketch this curve. dx 2 1 1 dy 2 3 = cot x(cos2 x − 3 sin2 x)2 (b) For the curve y = sin x cos x, where 0 ≤ x ≤ π, show that 2 dx 4 on the condition that x 6= 0. Sketch the curve. [Answer :

√ ! π π 3 3 , , ,0 , 6 16 2

13. [STPM ] A curve is given by its parametric equations

1 x = t2 , y = 1 − , (t > 0). t

√ ! 5π 3 3 ] ,− 6 16

The curve intersects the x-axis at P . Find the equation of the tangent to the curve at P .

14. [STPM ]

The parametric equations of a curve are x = t2 , y = t3 . Express of the tangent to the curve at the point P (p2 , p3 ).

15. [STPM ]

If y = 3x + sin x − 8 sin Deduce that

dy in terms of t. Find the equation dx

[Answer :

3 dy = t ; 2y = 3px − p3 ] dx 2

1 dy 1 x , find and express your answer in terms of cos x . 2 dx 2

dy ≥ 0 for all values of x. dx

16. [STPM ]

[Answer : 2y − x + 1 = 0]

1 2

2

[Answer : 2 cos x − 1

]

x−3 . (x − 2)(x + 1) (b) Find the points where the curve intersects the axes, and find the stationary points on this curve. (a) Find the equation of the asymptotes of the curve y =

(c) Sketch this curve.

(d) Find the values of k such that the equation (x − 3) = k(x − 2)(x + 1) does not have real roots. 1 9

[Answer : (a) x = 2, x = −1, y = 0 ; (b) (3, 0), (0, 1.5), (1, 1), (5, ) ; (d)

1 < k < 1] 9

17. [STPM ] Find the coordinates of the stationary points of the curve y = x − ln(1 + x) and sketch the curve. 54

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8: Differentiation

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18. [STPM ]

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[Answer : (0, 0)]

1 (a) Find the points on the x-axis intersected by the curve y = x3 − x2 − 2x. Find also the maximum 2 and minimum points, as well as any points of inflection on this curve. (b) Sketch this curve. 1 (c) Find the value of k if the equation x3 − x2 − 2x = k has a repeated root, and state this root. 2 √

√

[Answer : (a) (0, 0), (1 + 5, 0), (1 − 5, 0) ;

2 20 2 44 Maximum point (− , ) , Minimum point (2, −4) , Point of inflection ( , − ) ; 3 27 3 27 20 2 (c) k = −4, repeated root=2; k = , repeated root=− ] 27 3

19. [STPM ] If y = etan x , show that

dy d2 y = (1 + tan x)2 . dx2 dx

20. [STPM ] Show that the equation of the normal to the curve y = tan 2x at the point where the x-coordinate is √ π is 3x + 24y = π − 24 3. 3 21. [STPM ] The parametric equations of a curve are x = t2 − 2, y = t3 − 3. Find the equation of the normal to the curve at the point where the parameter t = 2.

22. [STPM ] Differentiate with respect to x (a) (x2 + 2x)ex 1 − x2 (b) √ , 1 + 2x

2 +2x

,

simplifying your answers.

[Answer : x + 3y = 17]

2

[Answer : (a) 2(x + 1)3 ex

+2x

; (b) −

3x2 + 2x + 1 3

(1 + 2x) 2

]

23. [STPM ] Find the equations of the tangent and normal to the curve x2 y + xy 2 = 12 at the point (1, −4).

24. [STPM ] A curve has the equation y 2 = x2 (x + 3).

[Answer : 8x − 7y = 36 ; 7x + 8y + 25 = 0]

Show that the curve is symmetric about the x-axis. Show that for all points on this curve, x ≥ −3.

Find coordinates of the turning points of this curve.

Sketch the curve. Show clearly the shape of the curve near the origin. Find the area bounded by the loop of this curve. 55

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8: Differentiation 24 √ 3] 5

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[Answer : (−2, 2), (−2, −2) ; Area=

25. [STPM ] Two parallel sides of a rectangle respectively lengthen at a rate of 2 cm per second, while the other two parallel sides shorten such that the area of the rectangle is always 50 cm2 . If, at the time t, the length of each lengthening side is x, the length of each shortening side is y, and the perimeter of the rectangle is p, show that dp y . =4 1− dt x Find the rate of change in the perimeter when (a) x=5 cm, (b) y=5 cm.

√ Show that the perimeter of the rectangle is the least when x = y = 5 2 cm.

26. [STPM ] Differentiate with respect to x 2

(a) (2x3 + 1)ex , (b) ln(x2 e−x ),

27. [STPM ]

[Answer : (a) −4 ; (b) 2]

2

[Answer : (a) 2xex (2x3 + 3x + 1) ; (b)

x2 . x2 − 4 Write the equations of the asymptotes of this curve. A curve has the equation y =

2 − 1] x

Find the coordinates of the turning point on this curve, and determine if this is a maximum or minimum point. Determine if there are any points of inflection on this curve. Sketch this curve.

[Answer : x = −2, x = 2, y = 1 ; (0,0) is a local maximum point ; No points of inflexion]

28. [STPM ] If y = ln(sin px + cos px), show that

d2 y + dx2

dy dx

2

29. [STPM ] Differentiate each of the following with respect to x. (a) (x2 + 1)e−x , √ (b) cos2 ( x).

+ p2 = 0.

2 −x

[Answer : (a) −(x − 1) e 56

√ sin 2 x ; (b) − √ ] 2 x

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30. [STPM ] dy x−1 Find if ey = . dx 3−x

Determine the gradient of the curve y = ln

31. [STPM ]

x−1 3−x

8: Differentiation

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at the point where it intersects the x-axis.

[Answer :

2 ; 2] (x − 1)(3 − x)

1 1 − . 2 x x Find the coordinates of the turning point of the curve, and determine if it is a maximum or minimum point. A curve has the following equation y =

Sketch this curve.

The tangent to the curve at the point A(1, 0) meets the curve once again at the point B. Find the coordinates of B. 1 4

[Answer : (2, − ) is minimum point, B = (−1, 2)]

32. [STPM ] The equation of a curve is x2 y + xy 2 = 2. Find the equations of both tangents to the curve at the point x = 1.

33. [STPM√] If y = sin x, show that

4y 3

[Answer : x + y = 2 ; y = −2]

d2 y + y 4 + 1 = 0. dx2

34. [STPM ] Differentiate each of the following with respect to x. (a) e−x ln x3 , 2x (b) . 1 + x4

35. [STPM ]

[Answer : (a)

2x [(1 + x4 ) ln 2 − 4x3 ] 3e−x ] (1 − x ln x) ; (b) x (1 + x4 )2

√ dy √ The variables x and y are connected by y x − y − x = 1. Find the values of y and when x = 1. dx


f (x) = cos x +

1 cos 2x, 0 ≤ x ≤ 2π. 2

57

4 3

[Answer : 4 , − ]

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8: Differentiation

(c) Sketch the graph f . (d) State the range of f . [Answer : (a) 0.4π, 1.6π ; (b)

3 0, 2

2 4 1 3 3 3 3 3 , π, − , 2π, , , ; (d) {y : − ≤ y ≤ }] π, − π, − 2 2 3 4 3 4 4 2

37. [STPM ] If y 2 = ln(x2 y) where x, y > 0, (a) show that (b) find

dy 2y = , dx x(2y 2 − 1)

dy when y = 1. dx

38. [STPM ]

If x = sin3 2θ, y = cos3 2θ, find

dy in terms of θ. dx

39. [STPM ] 1 If y = (2ex − 6x + 5) 2 , show that

40. [STPM ]

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(b) Find all the pairs (x, f (x)) when f 0 (x) = 0.

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(a) Find all values of x in the form of kπ, with k correct to one decimal place when f (x) = 0.

y

d2 y + dx2

dy dx

2

= ex .

2 e

[Answer : (b) √ ]

[6 marks]

[Answer : − cot 2θ]

[4 marks]

Figure above shows a composite solid which consists of a cuboid and a semicylindrical top with a common face ABCD.

58

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8: Differentiation

3

1 [9600 − (8 + 5π)x2 ]. 24x

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y=

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The breadth and length of the cuboid is x cm and 2x cm respectively and its height is y cm. Given that the total surface area of this solid is 2400 cm2 . Show that

[3 marks]

If the volume of this solid is V cm , express V in terms of x. Hence, show that V attains its maximum 40 . [9 marks] when x = √ 4+π Find this maximum volume. [3 marks] [Answer : V =

1 64000 [9600x − 2πx3 − 8x3 ] , √ ] 12 3 π+4

41. [STPM ] Find the gradient of the curve 2x2 + y 2 + 2xy = 5 at the point (2, −1).

42. [STPM ] The parametric equations of a curve are

[3 marks]

[Answer : -3]

x = sec t − tan t; y = cosec t − cot t,

dy 1 + sin t 1 =− . with 0 < t < π. Show that 2 dx 1 + cos t

[4 marks]

−1 3 Tangent to the curve at the point A, with t = tan , meets the tangent to the curve at the point 4 4 B, with t = tan−1 , at point N . Find the coordinates of N . [8 marks] 3

43. [STPM ] The equation of a curve is

[Answer : N =

x2 . x2 − 3x + 2 Find the asymptotes and the stationary points of the curve. Sketch the curve.

y=

[8 marks] [4 marks]

2

2

Determine the number of real roots of the equation k(x − 1) (x − 2) = x where k > 0.

The equation of a curve is y =

[3 marks]

[Answer : Asymptotes are x = 1, x = 2, y = 1 ; 1 root]

44. [STPM ] √ √ dy Find in terms of x if x = e t and y = et . dx

45. [STPM ]

7 7 , ] 17 17

e2kx − 1 where k is a positive constant. e2kx + 1 59

[4 marks]

[Answer :

e

(ln x)2 2

x

ln x]

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8: Differentiation

dy > 0 for all values of x. dx dy d2 y d2 y (b) Show that + ky 2 = k. Hence, show that ≤ 0 for x ≥ 0 and ≥ 0 for x ≤ 0. dx dx2 dx2 (c) Sketch the curve.

Given that y =

sin kx , where k is a positive integer, show that 1 + cos kx sin kx

d2 y = k2 y2 . dx2

[3 marks]

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46. [STPM ]

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(a) Show that

[8 marks]

[4 marks]

[6 marks]

47. [STPM ] The graphs of y = x3 + ax2 + bx + c passes through (3, −21) and has stationary points when x = 2 and x = −2. Find the values of a, b and c. [5 marks] Find the coordinates of these stationary points and determine if they are local extremums. Find also the point of inflexion of the curve. [7 marks] dy < 0. [3 marks] Determine the set of x so that dx [Answer : a = 0 , b = −12 , c = −12 ; (2,-28) is local minimum , (-2,4) is local maximum ; point of inflexion is (0,-12) ; {x : −2 < x < 2}]

48. [STPM ] A curve has parametric equations x = e2t − 2t and y = et + t. Find the gradient of the curve at the point with t = ln 2. [5 marks] [Answer :

1 ] 2

49. [STPM ] A curve with equation y = x3 + px2 + qx + r cuts the y-axis at y = −34 and has stationary points at x = 3 and x = 5. Find the values of p, q, and r. [6 marks] Show that the curve cuts the x-axis only at x = 1, and find the gradient of the curve at that point. Sketch the curve.

50. [STPM ] Given a curve with parametric equation

with a > 0 and t ∈ R.

[7 marks] [2 marks]

[Answer : p = −12 , q = 45 , r = −34 ; 24]

x = a(t − 3t3 ), y = 3at2 ,

Determine the values of t when the curve cuts the y-axis and sketch the curve. 2 2 dx dy Show that + = a2 (1 + 9t2 )2 . dt dt

[4 marks] [3 marks]

√

[Answer : t = 0, 60

√ 3 3 ,− ] 3 3


8: Differentiation

52. [STPM ]

Given that y = e−x cos x, find

53. [STPM ] Function f if defined by

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51. [STPM ] Find the equation of the normal to the curve x2 y + xy 2 = 12 at the point (3, 1).

dy d2 y and when x = 0. dx dx2

f (x) =

2x . (x + 1)(x − 2)

Show that f 0 (x) < 0 for all values of x in the domain of f .

[6 marks]

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[Answer : y =

38 15 x− ] 7 7

[4 marks]

[Answer :

d2 y dy = −1, = 0] dx dx2

[5 marks]

Sketch the graph of y = f (x). Determine if f is a one to one function. Give reasons to your answer. [6 marks]

Sketch the graph of y = |f (x)|. Explain how the number of the roots of the equation |f (x)| = k(x − 2) depends on k. [4 marks]

54. [STPM ]

[Answer : f is not one to one function. If k ≥ 0, 1 root. If k < 0, 3 roots.]

dy √ when y = 1. If y = ln xy, find the value of dx

55. [STPM ]

[5 marks]

[Answer :

1 ] e2

2 A curve is defined by the parametric equations x = 1 − 2t, y = −2 + . Find the equation of the t normal to the curve at the point A(3, −4). [7 marks] The normal of the curve at the point A cuts the curve again at point B. Find the coordinates of B.

56. [STPM ] cos x d2 y dy If y = , where x 6= 0, show that x 2 + 2 + xy = 0. x dx dx 57. [STPM ]

[4 marks]

[Answer : x + y + 1 = 0 ; B(−1, 0)]

[4 marks]

1 Find the coordinate of the stationary point on the curve y = x2 + where x > 0; give the x-coordinate x and y-coordinate correct to three decimal places. Determine whether the stationary point is a minimum point or a maximum point. [5 marks] [Answer : (0.794 , 1.890) , minimum]

58. [STPM ] If y = x ln(x + 1), find an approximation for the increase in y when x increases by δx. Hence, estimate the value of ln 2.01 given that ln 2 = 0.6931. 61

[6 marks]

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8: Differentiation

The function f is defined by f (t) =

4ekt − 1 where k is a positive constant, t > 0. 4ekt + 1

(a) Find the value of f (0). (b) Show that f 0 (t) > 0. 2

0

[1 marks] [5 marks]

00

(c) Show that k[1 − f (t) ] = 2f (t) and hence show that f (t) < 0. (d) Find lim f (t). t→∞

(e) Sketch the graph of f .

60. [STPM ] If y =

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59. [STPM ]

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[Answer : 0.698]

[2 marks] [2 marks]

[Answer : (a)

dy x , show that x2 = (1 − x2 )y 2 . 2 1+x dx

61. [STPM ]

Find the coordinates of the stationary points on the curve y = Sketch the curve.

[6 marks]

3 ; (d) 1] 5

[4 marks]

x3 and determine their nature. x2 − 1 [10 marks] [4 marks]

Determine the number of real roots of the equation x3 = k(x2 − 1), where k ∈ R, when k varies.[3 marks] √ √ √ 3 3 3 3 ) is local min. , (− 3, − ) is local max. [Answer : (0, 0) is inflexion point , ( 3, √ √ √2 √ 2 √ 3 3 3 3 3 3 3 3 3 3 1 real root for − ] 2 2 2 2 2 √

62. [STPM ] sin x − cos x d2 y dy If y = , show that = 2y . 2 sin x + cos x dx dx 63. [STPM ] Show that the curve y =

[6 marks]

x is always decreasing. [3 marks] −1 Determine the coordinates of the point of inflexion of the curve, and state the intervals for which the curve is concave upwards. [5 marks] x2

Sketch the curve.

64. [STPM ] A curve is defined by x = cos θ(1 + cos θ) , y = sin θ(1 + cos θ). (a) Show that

dx dθ

2

+

dy dθ

2

= 2(1 + cos θ).

[3 marks]

[Answer : (0, 0) ; (−1, 0) ∪ (1, ∞)]

[4 marks]

62

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8: Differentiation

[Answer : p = 3, q = 5]

66. [STPM ] A curve is defined by the parametric equations x=t−

where t 6= 0.

2 t

and y = 2t +

1 t

dy 5 1 dy =2− 2 , and hence, deduce that − < < 2. dx t +2 2 dx dy 1 (b) Find the coordinates of points when = . dx 3 (a) Show that

67. [STPM ]

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65. [STPM ] The line y + x + 3 = 0 is a tangent to the curve y = px2 + qx, where p 6= 0 at the point x = −1. Find the values of p and q. [6 marks]

[8 marks]

[3 marks]

[Answer : (b) (-1,3) , (1,-3)]

dy 1 1 , =√ Given that u = (ex + e−x ), where x > 0 and y = f (u) is a differentiable function f . If 2 du u2 − 1 dy show that = 1. [5 marks] dx 68. [STPM ] The functions f and g are defined by

f : x → x3 − 3x + 2, x ∈ R. g : x → x − 1, x ∈ R.

(a) Find h(x) = (f ◦ g)(x), and determine the coordinates of the stationary points of h.

[5 marks]

(b) Sketch the graph of y = h(x).

[2 marks]

1 (c) On a separate diagram, sketch the graph of y = . h(x) Hence, determine the set of values of k such that the equation i. one root, ii. two roots, iii. three roots.

[3 marks]

1 = k has h(x)

[1 marks] [1 marks]

[1 marks]

1 4

1 4 1 {k : k > }] 4

[Answer : (a) h(x) = x3 − 3x2 + 4 , (0,4) , (2,0) ; (c) (i) {k : k < 0, 0 < k < } ; (ii) {k : k = } ; (iii)

69. [STPM ] √ Given that y is differentiable and y x = sin x, where x 6= 0. Using implicit differentiation, show that 2 dy 1 2d y 2 x +x + x − y = 0. dx2 dx 4 [6 marks]

63

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f (x) =

(a) State all asymptotes of f .

8: Differentiation

ln 2x , where x > 0. x2

[2 marks]

(b) Find the stationary point of f , and determine its nature. (c) Obtain the intervals, where i. f is concave upwards, and ii. f is concave downwards.

[6 marks]

Hence, determine the coordinates of the point of inflexion.

(d) Sketch the graph y = f (x).

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[6 marks] [2 marks]

1 1 2 2 is a maximum point. e , [Answer : (a) x = 0, y = 0 ; (b) 2 e (c)(i) (1.15, ∞) ; (ii) (0, 1.15) ; (1.15, 0.630)]

71. [STPM ]

Given that y = (2x)2x , find

dy in terms of x. dx


f (x) = √

(a) Show that

[4 marks]

[Answer : (2x)2x (2 + 2 ln(2x))]

e−x , where x ∈ R, 1 + x2

f 0 (x) =

−e−x (x2 + x + 1) 3

(1 + x) 2

(b) Show that f is a decreasing function. (c) Sketch the graph of f .

.


73. [STPM ] A curve is defined by the parametric equations x = ke−t cos t and y = ke−t sin t, where k is a constant.

(a) Show that

dx dt

74. [STPM ]

2

+

dy dt

2

= 2k 2 e−2t .

[4 marks]

Find the equation of the normal to the curve with parametric equations x = 1 − 2t and y = −2 + at the point (3, −4).

2 t

[6 marks]

[Answer : y = −x − 1]

75. [STPM ] A right circular cone of height a + x, where −a ≤ x ≤ a, is inscribed in a sphere of constant radius a, such that the vertex and all points on the circumference of the base lie on the surface of the sphere. 64

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8: Differentiation

(c) Sketch the graph of V against x.

[2 marks]

[3 marks]

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[6 marks]

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1 (a) Show that the volume V of the cone is given by V = π(a − x)(a + x)2 . 3 (b) Determine the value of x for which V is maximum and find the maximum value of V .

1 1 (d) Determine the rate at which V changes when x = a if x is increasing at a rate of a per 2 10 minute. [4 marks] [Answer : (b) x =

1 a 32 3 , πa ; (d) − πa3 ] 3 81 40

76. [STPM ] Find the gradients of the curve y 3 + y = x3 + x2 at the points where the curve meets the coordinate axes. [6 marks] [Answer : 0 , 1]

77. [STPM ] The parametric equations of a curve are x = θ − sin θ and y = 1 − cos θ. Find the equation of the 1 [7 marks] normal to the curve at a point with parameter π. 2

78. [STPM ] A curve is defined implicitly by the equation x2 + xy + y 2 = 3.

π 2

[Answer : y = −x + ]

dy 2x + y + = 0. [3 marks] dx x + 2y (b) Find the gradients of the curve at the points where the curve crosses the x-axis and y-axis.[5 marks] (a) Show that

(c) Show that the coordinates of the stationary points of the curve are (-1,2) and (1, -2).

[5 marks]

(d) Sketch the curve.

[2 marks]

79. [STPM ] A rectangle with a width 2x is inscribed in a circle of constant radius r.

1 2

[Answer : (b) − , -2]

(a) Express the area A of the rectangle in terms of x and r. √ (b) Show that the rectangle is a square of side r 2 when A has a maximum value.

80. [STPM ] The graph of y = 2 cos x + sin 2x for 0 ≤ x ≤ 2π is shown below.

65

[2 marks]

[5 marks]

p

[Answer : (a) A = 4x r2 − x2 ]

8: Differentiation

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The points A and C are local extremum points. The points B, D, E and F are points of inflexion. (a) Determine the coordinates of i. the points of local extremum. ii. the points of inflexion.

(b) State the intervals where the graph is concave upward. (c) Calculate the area of the region bounded by the curve and the x-axis.

[5 marks] [5 marks]

[1 marks]

[4 marks]

√ ! √ ! π 3 3 5π 3 3 , ,− [Answer : (a) (i) A = ,C= ; 6 2 6 2 π π 3π 3π , 0 , D = (3.39, −1.45) , E = , 0 , F = (6.03, 1.45) ; (b) , 3.39 ∪ , 6.03 ; (c) 4] (ii) B = 2 2 2 2

81. [STPM ]

The graph of y =

3x − 1 is shown below. (x + 1)3

The graph has a local maximum at A and a point of inflexion at B. 66

kkleemaths.com


8: Differentiation

(a) Write the equations of the asymptotes of the graph.

[1 marks]

84. [STPM ] Differentiate with respect to t 2

(a) (t2 − 1)et −1 , r 1 (b) ln 1 + . t

[1 marks] [1 marks]

equation

[1 marks]

5 27 ; , 3 128 5 (i) {x : x < −1, −1 < x ≤ 1} ; (ii) (−∞, −1) ∪ ,∞ ; 3 1 (c) (i) {k : 0 < k < } ; (ii) {k : −1 < k ≤ 0}] 4

(a) Find the stationary points on the curve, and determine it’s nature.

83. [STPM ] p If y sin−1 2x = 1 − 4x2 , show that

[9 marks]

[2 marks]

[Answer : (a) x = −1 , y = 0 ; (b) A 1,

82. [STPM ] The equation of a curve is y = x3 e3−2x .

(b) Sketch the curve.

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i. has three distinct real roots, ii. has only one positive root.

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(b) Determine the coordinates of the points A and B. Hence, state dy i. the set of values of x when ≥ 0, dx ii. the intervals where the graph is concave upward. 3x − 1 (c) Using the above graph of y = , determine the set of values of k for which the (x + 1)3 3x − 1 − k(x + 1)3 = 0

,B

[7 marks]

dy + 4xy + 2y 2 = 0. dx

85. [STPM ] For the graph of y = 3x4 + 16x3 + 24x2 − 6,

[3 marks]

[Answer : (a) (0,0)=point of inflexion,

(1 − 4x2 )

1 4

3 27 , =local maximum] 2 8

[5 marks]

[3 marks]

[3 marks]

[Answer : (a) 2t3 et

2

−1

; (b) −

1 ] 2(t2 + 1)

(a) determine the intervals on which the graph is concave upward and concave downward, [6 marks] (b) find the points of inflexion,

[3 marks]

67


8: Differentiation

2 3

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(c) determine the extremum point and its nature. Hence, sketch the graph of y = 3x4 + 16x3 + 24x2 − 6.

[Answer : (a) (−∞, −2) ∪ − , ∞ , −2, −

86. [STPM ] A continuous function f is defined by

where c is a constant. (a) Determine the value of c.

2 3

; (b)

[3 marks]

2 14 − , 3 27

and (−2, 10) ; (c) (0, −6) minimum]

 x3 − 1, −1 ≤ x < 2, f (x) = 1 − (x − 3)2 + c, 2 ≤ x ≤ 8, 2

(b) Find the values of x such that f (x) = 0. (c) Sketch the graph of f .

[3 marks]

(d) Find the maximum and minimum values of f .

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(e) State whether f is a one-to-one function or not. Give a reason for your answer. [Answer : (a) c =


[3 marks] [2 marks]

√ 15 15 ; (b) x = 1, 3 + 15 ; (d) max= , min=-5] 2 2

87. [STPM ] A water storage tank ABCDEF GH is a part of an inverted right square based pyramid, as shown in the diagram below.

The complete pyramid OABCD has a square base of sides 12 m and height 15 m. The depth of the 1 tank is 9 m. Water is pumped into the tank at the constant rate of m3 min−1 . 3

68

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8: Differentiation

[3 marks]

(c) Calculate the time taken to fill up the tank if initially the tank is empty.

88. [STPM ] The equation of a curve is y = x(x − 2)3 .

[Answer : (b)

(a) Find the set of values of x for which y ≥ 0.

25 ; (c) 33.7] 3888

[9 marks]

[3 marks]

[Answer : (a) {x : x ≤ 0, x ≥ 2} ; (b) Extremum=

69

[3 marks]

[3 marks]

(b) Determine the extremum point and the points of inflexion on the curve. (c) Sketch the curve.

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(a) Show that the volume of water V m3 when the depth of water in the tank is h m is given by 16 V = h(h2 + 18h + 108). [3 marks] 75 (b) Find the rate at which the depth is increasing at the moment when the depth of water is 3 m.

1 27 ,− 2 16

, Inflexion=(1, −1), (2, 0)]

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9: Integration

1

Z

x dx, give your answer correct to two decimal places. 4 − x2 Z 1 x dx (b) Using the substitution t = tan , or otherwise, find 2 3 − 5 cos x √

(a) Evaluate

1 2

1 [Answer : (a) 0.20 ; (b) ln 4

2. [STPM ]

Z

(a) Find

π 4

tan3 θ dθ, giving your answer correct to two significant figures.

0

2

4

Z

√

(b) Using the substitution u = 2t + 1, or otherwise, evaluate

3. [STPM ]

Z

2a

(a) Find

a

x3 dx. x4 + a4

0

4. [STPM ]

(a) Show that

0

4

x−1 dx = ln 2 2x + 3x + 1

1

Z

Z

(sin x + 3 cos x)2 dx. Z 2 (x − 1)(5x + 2) 1 8 (b) Show that dx = ln . 2 + 2) (2x − 1)(x 2 3 1 (a) Find

0

r

70

2 tan x2 − 1 2 tan x2 + 1

+ c]

[Answer : (a)

Z 1

5

10 ] 3

π 1−x dx = − 1. 1+x 2

25 . 27

(b) Using the substitution u2 = 2x − 1, or otherwise, evaluate

5. [STPM ]

t dt. 2t + 1

[Answer : (a) 0.15 ; (b)

(b) Using the substitution x = cos 2θ, or otherwise, prove that

Z

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1. [STPM ]

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Integration

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9


√ x 2x − 1 dx.

1 ln 4

[Answer : (b)

17 ] 2

428 ] 15

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9: Integration

1

x3 dx, leaving your answer in a form involving logarithms. 2 0 x +2 Z 1p 1 1√ 4 − x2 dx = π + 3. (b) Show that 3 2 0 Z

(a) Evaluate

7. [STPM ]

Z

(a) Find

2 + cos x dx. sin2 x

1 [Answer : (a) + ln 2

Z

(b) Using the substitution x = 2 sin θ, or otherwise, evaluate correct to two significant figures.

3 cos 2x + c] 2

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6. [STPM ]

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[Answer : (a) 5x + 2 sin 2x −

0

1

√

2 ] 3

x2 dx, giving your answer 4 − x2

[Answer : (a) −2 cot x − cosec x + C ; (b) 0.118]

8. [STPM ] Sketch the corresponding curves for the following equations in separate diagrams, showing the turning points and any asymptotes parallel to the axes: (a) y = (x − 1)(x − 3) 1 (b) y = (x 6= 1, x 6= 3) (x − 1)(x − 3)

4 intersects the graphs (b) at the points A and B. 3 Calculate the finite area bounded by the line AB and the portion of the graph between A and B, giving your answer correct to three significant figures. The line y = −

[Answer : (a) (2,-1), minimum point ; (b) (2,-1), maximum point , asymptotes are x = 1, x = 3, y = 0 ;

9. [STPM ]

Z

1

Find the exact value of

√ x x + 3 dx.

−2

10. [STPM ]

Z (a) Evaluate

π 4

tan2 x dx.

0

71

Area=0.235]

8 5

[Answer : − ]

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9: Integration

18 − 4x − x2 B C A + + ≡ , show that A = 2 and determine the 2 (4 − 3x)(1 + x) 4 − 3x 1 + x (1 + x)2 values of B and C.Z 1 7 3 18 − 4x − x2 dx = ln 2 + Hence, show that 2 (4 − 3x)(1 + x) 3 2 0 [Answer : (a) 1 −

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(b) Given that

π ; (b) B = 1, C = 3] 4

11. [STPM ] A bowl in the shape of a hemisphere with a radius a and its rim horizontal, is filled with a liquid to a height of h unit. Show, through integration, that the volume of the liquid in the bowl is 1 2 πh (3a − h). 3

12. [STPM ]

Z

(a) Show that

0

π 6

√ 4π − 3 3 sin x cos x dx = . 192 2

2

(b) Using the substitution u = x2 , or otherwise, evaluate logarithmic form.

13. [STPM ]

Z

√ 2 2

Z 0

x dx, leaving your answer in 1 − x4

[Answer : (b)

2x + 1 √ dx. x+1 Z 4 11x2 + 4x + 12 (b) Show that dx = ln 675. 2 0 (2x + 1)(x + 4) (a) Find

4 3

1 ln 3] 4

3

1

[Answer : (a) (x + 1) 2 − 2(x + 1) 2 + c]

14. [STPM ] Using integration, find te area of the finite region bounded by the curve √ √ √ x+ y = a and the coordinate axes, if a is a positive constant.

[Answer :

1 2 a ] 6

15. [STPM ] Find the coordinates of the points P , Q where the curve 2y = x + 3 meets the parabola y 2 = 4x. Find the area bounded by the arc P Q of this parabola and the line P Q.

8 3

[Answer : P (1, 2) , Q(9, 6) ; ] 72


9: Integration

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kkleemaths.com

16. [STPM ] By using the substitution x = sin θ, or otherwise, evaluate 1 2

Z

1 3

x2

√

1 dx 1 − x2

leaving your answer in the forms of surds.

17. [STPM ]

(a) Express

√

√

[Answer : 2 2 − 3]

17 + x in partial fractions. Hence or otherwise, show that (4 − 3x)(1 + 2x) Z

1 2

− 13

17 + x 1 dx = (19 ln 2 + 9 ln 3). (4 − 3x)(1 + 2x) 6 Z

(b) Using the substitution x = sin θ, or otherwise, find

18. [STPM ]

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0

x2 1 dx. 2 (1 − x2 ) 12

3 1 5 + ; (b) [Answer : (a) 4 − 3x 1 + 2x 2

√ ! π 3 − ] 6 4

8 − 4 that intersects the x-axis at the point A. x The tangent to the curve at the point P (1, 4) intersects the x-axis at the point Q. Find the coordinates of Q. The diagram above shows a portion of the curve y =

Find the area of the shaded region bounded by the curve AP and the line segments P Q and QA.

19. [STPM ]

73

[Answer : Q =

3 , 0 ; A=8 ln 2 − 5] 2

9: Integration

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The diagram shows a hemispherical bowl with radius a and its rim horizontal. Show that when the depth of water in the bowl is y, the volume V of water in the bowl is given by 1 V = πy 2 (3a − y). 3

Initially, the bowl is empty. Water is then poured into the bowl at a constant rate. The time taken to 1 fill the bowl is T . Find, in terms of T , the time taken for y to become a. Find, in terms of T and a, 2 1 the rate at which the water level rises when y = a. 2 [Answer :

8a 5 T ; ] 16 9T

20. [STPM ] Determine the coordinates of the points of intersection between the graphs y 2 = x and y = −x + 2. Find the area bounded by the two graphs. Find also the volume of the solid generated when this area is rotated through 2π about the y-axis.

21. [STPM ]

(a) Using the substitution x = a tan θ, or otherwise, show that √ Z a x3 8−5 2 , 5 dx = 12a 0 (a2 + x2 ) 2

[Answer : (1, 1) , (4, −2) ;

9 72 ; π] 2 5

with a > 0. Z 8 (x − 1)2 (b) Evaluate , dx, giving your answer correct to three significant figures. x2 − 4 3

22. [STPM ]

[Answer : (b) 3.89]

(a) Find the values of the constants A and B such that cos x ≡ A(3 cos x + 4 sin x) + B(−3 sin x + Z π 2 cos x 4 cos x). Hence or otherwise, find the value of dx giving your answer correct 0 3 cos x + 4 sin x to three decimal places. 74


9: Integration Z

3

r

correct to three significant figures.

0

e2

(b)

e

2

5−x dx giving your answer x−1

[Answer : (a) A =

23. [STPM ] Evaluate Z π 6 tan 2θ dθ, (a) Z

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kkleemaths.com

(b) By using the substitution x = 3 − 2 cos 2θ, or otherwise, find

1 dx, x ln x

4 3 ,B= , 0.235; (b) 1.32] 25 25

[Answer : (a)

24. [STPM ] Find the area bounded by the curves y = 1 − x2 and y = x − 1.

25. [STPM ]

1 ln 2 ; (b) ln 2] 2

[Answer :

Z

(a) By using a suitable substitution or otherwise, find the value of answer correct to three significant figures.

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1

2

9 ] 2

x−1 dx, giving your (2x − 1)2

d (b) Show that (tan3 θ) = 3 tan4 θ + 3 sec2 θ − 3. Hence, determine the value of dx

Z

π 4

tan4 θ dθ.

0

[Answer : (a) 0.108 ; (b)

π 2 − ] 4 3

26. [STPM ] √ π Sketch the area R bounded by the y-axis, x-axis, line x = , and the curve y = 1 + sin x. Find the 2 volume of the solid formed when R is rotated through four right angles about the x-axis.

27. [STPM ] Show that

d (tan3 x) = 3 sec4 x − 3 sec2 x. dx Z

Hence, determine the value of 0

π 4

sec4 x dx.

28. [STPM ]

75

[Answer :

π (π + 2)] 2

[Answer :

4 ] 3

9: Integration

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kkleemaths.com


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1 The shaded area shown in the diagram above is bounded by the curve y = e−x + , the curve y = ex −1, 2 and the y-axis. Find the area of the shaded region. [Answer :

1 (3 ln 2 − 1)] 2

29. [STPM ] dy ln x , where x > 0, determine the set of values for x such that > 0 and the set of values for If y = x dx dy 1 x such that < 0. Hence, show that the maximum value of y is . dx e ln x x Sketch the curve y = , where x > 0. Hence sketch the curve y = , x > 0. Show that the area x ln x 1 ln x , the x-axis, and the line x = is equal to the area bounded by the bounded by the curve y = x e ln x curve y = , the x-axis, and the line x = e. x

30. [STPM ]

Z

Find the value of

1

√ x 1 + x dx.

0

31. [STPM ] Express

x2

[Answer : {x : 0 < x < e} ; {x : x > e}]

1 in the form of partial fractions. Hence, show that −1 Z 1 1 x−1 dx = ln + c, 2 x −1 2 x+1

where c is the constant.

Using integration by parts, show that Z Z 1 x 2x2 dx = + dx. x2 − 1 x2 − 1 (x2 − 1)2 Z Deduce the value of 2

4

[Answer :

x2 dx. Give your answer correct to three decimal places. (x2 − 1)2 76

√ 4 (1 + 2)] 15


9: Integration

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kkleemaths.com

[Answer :

1 1 − ; 0.347] 2(x − 1) 2(x + 1)

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kkleemaths.com

32. [STPM ] Calculate the volume of the solid generated when the area bounded by the curves y = x2 and y 2 = 8x is rotated completely about the x-axis. [Answer :

48 π] 5

33. [STPM ] Sketch the curve y = x(x + 1)(2 − x). Find the area bounded by the curve y = x(x + 1)(2 − x) and the x-axis.


Z

x ln x dx = 1

Z

Hence, find the value of

35. [STPM Z ] e

e

1

e

[Answer : 3

1 2 e +1 . 4

x(ln x)2 dx correct to three decimal places.

[Answer : 1.597]

(2x + 1) ln x dx. Give your answer in terms of e.

Find

1

1 ] 12

[Answer :

1 2 (e + 3)] 2

36. [STPM ] Sketch, on the same axes, the graphs of y 2 = x and y = 2 − x. Show the coordinates of the points of intersection between the graphs. Calculate the area bounded by y 2 = x and y = 2 − x.

If V1 is the volume of the solid formed when the area above the x-axis bounded by y 2 = x, y = 2 − x, and the x-axis is rotated completely about the y-axis, and V2 is the volume of the solid formed when the area under the x-axis bounded by y 2 = x, y = 2 − x, and the x-axis is rotated completely about the y-axis, find V1 : V2 .

37. [STPM ]

Z

By using a suitable substitution, find the value of

3

9 2

[Answer : Area= ; V1 : V2 = 4 : 23]

√ x 1 + x dx.

0

[Answer : 7

11 ] 15

38. [STPM ] Find the coordinates of all the points of intersection between the line y = 12 − 4x and the curve y = 12 − x3 . Show that there is a point of inflection on the curve y = 12 − x3 at x = 0. Sketch, in the same diagram, the line y = 12 − 4x and the curve y = 12 − x3 .

The area bounded by the line y = 12 − 4x and the curve y = 12 − x3 is rotated through four right angles about the x-axis. Calculate the volume of the solid of revolution formed. 77

kkleemaths.com


9: Integration

Find

√

x dx. 2x − 1

40. [STPM ] Find Z x √ (a) dx, 2 4 − x Z 3 (b) x2 e−x dx

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39. [STPM Z ]

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[Answer : (0, 12), (−2, 20), (2, 4) ; Volume=192π]

1

[Answer :

p

(2x − 1) 2 (x + 1) + C] 3

1 3

3

[Answer : (a) − 4 − x2 + C ; (b) − e−x + C]

41. [STPM ] The gradient of the tangent to a curve at the point (x, y) where x > 3, varies inversely with (x − 3), and the curve passes through the points (4, 0) and (6, ln 9). Show that the equation of the curve is y = 2 ln(x − 3). Sketch the curve.

Find the finite area bounded by the curve, the x-axis and the line x = 6.


43. [STPM ] Function f is defined as

(a) Sketch the graph of f . (b) Find the range of f .

Z 2

3

11 x2 dx = + ln 2. 3 (x − 1) 8

( x(x − π), 0 ≤ x < 2π, f (x) = 2 π sin(x − π), 2π ≤ x ≤ 3π.

[Answer : 2(3 ln 3 − 2)]

[6 marks]

[4 marks] [3 marks]

(c) Determine whether f is a one-to-one function. Give reasons for your answer.

[2 marks]

(d) Find the area of the region bounded by graph f and the x-axis.

[6 marks]

44. [STPM ]

If f (x) =

[Answer : (b) R={y : −π 2 ≤ y < 2π 2 } ; (d) π 3 + 2π 2 ]

x2 , find f 0 (x). Hence, evaluate 2x − 5 Z 2 1

x(x − 5) dx. (2x − 5)2

[5 marks]

78

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9: Integration 11 2x(x − 5) ;− ] (2x − 5)2 6

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45. [STPM ] Sketch the curve y = x(x − 3)(x + 2).

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kkleemaths.com

[Answer : f 0 (x) =

[2 marks]

If A1 and A2 respectively denote the area of the regions bounded by the curve and the x-axis above and below the x-axis, find A1 : A2 . [5 marks]

46. [STPM ]

[Answer :

64 ] 189

1 . [2 marks] x Calculate the volume of the solid of revolution when the region bounded by the above graphs is rotated through 360◦ about the y-axis. [5 marks] Sketch the graphs y = 4, y = 8x2 and y =


( 2 − |x − 1|, f (x) = x2 − 9x + 18,

(a) Sketch the graph of f . Z 6 f (x)dx. (b) Evaluate 0

48. [STPM ] A curve has equation y 2 = x2 (4 − x2 ).

[Answer :

x < 3, x ≥ 3.

1 π] 2

[5 marks]

[5 marks]

[Answer : (b) 8 ]

Show that for any point (x, y) lying on the curve, then −2 ≤ x ≤ 2 and −2 ≤ y ≤ 2.

[3 marks]

Sketch the curve.

[3 marks]

Calculate the area of the region bounded by this curve.

[4 marks]

Calculate the volume of the solid of revolution when the region bounded by this curve and y = x in the first quadrant is rotated through 360◦ about the x-axis. [5 marks]

49. [STPM ]

Z

Find the value of

4

x 1

0

(5x2 + 1) 2

.

79

√ 6 3 32 π] [Answer : Area= ; Volume= 3 5

[6 marks]

[Answer :

8 ] 5

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9: Integration

50. [STPM ]

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4 Sketch on the same coordinate system the curve y = , and y 2 = 4(x − 1). Find the coordinates of x the points of intersection of the two curves. [6 marks] 4 20 Show that the area of the region bounded by y = , y 2 = 4(x − 1) and y = 4 is − 4 ln 2 .[4 marks] x 3 Calculate the volume of the solid of revolution when this region is rotated through 360◦ about the y-axis. [5 marks] [Answer : Point of intersection=(2,2) ; Volume=

51. [STPM ]

Z

Evaluate the definite integral

π 2

x2 sin x dx.

0

52. [STPM ]

ln x Show that the curve y = has a stationary point at x local minimum point or a local maximum point. Sketch the curve.

296 π] 15

[6 marks]

[Answer : π − 2]

1 e, and determine whether this point is a e [6 marks] [3 marks]

1 ln x , the x-axis, and the line x = is Show that the area of the region bounded by the curve y = x e ln x , the x-axis, and the line x = e. [5 marks] equal to the area of the region bounded by the curve y = x 53. [STPM ]

By using suitable substitution, find

Z

3x − 1 √ dx. x+1

[5 marks]

3

1

[Answer : 2(x + 1) 2 − 8(x + 1) 2 + C]

54. [STPM ] Find the point of intersection of the curves y = −x2 + 3x and y = 2x3 − x2 − 5x. Sketch on the same coordinate system these two curves. [5 marks] Calculate the area of the region bounded by the curves y = −x2 + 3x and y = 2x3 − x2 − 5x. [6 marks]

55. [STPM ]

[Answer : Point of intersection=(0,0), (2,2), (-2,-10) ; Area=16 units2 .]

Z

Using the substitution u = 3 + 2 sin θ, evaluate

0

π 6

cos θ dθ. (3 + 2 sin θ)2

[5 marks]

[Answer :

1 ] 24

56. [STPM ] a The curve y = x(b − x), where a 6= 0, has a turning point at point (2, 1). Determine the values of a 2 and b. [4 marks] Calculate the area of the region bounded bt the x-axis and the curve.

[4 marks]

Calculate the volume of the solid formed by revolving the region about the x-axis.

[4 marks]

80

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9: Integration

Z

Show that

e

ln x dx = 1.

1

58. [STPM ]

1 8 32 ,b=4; ; π] 2 3 15

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57. [STPM ]

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[Answer : a =

[4 marks]

1 Sketch on the same coordinates axis y = x and the curve y 2 = x. Find the coordinate of the points 2 of intersection. [5 marks] 1 Find the area of region bounded by the line y = x and the curve y 2 = x. [4 marks] 2 Find the volume of the solid formed when the region is rotated through 2π radians about the y-axis. [4 marks]

[Answer : (0, 0) , (4, 2) ;

59. [STPM ] Sketch, on the same coordinate axes, the curves y = ex and y = 2 + 3e−x . Calculate the area of the region bounded by the y-axis and the curves.

60. [STPM ]

[2 marks]

[6 marks]

[Answer : 2 ln 3]

Ax + B C 2x + 1 in the form 2 + where A, B and C are constants. + 1)(2 − x) x +1 2−x Z 1 2x + 1 dx. Hence, evaluate 2 0 (x + 1)(2 − x) Express

(x2

61. [STPM ] Find Z 2 x +x+2 (a) dx, x2 + 2 Z x (b) dx. ex+1

62. [STPM ]

[Answer : (a) x +

[Answer :

[4 marks]

1 3 x + ; ln 2] +1 2−x 2

[4 marks]

1 1 x ln(x2 + 2) + C ; (b) − x+1 − x+1 + C] 2 e e

dy 3x − 5 = √ , where x > 0. If dx 2 x

the curve passes through the point (1, −4),

(b) sketch the curve,

x2

[3 marks]

[3 marks]

The gradient of the tangent to a curve at any point (x, y) is given by

(a) find the equation of the curve,

4 64 ; π] 3 15

[4 marks]

[2 marks]

(c) calculate the area of the region bounded by the curve and the x-axis.

[5 marks]

3

1

[Answer : (a) y = x 2 − 5x 2 ; (c) 81

20 √ 5] 3

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9: Integration

sec x(sec x + tan x)2 dx = 1 +

0

64. [STPM ]

Z

Show that

2

3

5 (x − 2)2 dx = + 4 ln 2 x 3

√

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π 4

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Z

Show that

2.

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63. [STPM ] [4 marks]

2 . 3

[4 marks]

65. [STPM ] Sketch, on the same coordinate axes, the curves y = 6 − ex and y = 5e−x , and find the coordinates of the points of intersection. [7 marks] Calculate the area of the region bounded by the curves.

[4 marks]

Calculate the volume of the solid formed when the region is rotated through 2π radians about the x-axis. [5 marks]

66. [STPM ]

Z

Using an appropriate substitution, evaluate

1

[Answer : (ln 5, 1) ; 6 ln 5 − 8 ; π(36 ln 5 − 48)]

1

x2 (1 − x) 3 dx.

[7 marks]

0

[Answer :

67. [STPM ] Given a curve y = x2 − 4 and straight line y = x − 2,

(a) sketch, on the same coordinates axes, the curve and the straight line, (b) determine the coordinate of their points of intersection,

[2 marks]

[2 marks]

(c) calculate the area of the region R bounded by the curve and the straight line, ◦

[4 marks]

(d) find the volume of the solid formed when R is rotated through 360 about the x-axis. [Answer : (b) (−1, 3) , (2, 0) ; (c)

68. [STPM ]

0

[5 marks]

9 108 ; (d) π] 2 5

Z

2e

ln x dx.

Given that f (x) = x ln x, where x > 0. Find f (x), and hence, determine the value of

69. [STPM ]

Use the substitution u = ln x, evaluate

70. [STPM ] 2 Differentiate ex with respect to x.

Z

1

e

(x + 1) ln x dx. x2

Hence, determine integers a, b and c for which Z 2 a 2 x3 ex dx = ec . b 1

27 ] 140

e

[6 marks]

[Answer : 2e ln(2)]

[6 marks]

[Answer :

3 2 − ] 2 e

[9 marks]

82

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9: Integration 2

71. [STPM ] Differentiate tan x with respect to x, and hence, show that π 3

Z 0

72. [STPM ]

π x sec2 xdx = √ − ln 2. 3

Using the substitution x = 4 sin2 u, evaluate

1r

Z

0


Z

e

1

Z

Hence, find the value of 1

e

x(ln x)2 dx.

[6 marks]

[Answer : sec2 x]

x dx. 4−x

x ln x dx =

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[Answer : 2xex , a = 3, b = 2, c = 4]

[6 marks]

[Answer :

1 2 e +1 . 4

√ 2 π − 3] 3

[4 marks]

[3 marks]

1 4

1 4

[Answer : − + e2 ]

74. [STPM ] Sketch, on the same axes, the curve y 2 = x and the straight line y = 2 − x, showing the coordinates of the points of intersection. [4 marks] (a) State whether the curve y 2 = x has a turning point. Justify your answer.

[2 marks]

(b) Calculate the area of the region bounded by the curve y 2 = x and the straight line y = 2 − x. [4 marks]

(c) Calculate the volume of the solid formed by revolving the region bounded by the curve y 2 = x and the straight line y = 2 − x completely about the y-axis. [5 marks]

75. [STPM ]

The equations of two curves are given by y = x2 − 1 and y =

[Answer : (b) 4.5 ; (c) 14.4π]

6 . x2

(a) Sketch the two curves on the same coordinate axes.

[3 marks]

(b) Find the coordinates of the points of intersection of the two curves.

[3 marks]

(c) Calculate the volume of the solid formed when the region bounded by the two curves and the line x = 1 is revolved completely about the y-axis. [6 marks] √

√

[Answer : (b) (− 3, 2), ( 3, 2) ; (c) 2π(−1 + 3 ln 3)] 83


9: Integration


Z

e

1

78. [STPM ]

Z

Find the value of

1

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76. [STPM ] Using the substitution u = x − 2, show that Z 4 1 π x−1 dx = ln 2 + . 2 2 8 2 x − 4x + 8

ln x 1 dx = x5 16

[8 marks]

5 1− 4 . e

[6 marks]

(1 + 2x) ln(1 + x)dx. 0

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[5 marks]

1 2

[Answer : 2 ln 2 − ]

79. [STPM ] The curve y = ln(4x) is shown in the diagram below.

The tangent to the curve at the point P passes through the origin O. e (a) Show that the coordinates of the point P is , 1 , and find the equation of the tangent to the 4 curve. [5 marks] (b) Calculate the area of the shaded region bounded by the curve, the tangent and the x-axis.[5 marks] (c) Calculate the volume of the solid formed when the shaded region is revolved completely about the y-axis. [5 marks]

80. [STPM ]

Show that

4 e

1 8

[Answer : (a) y = x ; (b) e −

d (cosec x) = − cosec x cot x. dx

1 1 ; (c) π(e2 − 3)] 4 96

[2 marks]

84

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9: Integration

x2

81. [STPM ] Evaluate the definite integrals Z

ln 2

(a)

0

Z

ex dx, 1 + ex

3

(b)

√

0

x dx. 1+x

82. [STPM ]

Z

1

Show that

0

2 x2 cos−1 x dx = . 9

83. [STPM ]

The equation of a curve is

dx √ . 4 + x2

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Z

x2 (y − 3)2 + = 1. 4 9

(a) Sketch the curve.

(b) Calculate the area of the region bounded by the curve.

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Using the substitution x = 2 tan θ, find

[6 marks]

[Answer : −

1 p 4 + x2 + C] 4x

[3 marks]

[6 marks]

[Answer : (a) ln

8 3 ; (b) ] 2 3

[6 marks]

[2 marks]

[9 marks]

(c) Calculate the volume of the solid formed when the region is revolved completely about the y-axis.

84. [STPM ]

Z

3

ln x dx = 3 ln 3 − 2.

Show that

1

√

Z

Hence, evaluate

0

2

x ln(1 + x2 ) dx.

85

[4 marks]

[Answer : (b) 6π ; (c) 16π]

[5 marks]

[4 marks]

[Answer :

3 ln 3 − 1] 2

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10: Differential Equations

1. [STPM ] Variables t and y, with t ≥ 0 and y > 0, are related by

1 dy = y(2 − y), dt 2

2et . 1 + et Show that y → 2 when t → ∞. Sketch the graph of y versus t. 6 9 Find the difference in the values of t when y changes from to . 5 5 with the condition y = 1 when t = 0. Show that y =

2. [STPM ]

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Differential Equations

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10


[8 marks] [4 marks]

[3 marks]

[Answer : 1.7917]

dy A curve passes through point (2, 0) such that its gradient at point (x, y) satisfies equation (x2 − dx dy 3) = 4x(3 + y). Show that the equation of the curve is y = 3(x2 − 2)(x2 − 4). [6 marks] dx Sketch the curve. [3 marks] Find the area of the region bounded by the curve and the x-axis.

3. [STPM ]

[6 marks]

[Answer :

(a) Variables t and v, with 0 < t < 2, is related by the differential equation t

dv = v 2 − v, dt

48 √ (6 2 − 4)] 5

with the condition v = 2 when t = 1. Find v in terms of t and sketch the graph of v versus t.

(b) Show that

[8 marks]

dy y √ = (1 + x2 ) − xy, 2 dx 1+x where y is a function of x. Hence, solve the differential equation d (1 + x ) dx 2

3 2

(1 + x2 )

dy − xy = x(1 + x2 ), dx

with the condition y = 1 when x = 0.

4. [STPM ]

(a) Solve the differential equation

(1 − x)(1 + x2 )

[7 marks]

[Answer : (a) v =

dy + (2 − x + x2 )y = (1 − x)2 , x < 1, dx

with the condition y = 3 when x = 0.

2 ; (b) y = 1 + x2 ] 2−t

[8 marks]

86

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5. [STPM ] Solve the differential equation


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dy = (x + 1)(y + 1), dx with x > 0 and y > −1, and y = 2 when x = 1. Give your answer in the form y = f (x). x

[Answer : y = 3xex−1 − 1]

6. [STPM ] In a biochemical process, enzyme A changes continuously to enzyme B. Throughout the process, the total amount of A and B is constant. At any time, the rate that B is produced is directly proportional to the product of the amount of A and the amount of B at that time. At the beginning of the process, the amount of A and the amount of B are a and b respectively. If x denotes the amount of B that has been produced at time t after the process has begun, form a differential equation relating x and t to describe the process. [2 marks] Show that the solution of the differential equation is x=

where k is a positive constant.

ab(1 − e−(a+b)kt ) , b + ae−(a+b)kt

[8 marks]

Sketch the graph of x against t. [There is a point of inflection on the graph.]

[2 marks]

7. [STPM ] The rate of change of water temperature is described by the differential equation dθ = −k(θ − θs ) dt

where θ is the water temperature at time t, θs is the surrounding temperature, and k is a positive constant. A boiling water at 100◦ C is left to cool in kitchen that has a surrounding temperature of 25◦ C. The 3 water takes 1 hour to decrease to the temperature of 75◦ C. Show that k = ln . [6 marks] 2 ◦ ◦ When the water reaches 50 C, the water is placed in a freezer at −10 C to be frozen to ice. Find the time required, from the moment the water is put in the freezer until it becomes ice at 0◦ C. [6 marks] [Answer : Time = 4 hours 25 minutes]

8. [STPM ] Find the particular solution for the differential equation

x−2 1 dy + y=− 2 . dx x(x − 1) x (x − 1)

that satisfies the boundary condition y =

3 when x = 2. 4

[8 marks]

[Answer : y =

2x − 1 ] x2

9. [STPM ] The rate of increase in the number of a species of fish in a lake is described by the differential equation dP = (a − b)P dt

where P is the number of fish at time t weeks, a is the rate of reproduction, and b is the mortality rate, with a and b as constants. 87

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(a) Assuming that P = P0 at time t = 0 and a > b, solve the differential equation and sketch its solution curve. [7 marks] (b) At a certain instant, there is an outbreak of an epidemic of a disease. The epidemic results r in no 1 more offspring of the fish being produced and the fish die at a rate directly proportional to . P There are 900 fish before the outbreak of the epidemic and only 400 fish are alive after 6 weeks. Determine the length of time from the outbreak of the epidemic until all the fish of that species die. [9 marks]

10. [STPM ] The variables t and x are connected by

[Answer : (a) P = P0 e(a−b)t ; (b) 18 weeks]

dx = 2t(x − 1), dt

where x 6= 1. Find x in terms of t if x = 2 when t = 1.

[5 marks] 2

[Answer : x = et

−1

+ 1]

11. [STPM ] A canal of width 2a has parallel straight banks and the water flows due north. The points A and B are on opposite banks and B is due east of A, with the point O as the midpoint of AB. The x-axis and y-axis are taken in the east and north directions respectively with O as the origin. The speed of the current in the canal, vc , is given by x2 vc = v0 1 − 2 , a where v0 is the speed of the current in the middle of the canal and x is the distance eastwards from the middle of the canal. A swimmer swims from A towards the east at speed vr relative to the current inthe canal. Taking y to denote the distance northwards travelled by the swimmer, show that x2 dy v0 [3 marks] 1− 2 . = dx vr a If the width of the canal is 12 m, the speed of the current in the middle of the canal is 10 m s−1 and the speed of the swimmer is 2 m s−1 relative to the current in the canal, (a) find the distance of the swimmer from O when he is at the middle of the canal and his distance from B when he reaches the east bank of the canal, [7 marks] (b) sketch the actual path taken by the swimmer.

12. [STPM ] v Using the substitution y = 2 , show that the differential equation x

may be reduced to

2y dy + y2 = − dx x dv v2 = − 2. dx x

[3 marks]

[Answer : (a) 20 , 40]

[3 marks]

Hence, find the general solution of the original differential equation. 88

[4 marks]

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10: Differential Equations 1 ] Ax2 − x

d 2 (ln tan x) = . dx sin 2x Hence, find the solution of the differential equation

Show that

for which y =

(sin 2x)

[2 marks]

dy = 2y(1 − y) dx

1 1 when x = π. Express y explicitly in terms of x in your answer. 3 4

14. [STPM ] Find the general solution of the differential equation x

dy − 3y = x3 . dx

Find the particular solution given that y has a minimum value when x = 1. Sketch the graph of this particular solution.

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13. [STPM ]

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[Answer : y =

[Answer : y =

[8 marks]

tan x ] 2 + tan x


1 3

[Answer : y = x3 ln x + Cx3 ; y = x3 ln x − x3 ]

15. [STPM ] Find the general solution of the differential equation x

dy = y 2 − y − 2. dx

[6 marks]

[Answer : y =

2 + Ax3 ] 1 − Ax3

16. [STPM ] A particle moves from rest along a horizontal straight line. At time t s, the displacement and velocity of the particle are x m and v ms−1 respectively and its acceleration, in ms−2 , is given by √ dv = sin(πt) − 3 cos(πt). dt

Express v and x in terms of t.

[7 marks]

Find the velocities of the particle when its acceleration is zero for the first and second times. Find also the distance traveled by the particle between the first and second times its acceleration is zero.[7 marks] [Answer : v =

i √ √ i 1h 1 h√ 1 − cos(πt) − 3 sin(πt) , x = 2 3 cos(πt) + πt − sin(πt) + 3 ; π π √ 1 1 4 3 1 2 3 t= ,v=− ;t= ,v= ; + 2 ] 3 π 3 π 3π π

89



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17. [STPM ] The variables x and y, where x > 0, satisfy the differential equation x2

dy = y 2 − xy. dx

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Using the substitution y = ux, show that the given differential equation may be reduced to x

du = u2 − 2u. dx

Hence, show that the general solution of the given differential equation may be expressed in the form 2x y= , where A is an arbitrary constant. [10 marks] 1 + Ax2 Find the equation of the solution curve which passes through the point (1,4) and sketch this solution curve. [4 marks] [Answer : y =

18. [STPM ] Show that the substitution u = x2 + y transforms the differential equation

into the differential equation

(1 − x)

dy + 2y + 2x = 0 dx

(1 − x)

du = −2u. dx

4x ] 2 − x2

[3 marks]

19. [STPM ] A 50 litre tank is initially filled with 10 litres of brine solution containing 20 kg of salt. Starting from time t = 0, distilled water is poured into the tank at a constant √ rate of 4 litres per minute. At the same time, the mixture leaves the tank at a constant rate of k litres per minute, where k > 0. The time taken for overflow to occur is 20 minutes. (a) Let Q be the amount of salt in the tank at time t minutes. Show that the rate of change of Q is given by √ Q k dQ √ . =− dt 10 + (4 − k)t Hence, express Q in terms of t.

[7 marks]

(b) Show that k = 4, and calculate the amount of salt in the tank at the instant overflow occurs. [6 marks]

(c) Sketch the graph of Q against t for 0 ≤ t ≤ 20.

[2 marks]

h

√

[Answer : (a) Q = C 10 + (4 − k)t

20. [STPM ] Find the particular solution of the differential equation ex

dy − y 2 (x + 1) = 0 dx

for which y = 1 when x = 0. Hence, express y in terms of x. 90

√

k+k i− 416−k

; (b) 4kg]

[7 marks]

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10: Differential Equations ex ] 2 + x − ex

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[Answer : y =

21. [STPM ] One of the rules at a training camp of 1000 occupants states that camp activities are to be suspended if 10 of the occupants are infected with a virus. A trainee infected with a flu virus enrolls in the camp causing an outbreak of flu. The rate of increase of the number of infected occupants x at t days is given by differential equation dx = kx(1000 − x), dt where k is a constant. Assume that the outbreak of flu begins at the time the infected trainee enrolls and no one leaves the camp during the outbreak, 1000e1000kt , [9 marks] 999 + e1000kt (b) Determine the value of k if it is found that, after one day, there are five infected occupants,[3 marks] (a) Show that x =

(c) Determine the number of days before the camp activities will be suspended.

22. [STPM ]

Using the substitution z =

may be reduced to

[Answer : (b) k = 0.0016134 or

1 , show that the differential equation y dy 2y − = y2 dx x

2z dz + = −1. dx x

1 ln 1000

[4 marks]

999 199

; (c) 3 days]

[2 marks]

Hence, find the particular solution y in terms of x for the differential equation given that y = 3 when x = 1. [6 marks] Sketch the graph y.

23. [STPM ] Find the general solution of the differential equation

2 ln x 1 dy = . x dx cos y

91

[3 marks]

[Answer : y =

3x2 ] 2 − x3

[5 marks]

1 2

[Answer : sin y = x2 ln x − x2 + c]



may be reduced to

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24. [STPM ] Using the substitution y = vx, show that the differential equation xy

dy − x2 − y 2 = 0 dx vx

dv = 1. dx

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[3 marks]

Hence, find the particular solution that satisfies y = 2 and x = 1.

[6 marks]

[Answer : y 2 = 2x2 (ln x + 2)]

25. [STPM ] The variables x and y, where x > 0 and y > 0, satisfy the differential equation y(y + x) dy = . dx x(y − x)

Show that the substitution y = ux transforms the given differential equation into the differential equation 2u du = . dx x(u − 1) [3 marks]

1 Hence, find the solution of the given differential equation for which y = 2 when x = . 2

[6 marks] y

[Answer : y = 4x + x ln xy or xy = e x −4 ]

26. [STPM ] Differentiate ye−x with respect to x. Hence, find the solution of the differential equation

for which y = 1 when x = 0.

dy − y = ex cos x dx

27. [STPM ] The variables x and y, where x > 0, satisfy the differential equation x2

dy = 2xy + y 2 . dx

[6 marks]

[Answer : e−x

dy ; ye−x = sin x + 1] dx

Using the substitution y = ux, show that the given differential equation can be transformed into x

du = u + u2 . dx

[3 marks]

Show that the general solution of the transformed differential equation can be expressed as u = where A is an arbitrary constant.

x , A−x

[7 marks]

Hence, find the particular solution of the given differential equation which satisfies the condition that y = 2 when x = 1. [3 marks] 92

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28. [STPM ] Find the solution of the differential equation

2xy dy − = ex (1 + x2 ) dx 1 + x2

given that y = 3 when x = 0.

29. [STPM ]

[8 marks]

[Answer : y = (ex + 2)(1 + x2 )]

x2 . x−1 Hence, find the particular solution of the differential equation Show that e

R

x−2 dx x(x−1)

=

x−2 1 dy + y=− 2 dx x(x − 1) x (x − 1)

which satisfies the boundary condition y =

3 when x = 2. 4

30. [STPM ] The variables x and y, where x, y > 0, are related by the differential equation

Using the substitution y =

2y dy + y2 = − . dx x

2x2 ] 3 − 2x

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[Answer : y =

[4 marks]

[4 marks]

[Answer : y =

2x − 1 ] x2

u , show that the differential equation may be reduced to x2 u2 du = − 2. dx x

[3 marks]

Solve this differential equation, and hence, find y, in terms of x, with the condition that y = 1 when x = 1. [6 marks]

31. [STPM ] Show that the substitution y = ux transform the differential equation y dy x = y − 2x cot dx x into the differential equation

x

du = −2 cot u. dx

[Answer : y =

1 ] x(2x − 1)

[3 marks]

Hence, find the solution of the given differential equation satisfying the condition y = 0 when x = 1. Give your answer in the form y = f (x). [5 marks] 93

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[Answer : y = x cos−1 x2 ] 32. [STPM ] The number of rabbits in a farm increases at a rate proportional to the number of rabbits at a certain time. The number of rabbits doubled to 10000 from the beginning of the year 1985 until the beginning of 1990. At that time, an outbreak of a certain disease occurred in the farm which caused the death of rabbits at the rate of 100 rabbits per month. No vaccine was found until the beginning of the year 1992. Find the remaining number of rabbits that survived just before the vaccine was found. [9 marks] [Answer : unable to solve]

33. [STPM ] Using the substitution u = ln y, show that the non-linear differential equation x

dy + (3x + 1)y ln y = ye−2x dx

can be transformed into the linear differential equation x

du + (3x + 1)u = e−2x . dx

[4 marks]

Solve this linear differential equation, and hence, find the solution of the original non-linear differential equation, given that y = 1 when x = 1. [9 marks] Find the limiting value of y as x → ∞.

[2 marks]

[Answer : y = e

34. [STPM ] The variables x and y, where x, y > 0 are related by the differential equation

Show that the substitution u =

and find u2 in terms of x.

xy

dy + y 2 = 3x4 . dx

e−2x x

−e

1−3x x

; 1]

y transforms the above differential equation into x2 du 1 − u2 =3 , x dx u

[9 marks]

Hence, find the particular solution of the original differential equation which satisfies the condition y = 2 when x = 1. [3 marks] [Answer : u2 = 1 −

A 3 ; y 2 = x4 + 2 ] x6 x

35. [STPM ] Using the substitution z = cos y, find the general solution of the differential equation dy 1 1 + cot y = 2 cosec y. dx x x

[7 marks]

94

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Show that the substitution u =

1 transforms the non-linear differential equation y

into the linear differential equation

y dy + = y 2 ln x dx x du u − = − ln x dx x

1 + Cx] 2x

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36. [STPM ]

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[Answer : cos y = −

[4 marks]

1 Solve this linear differential equation, and hence, obtain y in terms of x, given that y = when x = 1. 2

37. [STPM ] Solve the differential equation

38. [STPM ]

[8 marks]

[Answer : y =

2 dy − y = x2 e2x . dx x

2 ] 4x − x(ln x)2

[6 marks]

1 2

[Answer : y = x2 e2x + Cx2 ]

dy Find the solution of the differential equation x − y = 2 which satisfies the condition y = 0 when dx x = 1. [5 marks] 39. [STPM ] The rate of elimination of a certain drug from a bloodstream is k times the mass, x mg. of the drug still present at time t. The half-life of the drug in the bloodstream is 100 minutes. (a) A dose of x0 mg of the drug is injected directly into the bloodstream.

i. Write down a differential equation relating x and t, and solve this differential equation.[4 marks] ii. Determine the value of k. [2 marks]

(b) The drug is intravenously fed into the bloodstream at an infusion rate of r mg per minute such dx that = −kx + r. Assuming that x = 0 when t = 0. dt i. express x in terms of t. [6 marks] ii. estimate the infusion rate that results in a long-term amount of 50 mg of drug in the bloodstream. [3 marks] 40. [STPM ] By making the change of variable z = x − 3y, show that the differential equation x − 3y + 5 dy = dx x − 3y − 1

may be reduced to a differential equation with separable variables.

[3 marks]

Solve this differential equation, and hence, obtain an equation which defines the relation between x and y. [4 marks] 95

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41. [STPM ] Find the general solution of the differential equation 2x

1 dy + y = x 2 e−2x , x > 0. dx

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[Answer : x − y − 3 ln |x − 3y + 5| = A]

[5 marks]

1 Hence,find the particular solution of the differential equation which satisfies the condition y = e−2 2 when x = 1. [3 marks]

96

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11: Maclaurin Series

1. [STPM ] Given that y = ln(1 + cos x), where −π < x < π, show that d2 y + e−y = 0. dx2

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Maclaurin Series

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11


[4 marks]

6

Using Maclaurin theorem, find the expansion of y in ascending powers of x up to the term x .[7 marks] Hence,

Z

(b) taking x =

1

ln(1 + cos x) dx, correct to three decimal places,

(a) estimate the value of 0

π , estimate the value of ln 2 correct to three decimal places. 2 1 4

[Answer : ln 2 − x2 −

2. [STPM ]

1 1 Given ln y = sin−1 x, where − π < sin−1 x < π, show that 2 2 2 dy (a) (1 − x2 ) − y 2 = 0, dx (b) (1 − x2 )

d2 y dy −x − y = 0. 2 dx dx

−1

[2 marks] [2 marks]

1 4 1 6 x − x + . . . ; (a) 0.608 ; (b) 0.691] 96 1440

[2 marks]

[2 marks]

Hence, find the Maclaurin expansion of esin x in ascending powers of x up to the term in x5 . State the range of values of x for the expansion valid. [7 marks] Using a suitable value of x in the expansion, estimate the value of π correct to four significant figures. Find the percentage error of estimation if π = 3.142. [4 marks]

3. [STPM ] Using Maclaurin expansion, find

4. [STPM ]

1 2

1 3

[Answer : 1 + x + x2 + x3 +

5 4 1 5 x + x + . . . , {x : −1 < x < 1} ; 3.130 ; 0.382%] 24 6

1 − cos2 x . x→0 x(1 − e−x ) lim

[4 marks]

[Answer : 1]

(a) Find the expansion for ex cos x and ex sin x in ascending powers of x up to the term x3 .

(b) Using Maclaurin theorem, find the expansion for sec x in ascending powers of x up to the term x4 . [5 marks] Deduce the first three non zero terms in the expansion sec2 x and tan x in ascending powers of x. [4 marks]

97

STPM MATHEMATICS (T) 1 3

1 3

1 2

5. [STPM ]

−1

If y = sin

d2 y x, show that =x dx2

dy dx

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[Answer : (a) 1 + x − x3 + . . . , x + x2 + x3 ; (b) 1 + x2 +

3

d3 y and = dx3

dy dx

11: Maclaurin Series 5 4 2 1 2 x ; 1 + x2 + x4 , x + x3 + x5 ] 24 3 3 15

3 + 3x

2

dy dx

5 .

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[4 marks]

Using Maclaurin theorem, express sin−1 x as a series of ascending powers of x up to the term in x5 . State the range of values of x for the expansion valid. [7 marks] Hence,

(a) taking x = 0.5, find the approximation of π correct to two decimal places, x − sin−1 x (b) find lim . x→0 x − sin x

6. [STPM ]

Use the expansion of x7 .

1 6

[Answer : x + x3 +

[2 marks]

[2 marks]

3 5 x , {x : |x| < 1} ; (a) 3.14 ; (b) −1] 40

1 to express tan−1 x as a series of ascending powers of x up to the term in 1 + x2 [4 marks]

Hence find, in terms of π, the sum of the infinite series

1 1 1 − + − ... 2 3×3 5×3 7 × 33

[2 marks]

√ 3 1 3 1 5 1 7 π] [Answer : x − x + x − x ; 1 − 3 5 7 6

7. [STPM ] If y = (sin−1 x)2 , show that, for −1 < x < 1,

(1 − x2 )

Find the Maclaurin series for (sin

−1

2

d2 y dy −x − 2 = 0. 2 dx dx

[3 marks]

6

x) up to the term in x .

[6 marks]

1 3

[Answer : x2 + x4 +

8 6 x ] 45

8. [STPM ] √ Find the Maclaurin expansion of x cos x up to the term in x3 . State the range of values of x for which the expansion is valid. [7 marks]

9. [STPM ]

Using the result that tan

Hence, find lim

x→0

sin x

x

Z x= 0

1 dt, show that 1 + t2

tan−1 x = x −

where |x| < 1.

tan−1 x

−1

x3 x5 x7 + − + ... 3 5 7

.

1 4

[Answer : x − x3 ; {x : |x| < 1}]

[2 marks] [2 marks]

98

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10. [STPM ] Given that y = x − cos−1 x, where −1 < x < 1. d2 y (a) Show that =x dx2

dy −1 dx

3 .

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[Answer : 1]

[2 marks]

(b) Find the Maclaurin series for y in ascending powers of x up to the term in x5 . 1 2

[5 marks]

1 6

[Answer : − π + 2x + x3 +

3 5 x ] 40

11. [STPM ] Using Maclaurin’s theorem, obtain the expansion of ln(1 + cos x) up to the term in x4 . [6 marks] √ π sin x π3 Hence, find the expansion of up to the term in x3 , and show that 3 ≈ 2 − − . 1 + cos x 12 5184

If y = sin

d2 y x, show that =x dx2

1 4

[Answer : ln(2) − x2 −

12. [STPM ]

−1

[5 marks]

dy dx

3

d3 y and = dx3

dy dx

3 + 3x

2

dy dx

1 4 1 1 x + . . . ; x + x3 + . . .] 96 2 24

5 .

[5 marks]

Using Maclaurin’s theorem, express sin−1 x as a series of ascending powers of x up to the term in x5 . State the range of values of x for which the expansion is valid. [7 marks]

13. [STPM ] Using the Maclaurin series for ln(1 + x), evaluate

14. [STPM ] Given that y = sin−1 x. (a) Show that

x − ln(1 + x) x→0 x2 lim

(1 − x2 )

1 6

[Answer : x + x3 +

d2 y dy −x = 0. dx2 dx

3 5 x , {x : |x| < 1}] 40

[3 marks]

[Answer :

1 ] 2

[4 marks]

1 3 (b) Using Maclaurin’s theorem, show that the series expansion for sin−1 x is x + x3 + x5 + . . .. 6 40 State the range of values of x for which the expansion is valid. [8 marks] 1 (c) Using the series expansion in (b), where x = , estimate the value of π correct to three decimal 2 places. [3 marks] 99

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[Answer : (b) {x : |x| < 1} ; (c) 3.139] 15. [STPM ] Using the Maclaurin theorem, find the expansion of ln(1 + sin x) up to the terms in x3 . 2x − ln(1 + sin x)2 . x→0 2x2 − x4

Hence, evaluate lim

16. [STPM ] −1 If y = ecos x , show that (1 − x2 )

17. [STPM ] The function g is defined by

for all values of x.

[2 marks]

1 2

−1

x

up to and including the term in x4 . π

√

g(x) = e

sin

! 3 x . 2

(a) Show that g 00 (x) + g 0 (x) + g(x) = 0. Hence, show that the Maclaurin series for g(x) is √ √ √ 3 3 2 3 4 x− x + x − .... 2 4 48

π

1 2

π

1 3

π

5 π 4 e2x ] 24

[5 marks]

[5 marks]

(b) Use the Maclaurin scries obtained in (a) to g(x) i. find the expansion of ascending powers of x up to the term in x3 . 1 + 2x g(x) ii. find the value of lim . x→0 x

18. [STPM ] If y = (cos−1 x)2 , show that

1 2

[8 marks]

[Answer : y = e 2 − e 2 x + e 2 x2 − e 2 x3 +

− 12 x

1 6

[Answer : x − x2 + x3 ; ]

d2 y dy − y = 0, for − 1 < x < 1. −x 2 dx dx

Hence, find the Maclaurin series for ecos

[5 marks]

[3 marks] [2 marks]

√ √ √ 5 3 2 5 3 3 3 3 [Answer : (b)(i) x− x + x + . . . ; (ii) ] 2 4 2 2

2

(1 − x )

dy dx

2

= 4y.

Show that the Maclaurin’s series for (cos−1 x)2 is

π2 π − πx + x2 − x3 + . . . . 4 6

√

[3 marks]

[7 marks]

100

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[6 marks]

Hence, find the first three non-zero terms of the series expansion of e

−2x

tan x.

1 3

4 3

[2 marks]

7 3

[Answer : 1 − 2x + 2x2 − x3 + . . . ; tan x = x + x3 + . . . ; x − 2x2 + x3 + . . .]

20. [STPM ] State the Maclaurin series for cos 2x up to four non-zero terms. 2x2

cos 2x + x→0 6x4

Hence, find the value lim

−1

[1 marks]

.

[3 marks]

2 3

[Answer : 1 − 2x2 + x4 −

21. [STPM ] −1 Show that y = esin x − 1 satisfies the differential equation 2

(1 − x )

Deduce that

(1 − x2 )

and

(1 − x2 )

Find the Maclaurin series for e

sin−1

x

−1

esin x − 1 , (a) determine lim x→2 sin x Z (b) approximate the value of

0.1

(esin

0

dy dx

2

= (y + 1)2 .

d2 y dy −x =y+1 2 dx dx

d4 y d3 y d2 y − 5x − 5 = 0. dx4 dx3 dx2

−1

x

− 1)dx, correct to five decimal places.

(1 + x2 )

(1 + x2 )

1 2

dy =y dx

d2 y dy + (2x − 1) = 0. 2 dx dx −1

Hence, find the Maclaurin expansion of etan

x

[3 marks]

[2 marks]

1 3

[Answer : x + x2 + x3 +

22. Given that ln y = tan−1 x, show that

4 6 1 x + ... ; ] 45 9

[7 marks]

− 1 in ascending powers of x up to the term in x4 .

Hence,

and

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19. [STPM ] State the Maclaurin series for e−2x up to the term in x3 . [1 marks] dy If y = tan x, show that = 1 + y 2 . Obtain the Maclaurin series for tan x up to the term in x3 . dx

[3 marks]

5 4 x + . . . ; (a) 1 ; (b) 0.00518] 24

in ascending powers of x up to the term in x3 .

101

STPM MATHEMATICS (T) dy in terms of tan x, and hence show that dx

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23. Given that y = tan2 x, express

d2 y = 2 + 8y + 6y 2 . dx2


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By further differentiation, or otherwise, show that, if x is so small that x7 and higher powers of x can be neglected, 2 17 tan2 x = x2 + x4 + x6 . 3 45 √

24. Given that y = e

1+x

, show that

4(1 + x)

d2 y dy +2 = y. 2 dx dx

By further differentiation, or otherwise, show that Maclaurin series of y up to the term in x4 is 1 1 y = e 1 + x + x2 + kx4 , 2 48 where k is a constant to be determined.

1 1 25. If y = (cos x)x , where − π ≤ x ≤ π, prove that 2 2

dy = y(ln cos x − x tan x). dx

Express y as a series of ascending powers of x up to the term in x3 . 1 1 4 , correct to 3 decimal places. Hence, estimate the value of cos 4

102


(b) (1 − x2 )

d2 y dy −x − y = 0. 2 dx dx

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1 1 Given ln y = sin−1 x, where − π < sin−1 x < π, show that 2 2 2 dy (a) (1 − x2 ) − y 2 = 0, dx

−1


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[2 marks]

[2 marks]

Hence, find the Maclaurin expansion of esin x in ascending powers of x up to the term in x5 . State the range of values of x for the expansion valid. [7 marks] Using a suitable value of x in the expansion, estimate the value of π correct to four significant figures. Find the percentage error of estimation if π = 3.142. [4 marks]

103

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12: Numerical Methods

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Numerical Methods

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12


1. [STPM ] Use the Newton Raphson method with initial estimate, x0 = 0.5, find the root for equation 10x − 2 sin x = 5 correct to three decimal places. [5 marks] [Answer : 0.615]

2. [STPM ] 1 Draw, on the same axes, the graphs of y = e− 2 x and y = 4 − x2 . State the integer which is nearest to the positive root of the equation 1 x2 + e− 2 x = 4. [3 marks]

Find an approximation for this positive root by using the Newton-Raphson method until two successive iterations agree up to two decimal places; give your answer correct to two decimal places. [5 marks]

3. [STPM ]

Sketch, on the same coordinate axes, the graphs y = ex and y = (1 + x)ex − 2 = 0 has a root in the interval [0, 1].

[Answer : 2 ; 1.90]

2 . Show that the equation 1+x [7 marks]

Use the Newton-Raphson method with the initial estimate x0 = 0.5 to estimate the root correct to three decimal places. [6 marks] [Answer : 0.375]

4. [STPM ] Using the sketch graphs of y = x3 and x + y = 1, show that the equation x3 + x − 1 = 0 has only one real root and state the successive integers a and b such that the real root lies in the interval (a, b). Use the Newton-Raphson method to find the real root correct to three decimal places.

5. [STPM ]

[4 marks] [5 marks]

[Answer : 0.683]

1 Find the coordinate of the stationary point on the curve y = x2 + where x > 0; give the x-coordinate x and y-coordinate correct to three decimal places. Determine whether the stationary point is a minimum point or a maximum point. [5 marks] 1 1 The x-coordinate of the point of intersection of the curves y = x2 + and y = 2 , where x > 0, is x x p. Show that 0.5 < p < 1. Using the Newton-Raphson method to determine the value of p correct to three decimal places and, hence, find the point of intersection. [9 marks]

6. [STPM ]

[Answer : (0.794 , 1.890) , minimum ; p = 0.724 , (0.724 , 1.908)]

Z

Using trapezium rule, with five ordinates, evaluate

0

1p

4 − x2 dx.

[4 marks]

[Answer : 1.91] 104

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7. [STPM ] x + 4ex = 2 has a root in the interval [-1,0].

4 . Show that the equation 2−x

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Sketch, on the same coordinate axes, the graphs y = e−x and y =

[6 marks]

Estimate the root correct to three decimal places by using Newton-Raphson method with initial estimate x0 = −0.4. [5 marks] [Answer : -0.479]

8. [STPM ] Use the trapezium rule with subdivisions at x = 3 and x = 5 to obtain an approximation to Z 7 x3 dx, giving your answer correct to three places of decimals. [4 marks] 4 1 1+x By evaluating the integral exactly, show that the error of the approximation is about 4.1%. [4 marks]

9. [STPM ]

[XAnswer : 1.701]

Two iterations suggested to estimate a root of the equation x3 − 4x2 + 6 = 0 are xn+1 = 4 − 1 1 xn+1 = (x3n + 6) 2 . 2

(a) Show that the equation x3 − 4x2 + 6 = 0 has a root between 3 and 4.

6 and x2n

[3 marks]

(b) Using sketched graphs of y = x and y = f (x) on the same axes, show that, with initial approximation x0 = 3, one of the iterations converges to the root whereas the other does not. [6 marks]

(c) Use the iteration which converges to the root to obtain a sequence of iterations with x0 = 3, ending the process when the difference of two consecutive iterations is less than 0.05. [4 marks] (d) Determine whether the iteration used still converges to the root if the initial approximation is x0 = 4. [2 marks] 10. [STPM ] Show that the equation x3 − 15x2 + 300 = 0 has a root between 5 and 6.

[3 marks]

Given that x0 = 5 as an initial approximation, use the Newton-Raphson method to find the root correct to three decimal places. [5 marks] [XXAnswer : 5.671]

11. [STPM ] By sketching the graphs of y = 3ex and y = 2x − 8 on the same diagram, show that the equation 3ex + 2x − 8 = 0 has only one real root. [3 marks] Use the Newton-Raphson method, with the initial approximation x0 = 1, to find the root correct to three decimal places. [5 marks] [XAnswer : 0.768]

12. [STPM ] Use the trapezium rule with five ordinates to estimate, to three decimal places, the value of Z 1p 2 − x3 dx. 0

[5 marks]

105

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[XXAnswer : 1.310] 13. [STPM ] p The curve y = x x2 + 5 and the straight line y = 3x in the first quadrant is shown in the diagram below.

Using the trapezium rule with four intervals, determine the area of the region bounded curve and the straight line. [5 marks]

14. [STPM ]

[XAnswer : 0.683]

1 − 1 and y = −x3 − 2 on the same coordinate axes. Hence, show Sketch the graphs of y = 2−x that the equation x4 − 2x3 + x − 3 = 0 has two real roots. [7 marks]

Using the Newton-Raphson method with the initial approximation x0 = 2, find the positive real root of the equation x4 − 2x3 + x − 3 = 0, correct to four decimal places. [6 marks] State, with a reason, a situation in which the Newton-Raphson method fails.

[2 marks]

[XAnswer : 2.0977]

15. [STPM ] Show that the equation x4 − 2x3 − x + 1 = 0 has at least a real root in the interval [2, 3].

[2 marks]

2x3n − 1 is more likely to give a x3n − 1 convergent sequence of approximation to a root in the interval [2, 3]. Use your choice with x0 = 2.5 to determine the root corrects to three decimal places. [9 marks] Determine which of the iterations xn+1 = x4n − 2x3n + 1 and xn+1 =

[XAnswer : 2.118]

16. [STPM ] Using the Newton-Raphson method with the initial approximation x0 = 4, find the positive real root of the equation ex = 12.5x + 1, correct to three decimal places. [6 marks] [XAnswer : 3.909]

106

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13: Data Description

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Data Description

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13


1. [STPM ] The following is the systolic blood pressure, in mm Hg. of 10 patients in a hospital. 165 135 151 155 158 146 149 124 162 173

If a patient is selected randomly, find the probability his/her systolic blood pressure exceeds one standard deviation above or below the mean. [6 marks] [Answer :

3 ] 10

2. [STPM ] The following table shows the age, in years, of 121 participants in a conference relating to health. Age 18-24 25-31 32-38 39-45 46-52 53-59

Frequency 6 11 19 26 45 14

(a) Plot a histogram and a frequency polygon for the given grouped data. Give comments on the distribution of the age of the participants. [5 marks] (b) Find the estimates for the median and the semi-interquartile range for the participants ages. [8 marks]

(c) State whether the median is more suitable than the mean as the measure of base (central tendency) for the age distribution of the participants. Give reason(s) for your answer. [2 marks] [Answer : (a) negatively skewed ; (b) 45. , 6.8 ; (c) median]

3. [STPM ] The table below indicates the number of cars belonging to 30 houses in a housing area. Number of cars Frequency

0 2

1 15

2 10

3 2

4 1

(a) Find mode, median, and mean. [3 marks] (b) Determine, if the majority of the houses in the housing area have number of cars exceeding the mean. [2 marks] [Answer : (a) 1 , 1 , 1.5 ; (b) No]

4. [STPM ] The table below shows the number of audiences according to age, in years, that watch a horror film for a session at a mini theater. Age, x 15≤ x <18 18≤ x <21 21≤ x <27 27≤ x <33 33≤ x <39 39≤ x <45 107

Frequency 14 38 44 16 6 2

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(a) Plot a histogram for the data above.

[3 marks]

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(b) Plot a cumulative frequency curve for the data above. From the curve, estimate the median and the semi-interquartile range. [8 marks] (c) If there were 8 audiences between the ages of 15 to 17 that were mistakenly entered into the age group of 18 to 20 years, calculate the actual median. [4 marks] [Answer : (b) 21.8 , 2.6 ; (c) 22.09]

5. [STPM ] The table below shows the scores obtained by a group of students in a mathematics quiz. Score Number of students

If the median is 1,

(a) find the value of x, (b) find the mode and the mean score.

0 8

1 1

2 1

3 0

4 x

5 3

[1 marks]

[3 marks]

[Answer : (a) 4 ; (b) 0 , 2]

6. [STPM ] The number of personal computers sold in a week by a company for a period of 25 weeks is given as follows: 10 7

8 10

7 12

4 14

4 10

18 9

9 29

14 7

17 6

11 5

5 20

4 15

5

(a) Find the mean and the standard deviation of the number of personal computers sold in a week in the given period. [5 marks] (b) Find the percentage of the weeks that the weekly sales of the personal computers are within one standard deviation from the mean. [3 marks] (c) Construct a frequency distribution by grouping the number of personal computers sold per week according to class intervals 1-5, 6-10, 11-15, 16-20 and 21-30. Hence, plot a histogram corresponding to this grouped data and state the shape of the distribution. [7 marks] [Answer : (a) 10.4 , 5.88 ; (b) 72% ; (c) positively skewed]

7. [STPM ] The number of hand phones sold each day by a firm during a promotion period of 14 days is as follows: 28 27

23 35

31 23

26 33

33 28

23 27

Find the median and semi-quartile range for the above data.

24 31

[4 marks]

[Answer : 27.5 , 3.5]

8. [STPM ] Wage per hour, RMx, paid to temporary workers in the Production Department and Marketing Department of a factory are as follows. 108


8 ≤ x < 10 10 ≤ x < 12 12 ≤ x < 14 14 ≤ x < 16 16 ≤ x < 20


Number of temporary workers Production Department Marketing Department 15 8 18 31 35 58 25 32 12 6

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Wages

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(a) Plot two histograms for the above data on separate diagrams. Based on your histograms, state the spreads of the distributions of the temporary workers wages in the Production Department and Marketing Department of the factory. [6 marks] (b) Calculate the estimated median, mean, and standard deviation of the hourly wages of all the temporary workers in that factory. [9 marks] [Answer : (a) Positively skewed , symmetrical ; (b) 13.03 , 13.06 , 2.27]

9. [STPM ] Find the median and the semi-interquartile range for the following numbers: 550, 450, 400, 750, 650, 700, 850, 500. [4 marks] [Answer : 600 , 125]

10. [STPM ] The table below shows the number of women based on the number of their children according to result of a survey conducted on 50 000 married women in the country. Number of children 0 1 2 3 4 5 6

Number of women 5011 8698 11633 19418 4848 307 85

(a) Determine the percentage of women whose number of children are within the range of one standard deviation from the mean. [7 marks] (b) Let X represent the number of children of a woman that is chosen randomly. Tabulate the probability distribution of X with each probability given to three decimal places. [3 marks] (c) If five women are chosen randomly, find the probability that three of them have more than two children. [5 marks] [Answer : (a) 62.1% ; (c) 0.308]

11. [STPM ] The table below indicates the number of children of 1440 families in a town. Number of children Number of families

0 410

1 310

2 290

3 300

4 110

(a) Find mode, median, and mean number of children of the families. 109

5 20

[4 marks]

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[2 marks]

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(b) Find the percentage of the number of families that have number of children exceeding the mean.

[Answer : (a) 0 , 1.5 , 1.618 ; (b) 50%]

12. [STPM ] The table below shows the lengths of 200 fish of a certain type of tropical fish that are randomly selected in a study carried out at a fresh-water fish rearing centre. Length of fish (x mm) 0 < x ≤ 29 29 < x ≤ 34 34 < x ≤ 39 39 < x ≤ 44 44 < x ≤ 49 49 < x ≤ 54 54 < x ≤ 59 59 < x ≤ 64

Frequency 0 8 19 42 69 34 23 5

(a) Plot a histogram for the data above, and draw the frequency polygon on the histogram. State the shape of the distribution of the data above. [5 marks] (b) Calculate the approximate median length of the fish.

[3 marks]

(c) By using the coding method and taking 46.5 mm as the assumed mean, find the mean and the standard deviation of the length of the fish. [7 marks] [Answer : (a) negatively skewed ; (b) 46.25 ; (c) 46.28 , 6.71]

13. [STPM ] The data shown in the stemplot below are the marks in a statistics course obtained by a group of students at a local institution of higher learning. 3 4 5 6 7 8 9

1 0 1 1 0 2 1

4 2 2 3 2

3 2 3

7 6 4

9 8

Key: 9|1 means 91 marks

(a) Find the percentage of students who obtain less than 40 marks and the percentage of students who obtain at least 80 marks. [2 marks] (b) Find the mean and standard deviation of the students marks. (c) Find the median and semi-interquartile range of the students marks. (d) Construct a boxplot for the above data.

[5 marks]

[4 marks]

[4 marks]

[Answer : (a) 10% , 10% ; (b) 56.05 , 14.9 ; (c) 54 , 9.25]

110

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14. [STPM ] Explain, with graphical illustrations, the relationship between the mean, median, and mode in a negatively skewed frequency distribution and in a positively skewed frequency distribution. [3 marks] 15. [STPM ] The following table shows the lengths, in minutes, of 90 telephone calls made in a month by a trading company. Length (minutes) 0.0 − 2.9 3.0 − 5.9 6.0 − 8.9 9.0 − 11.9 12.0 − 14.9 15.0 − 20.9

Frequency 12 19 35 13 8 3

(a) Draw a histogram for the above grouped data.

[3 marks]

(b) Calculate the median and the semi-interquartile range of the lengths of telephone calls made by the trading company. [7 marks] [Answer : (b) 7.15 , 2.34]

16. [STPM ] The table below shows the number of defective electronic components per lot for 500 lots that have been tested. Numbers of defective components per lot Relative frequency

0 0.042

1

2

3

4

5

6 or more

0.054

0.392

0.318

0.148

0.014

0.032

(a) State the mode and the median number of defective electronic components per lot.

[2 marks]

(b) For the lots with defective components of more than 5, the mean number of defective components per lot is 6.4. Find the mean number of defective electronic components per lot for the given 500 lots. [2 marks] [Answer : (a) 2 , 3 ; (b) 2.7]

17. [STPM ] The number of teenagers, according to age, that patronize a recreation centre for a certain period of time is indicated in the following table. Age in Years 12 13 14 15 16 17 18 -

Number of teenagers 4 10 27 110 212 238 149

[ Age 12 - means age 12 and more but less than 13 years ]

111

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(a) Display the above data using histogram.

[3 marks]

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(b) Find the median and semi-interquartile range for the age of teenagers who patronize the recreation centre. Give your answer to the nearest months. [7 marks] [Answer : (b) 17 years 1 month , 10 months]

18. [STPM ] The number of hand phones that are sold in a week by 15 representatives in a town is as follows: 5

10

8

7

25

12

5

14

11

10

21

9

8

11

18

(a) Find the median, lower quartile, and upper quartile for this distribution.

[2 marks]

(b) Draw a box plot to represent the data.

[3 marks]

[Answer : (a) 10 , 8 , 14]

19. [STPM ] The frequency distribution of the final examination marks for statistics course at an institute of higher learning is as follows. Marks 10-29 30-39 40-49 50-59 60-69 70-79 80-99

Number of students 6 7 12 19 15 13 8

(a) Plot a histogram for the above data.

[3 marks]

(b) Plot a cumulative frequency curve. Hence, estimate the median, semi-quartile range, and the percentage of students who obtained 45 to 70 marks. [10 marks] [Answer : (b) 57.5 , 12.25 , 26.25%]

20. [STPM ] Show that, for the numbers x1 , x2 , x3 , . . . , xn with mean x ¯, X X (x − x ¯ )2 = x2 − n¯ x2 .

[2 marks]

The numbers 4, 6, 12, 5, 7, 9, 5, 11, p, q, where p < q, have mean x ¯ = 6.9 and Calculate the of p and q.

X

(x − x ¯)2 = 102.9. [6 marks]

[Answer : 1, 9]

21. [STPM ] The number of ships which anchor at a port every week for 26 particular weeks are as follows. 32 26

28 27

43 38

21 42

35 18

19 37

25 50

112

45 46

35 23

32 40

18 20

26 29

30 46

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(a) Display the data in a stemplot.

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(b) Find the median and interquartile range.

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[2 marks] [4 marks]

(c) Draw a boxplot to represent the data.

[3 marks]

(d) State the shape of the frequency distribution. Give a reason for your answer.

[2 marks]

[Answer : (b) 31 , 15 ; (d) positively skewed]

22. [STPM ] The time taken by 50 customers to browse through books in a bookshop is shown in the histogram below.

(a) State the modal class.

[1 marks]

(b) Calculate the mean time taken by the customers to browse through books in the bookshop.[2 marks] (c) If 25% of the customers take more than x minutes to browse through books in the bookshop, determine the value of x. [3 marks] [Answer : (a) 12-15 ; (b) 13.3 ; (c) 17.5]

23. [STPM ] The following data show the masses, in kg, of fish caught by 22 fishermen on a particular day. 23 69

48 22

51 42

25 46

39 23

37 52

(a) Display the above data in an ordered stemplot. (b) Find the mean and standard deviation.

(c) Find the median and interquartile range.

(d) Draw a boxplot to represent the above data.

41 41

38 40

37 59

20 68

88 59

[2 marks]

[5 marks]

[4 marks]

[3 marks]

(e) State whether the mean or the median is more suitable as a representative value of the above data. Justify your answer. [2 marks] [Answer : (b) 44 , 16.8 ; (c) 41 , 15 ; (e) median]

24. [STPM ] The stemplot below shows the driving experience (in thousand km) of 15 express bus drivers. 113

0 1 2 3 4 5 6

3 2 5 8 1

6 8 1

9 5 6

1

0


5

2

Key: 6k2 means 62 000 km

(a) Find the median, the first quartile and the third quartile.

[2 marks]

(b) Draw a boxplot to represent the data.

[3 marks]

[Answer : (a) 15000 , 10000, 26000]

25. [STPM ] The ages of 75 persons under the age of 50 years are shown in the table below. Age at last birthday (in years) Number of persons

0-9 3

10 - 14 6

15 - 19 15

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20 - 24 25

25 - 29 16

(a) Calculate the mode and the median.

30 - 49 10

[4 marks]

(b) Calculate the percentage of persons whose ages are between 18 and 35 years.

[6 marks]

[Answer : (a) 22.1 , 22.2 ; (b) 64.3%]

26. [STPM ] Overexposure to a certain metal dust at the workplace of a factory is detrimental to the health of its workers. The workplace is considered safe if the level of the metal dust is less than 198 µ g m−3 . The level of the metal dust at the workplace is recorded at a particular time of day for a period of 90 consecutive working days. The results are summarised in the table below. Metal dust level (µ g m−3 ) 170 - 174 175 - 179 180 - 184 185 - 189 190 - 194 195 - 199 200 - 204

Number of days 8 11 25 22 15 7 2

(a) State what the number 11 in the table means.

[1 marks]

(b) Calculate estimates of the mean and standard deviation of the levels of the metal dust. [5 marks] (c) Plot a cumulative frequency curve of the above data. Hence, estimate the median and the interquartile range. [7 marks] (d) Find the percentage of days for which the workplace is considered unsafe.

[3 marks]

[Answer : (b) 185 , 7.22 ; (c) 184.7 , 9.8 ; (d) 4.44%]

114



21

8

17

22

19

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27. [STPM ] Find the median and the interquartile range of the following numbers. 10

29

6

6

21

20

12

18

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25

[3 marks]

[Answer : 18.5 , 11]

28. [STPM ] An advertising firm conducted a survey on television viewing habits in urban and rural areas. The table below shows the number of hours per week spent watching television by 20 persons in urban areas and 18 persons in rural areas. Urban 35 36 34 45 39 40 40 31 43 36

35 34 29 26 47

38 30 40 35 35

Rural 16 48 24 50 48 34 34 40 8

45 47 40 40 25

31 32 42 44

(a) Construct a suitable stemplot for each of the above data set.

[3 marks]

(b) Comment on the skewness of the two distributions.

[2 marks]

(c) Calculate the mean and the standard deviation of the number of hours spent watching television for each area. [6 marks] (d) Compare the dispersion of the two distributions.

[1 marks]

[Answer : (c) 36.4 , 5.15 ; 36 , 11.4]

29. [STPM ] The times taken by 22 students to breakfast are shown in the following table.

Time (x minutes) Number of students

2≤x<5 1

5≤x<8 2

8 ≤ x < 11 4

11 ≤ x < 14 8

14 ≤ x < 17 5

17 ≤ x < 20 2

(a) Draw a histogram of the grouped data. Comment on the shape of frequency distribution.[4 marks] (b) Calculate estimates of the mean, median, and mode of the breakfast times. Use your calculations to justify your statement about the shape of the frequency distribution. [7 marks] [Answer : (b) 12.23 , 12.50 , 12.71]

30. [STPM ] The ages of patients visiting a clinic on a particular day are as follows. 18 19 21 19 21

18 20 51 20 62

20 8 20 19 19

115

30 19 19 64 20

26 21 20 19 22

28 43 22 18 19

18 19 20 19 23

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(a) Construct an ordered stemplot to display the above data.

[2 marks]

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(b) State a measure of central location that best describes the data. Give a reason for your answer. [2 marks]

31. [STPM ] The table below shows the duration, in seconds, taken by 100 workers to finish a task. Duration (x seconds) 0 < x ≤ 100 100 < x ≤ 200 200 < x ≤ 250 250 < x ≤ 300 300 < x ≤ 350 350 < x ≤ 400 400 < x ≤ 500

(a) Calculate an estimate of the mean.

Number of workers 2 10 20 26 24 10 8

[2 marks]

(b) Plot a histogram for the above data. Hence, estimate the mode.

[5 marks]

(c) Plot a relative cumulative frequency curve for the above data. Hence, determine the median and the percentage of workers who finish the task in more than 270 seconds. [7 marks] [Answer : (a) 284 ; (b) 287.5 ; (c) 285 , 58%]

32. [STPM ] The mean mark for a group of students taking a statistics test is 70.6. The mean marks for male and female students are 68.5 and 72.0 respectively. Find the ratio of the number of male to female students. [4 marks] [Answer : 2:3]

33. [STPM ] The masses (in thousands of kg) of solid waste collected from a town for 25 consecutive days are as follows: 41 48

53 33

44 46

55 55

48 49

57 50

50 52

(a) Construct a stemplot to represent the data. (b) Find the median and interquartile range.

(c) Calculate the mean and standard deviation. (d) Draw a boxplot to represent the data.

38 47

53 39

50 51

43 49

56 52

51

[2 marks]

[4 marks]

[5 marks]

[3 marks]

(e) Comment on the shape of the distribution and give a reason for your answer.

[2 marks]

[Answer : (b) 50 , 6 ; (c) 48.4 , 5.84 ]

34. [STPM ] The following table shows the age distribution of drivers who are at fault in accidents during a onemonth period in a country.

116

Age (years) 18-25 26-35 36-45 46-55 56-65 66-80

(a) Draw a histogram for the data.


Number of drivers 20 24 14 18 18 26

[3 marks]

(b) Estimate the mean and standard deviation of the ages of the drivers.

[5 marks]

[Answer : (b) 46.9 , 18.5]

35. [STPM ] The marks for 26 students in a test are as follows: 22 22

90 51

13 83

43 11

59 32

52 43

32 34

40 73

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58 81

68 65

76 62

(a) Construct a stemplot to represent the data.

53 38

37 45

[2 marks]

(b) Find the probability that a randomly selected student has a mark between 53 and 65, inclusively. (c) Determine the interquartile range.

(d) Draw a boxplot to represent the data.

[3 marks]

[4 marks]

[3 marks]

[Answer : (b) 5/26 ; (c) 31]

36. [STPM ] A sample of 100 fuses, nominally rated at 13 amperes, are tested by passing increasing electric current through them. The current at which they blow are recorded and the following cumulative frequency table is obtained. Currents (amperes) <10 <11 <12 <13 <14 <15 <16 <17

Cumulative frequency 0 8 30 63 88 97 99 100

Calculate the estimates of the mean, median and mode. Comment on the distribution.

[8 marks]

[Answer : 12.65 , 12.61 , 12.58 ; positively skewed]

37. [STPM ] The time taken by 128 workers to complete a task is summarised in the table below.

117

Time (minutes) 40 − 44 45 − 49 50 − 54 55 − 59 60 − 64 65 − 74


Number of works 6 14 23 42 30 13

Plot a cumulative frequency curve for the data.

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[3 marks]

Hence, estimate the number of workers who take more than 66 minutes to complete the task. [2 marks] [Answer : 10]

38. [STPM ] The distribution of the selling prices of 80 houses in a city is shown below. Selling price (RM ’000) 120− 150− 180− 210− 240− 270− 300− 330− 360−

Number of houses 8 23 17 18 8 4 1 1 0

The interval ‘120−’, for example, means RM 120 000 ≤ selling price < RM 150 000. (a) Construct a histogram for the data above.

[3 marks]

(b) State the most appropriate measure of location to describe the selling prices. Give a reason. [2 marks]

(c) Calculate the measure that you have selected in (b), and interpret your answer.

[4 marks]

[Answer : (b) median. Positively skewed. (c) 195882]

39. [STPM ] The mean and standard deviation of Physics marks for 25 school candidates and 5 private candidates are shown in the table below. Number of candidates Mean Standard deviation

School candidates 25 55 4

Private candidates 5 40 5

Calculate the overall mean and standard deviation of the Physics marks.

[5 marks]

[Answer : 52.5 ; 6.98]

40. [STPM ] The duration of telephone calls made by 175 individuals at a call center on a particular day is shown in the table below: 118

Duration (minutes) 1−7 8 − 14 15 − 21 22 − 28 29 − 35 36 − 42 43 − 49 50 − 56

Frequency 15 32 34 22 16 12 9 5

(a) Construct a histogram to display the data. (b)

i. Draw a frequency polygon on the histogram in (a). ii. Comment on the skewness of the distribution obtained.


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[2 marks]

[1 marks] [1 marks]

[Answer : (b) (ii) Positively skewed]

41. [STPM ] Forty soil samples are collected from a certain area and tested for their pH values. The pH values and the number of soil samples tested are given in the table below. pH value 4.0 − 4.5 4.5 − 5.0 5.0 − 5.5 5.5 − 6.0 6.0 − 6.5 6.5 − 7.0 7.0 − 7.5 7.5 − 8.0

Number of soil samples 1 3 5 9 11 7 3 1

(a) Construct the cumulative frequency table for this distribution, and plot the cumulative frequency curve. [3 marks] (b) Using the cumulative frequency curve, estimate the median and semi-interquartile range of the distribution. [4 marks] (c) The pH value of a sample is wrongly recorded as 5.8, while its actual value is 5.0. State whether the wrong pH value affects the median and semi-interquartile range. Justify your answers.[2 marks] (d) Another four soil samples are collected from the same area, and their pH values are found to be greater than 8.0. i. Out of the 44 samples, find the percentage of samples which have a pH value greater than 7.0. [2 marks] ii. Using the curve in (a), estimate the median of the 44 samples. State a reason why the same curve is used. [2 marks] [Answer : (b) 6.10, 0.5 ; (d)(i) 18.18% ; (ii) 6.20]

42. [STPM ] The prices of houses sold, x, in hundred thousand RM, in a municipality in the year 2007 is shown below.

119

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Price (hundred thousand RM) 2.5 < x ≤ 3.0 3.0 < x ≤ 3.4 3.5 < x ≤ 4.0 4.0 < x ≤ 4.5 4.5 < x ≤ 5.0 5.0 < x ≤ 5.5 5.5 < x ≤ 6.0 6.0 < x ≤ 6.5


Frequency 15 10 14 8 10 7 5 3

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(a) Plot a cumulative frequency curve for the above data. Hence, estimate the median and the quartiles. [5 marks] (b) Draw a boxplot to represent the above data. Comment on the prices of the houses sold. [4 marks] [Answer : (a) 3.90 hundred thousand , Q1 =3.15 hundred thousand , Q3 =4.85 hundred thousand]

43. [STPM ] The lifespan, (in hours), of 50 light bulbs are summarised in the table below. Lifespan (x hours) 1500 < x ≤ 1550 1550 < x ≤ 1575 1575 < x ≤ 1600 1600 < x ≤ 1625 1625 < x ≤ 1650 1650 < x ≤ 1675 1675 < x ≤ 1700 x > 1700

Number of bulbs 2 4 4 14 12 8 4 2

(a) Calculate the estimates of the median and mode, giving your answer to the nearest hour.[6 marks] (b) If the upper boundary of the last class interval is k hours and the mean is 1630 hours, determine the value of k. [3 marks] [Answer : (a) 1627 , 1621 ; (b) 1875]

44. [STPM ] Some summary statistics about the annual expenditure, in RM, of a sample of recreational golfers on golf activities are shown in the table below. Minimum 200.00

Maximum 15500.00

Mean 6285.67

Median 3500.00

(a) Comment on the shape of the distribution of expenditure. Give a reason for your answer.[2 marks] (b) State, with a reason, the appropriate measure of central tendency to describe the expenditure of the golfers. [2 marks] 45. [STPM ] The lengths (in minutes) of 200 telephone calls made by customers to a pizza customer call centre in a particular day is shown in the table below.

120

Length of call (minutes) 0.0 - 3.0 3.0 - 4.5 4.5 - 6.0 6.0 - 7.5 7.5 - 9.0 9.0 - 10.5 10.5 - 12.0 12.0 - 15.0


Number of calls 7 19 34 67 38 18 13 4

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(a) Construct a histogram to display the data. Comment on the shape of the distribution obtained. [4 marks]

(b) Use your histogram to estimate the mode and the median.

[5 marks]

(c) Calculate the mean of the length of calls made by the customers. (d) Determine the percentage of calls whose lengths are at least 10 minutes.

[2 marks]

[3 marks]

[Answer : (b) 6.80, 6.90 ; (c) 7.00875 ; (d) 11.5%]

46. [STPM ] X The mass x, in gram, of each of 250 cat fish is observed. It is found that (x − 300) = 1000 and X (x − 300)2 = 5000. Determine the mean and the standard deviation of the masses of the cat fish. [7 marks]

[Answer : 304, 2]

47. [STPM ] The frequency distribution table below shows the amount of rain x, in millimetres, for 60 days in a particular year. Amount of rain, x (mm) 5.6 ≤ x < 5.8 5.8 ≤ x < 6.0 6.0 ≤ x < 6.2 6.2 ≤ x < 6.4 6.4 ≤ x < 6.6 6.6 ≤ x < 6.8

(a) Calculate the mean of the amount of rain.

Number of days 2 7 16 21 12 2

[2 marks]

(b) Find the median, the first quartile and the third quartile of the amount of rain.

[4 marks]

(c) Using the information in (b), comment on the shape of the frequency distribution of the amount of rain. [2 marks] [Answer : (a) 6.23 ; (b) 6.25 , 6.08, 6.39]

48. [STPM ] The cumulative frequency distribution of monthly charges, to the nearest RM, of 200 randomly chosen postpaid subscribers to a particular telecommunication company is shown below.

121

Monthly charge ≤ 30 ≤ 50 ≤ 70 ≤ 90 ≤ 110 ≤ 150 ≤ 200 ≤ 250


Cumulative frequency 0 10 40 76 120 170 192 200

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(a) Construct a histogram to display the data and draw a frequency polygon in the histogram. (b) Calculate the mean of the monthly charges.

[3 marks]

(c) Determine the percentage of subscribers whose month charges are more than the mean. [4 marks] [Answer : (b) RM 108.15 ; (c) 42%]

49. [STPM ] Three groups of sales trainees of a company undergo different sales training programmes which are Programme A, Programme B and Programme C. The boxplots below represent the sales (RM ’000) earned by each group in the first month after the completion of the training programmes.

(a) Estimate the median of the three sales distributions. State the training programme which is most effective. [3 marks] (b) Which of the sales distributions has a mean closest to its median? Give a reason for your answer. (c) Which of the sales distributions has the largest dispersion? (d) Comment on the skewness of the three sales distributions.

[2 marks]

[1 marks]

[3 marks]

(e) State the effect on the mean and median sales when the outlier of the sales distribution for programme A is removed. [2 marks] 122

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(c) Program B ;(d) Skewed to the left, skewed to the right, symmetrical ; (e) Mean is larger, median almost unaffected]

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[Answer : (a) RM 77000 , RM 65000 , RM 65000, Program A ; (b) Program C, symmetrical;

50. [STPM ] The number of ships anchored at a port is recorded every week. The results for 26 particular weeks are as follows:

(a) (b) (c) (d)

32 26

28 27

43 38

21 42

35 18

19 37

25 50

45 46

35 23

32 40

18 20

26 29

30 46

Display the data in a stem-and-leat diagram. Find the median and interquartile range. Draw a box-and-whisker plot to represent the data. State the shape of the frequency distribution, giving a reason for your answer.

[2 marks]

[4 marks]

[3 marks]

[2 marks]

[Answer : (b) 31 , 15]

51. [STPM ] The lengths (in seconds) of 60 songs recorded by a certain group of singers are summarised in the table below. Song length, x 0 < x ≤ 120 120 < x ≤ 180 180 < x ≤ 240 240 < x ≤ 300 300 < x ≤ 360 360 < x ≤ 600

Number of songs 1 9 15 17 13 5

(a) Display the data on a histogram. (b) Calculate the mean song length to the nearest half second. (c) Calculate the standard deviation of the song lengths to the nearest half second.

[3 marks]

[2 marks]

[3 marks]

[Answer : (b) 264.0 ; (c) 90.5]

52. [STPM ] A set of data has a mean of 6.1, a median of 5.4 and a standard deviation of 1.2. Calculate the Pearson coefficient of skewness, and comment on your answer. [3 marks] [Answer : 1.75]

53. [STPM ] The stem-and-leaf diagram below shows the average intake of fat weekly, in grams, of 60 adult volunteers in a medical research. Stem 8 9 10 11 12 13 14

| | | | | | |

Leaf 1 2 2 3 0 2 0 0 2 3 3 3 0 1

5 4 2 1 3 6 7

5 5 3 2 4 6

7 6 3 2 4 7

8 6 4 3 5

8 7 5 4 6

9 5 5 6

6 5 7

7 5

8 5

9 7

8

8

9

9

Key: 10|0 means 100 123

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(a) Determine the median and the interquartile range of the distribution.

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(b) Draw a box-and-whisker plot to represent the data.

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[4 marks] [3 marks]

(c) The above circled observation was recorded wrongly from its actual value of 173. Show how this actual value affect your box-and-whisker plot? [3 marks] [Answer : (a) 111.5 , 23.5]

54. [STPM ] The numbers of letters received by a company for 24 days are shown below. 5 23

13 10

7 6

14 4

7 4

8 6

20 8

12 7

7 4

4 16

10 4

7 10

(a) Determine the median and the mean of the number of letter received.

[3 marks]

(b) Calculate Pearson’s coefficient of skewness.

[4 marks]

(c) State whether the median or the mean is a more appropriate measure of central tendency. Give a reason. [2 marks] [Answer : (a) 7 , 9 ; (b) 1.2 ; (c) median]

55. [STPM ] The number of graduates employed within six months of graduation against the duration of industrial training are summarised as follows: Durations (weeks) 1−4 5−8 9 − 12 13 − 16 17 − 20 21 − 24

Number of graduates employed 2 7 19 16 15 16

(a) Calculate the mean and median of the duration of training of the graduates.

[5 marks]

(b) Determine the Pearsons coefficient of skewness, and interpret your answer.

[6 marks]

[Answer : (a) 14.9 , 14.875]

56. [STPM ] In a survey, each respondent is required to rank the hygienic status of the school canteen based on five aspects: food, drink, dishware, kitchen utensils and surrounding. The ranking score ranges from 1 to 5, where 1 implies extremely unhygienic and 5 implies extremely hygienic. The average ranking score of each respondent for teachers and students are summarised by the box-and-whisker plots below.

124


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Comment on the measure of central tendency, measure of dispersion and the skewness of the average ranking scores of the teachers and the students. [3 marks] [XXAnswer : The median of two groups are the same. The interquartile range or semi-interquartile range of the average ranking score of the students is larger than teachers. Both are positively skewed.] 57. [STPM ] A man has to take a train from station X to his office on every working day. The train is scheduled to arrive at station X at 0830 hours every day. The actual time of arrival of the train is recorded over a period of 90 working days and the results are shown in the following table. Time of arrival (t) 0824 < t ≤ 0826 0826 < t ≤ 0828 0828 < t ≤ 0830 0830 < t ≤ 0832 0832 < t ≤ 0834 0834 < t ≤ 0836

Number of days 2 10 15 33 23 7

(a) If the man reaches the station at exactly 0825 hours every morning, calculate the average minutes, he has to wait before the train arrives for the 90 working days. [2 marks] (b) Construct a cumulative frequency table for the time of arrival of the train. (c) Calculate the median and the mode for the time of arrival of the train. (d) Calculate the semi-interquartile range for the lime of arrival of the train. (e) Determine the percentage of the days that the train is late.

[2 marks]

[4 marks]

[5 marks]

[2 marks]

[XXAnswer : (a) 5.91 ; (c) 0831.09, 0831.29 ; (d) 1.625 minutes (e) 70%]

58. [STPM ] The height of a group of 118 participants who take part in the National Service Training Programme are summarised in the frequency table below.

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Height (cm) 158 − 162 163 − 167 168 − 172 173 − 177 178 − 182 183 − 187


Frequency 6 9 17 33 45 8

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Given that the approximate values of the mean and the standard deviation for the heights of participants are 175.34 cm and 6.23 cm respectively. (a) Calculate the median for the heights of the participants.

[5 marks]

(b) Find the Pearson coefficient of skewness and comment on the shape of this distribution. [3 marks] [XAnswer : (a) 176.59 ; (b) −0.602]

59. [STPM ] The distances travelled (x km) by cars brought in for a 20 000 kilometre service centre in a particular week are summarised in the table below. Distance (km) 19000 ≤ x < 20000 20000 ≤ x < 21000 21000 ≤ x < 22000 x ≥ 22000

Number of cars 4 22 18 6

(a) Calculate, to the nearest km, the median and semi-interquartile range of the distance travelled. [6 marks]

(b) Give a reason why the median and semi-interquartile range might be more appropriate summary measures for these data than that of the mean and standard deviation. [1 marks] [XXAnswer : (a) median=20955 km, semi-interquartile range=626 km]

60. [STPM ] The student affairs officer of a college keeps a record on the number of hours spent on part-time jobs in a semester amongst students. The number of hours spent on part-time jobs for a sample of 20 students are displayed in the stem-and-leaf diagram below. 2 3 4 5 6 7 8

3 4 0 1 1 8 9

8 4 3 2 9

5 4 4

7 5 5

6

8

Key: 5|3 means 53 hours

(a) Find the mean and standard deviation of the distribution.

[4 marks]

(b) Determine the quartiles and interquartile range of the distribution.

[4 marks]

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(c) Determine whether there is any outlier.

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(d) Construct a box-and-whisker plot to represent the data.

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[2 marks] [3 marks]

(e) State a measure that best describes the central tendency, and a measure that best describes the dispersion of the data. Give a reason for your answer. [2 marks] [XXAnswer : (a) 48.30, 16.01 ; (b) 36, 54.50, 18.50 ; (c) Yes. There is 1 outlier ; (e) Median, interquartile range]

61. [STPM ] The time, to the nearest minute, taken by 200 participants to complete a cross-country are recorded and summarised in the table below. Time (minutes) 50 to less than 55 55 to less than 60 60 to less than 65 65 to less than 70 70 to less than 75 75 to less than 80 80 to less than 90

Number of participants 10 15 35 56 45 27 12

(a) Construct a cumulative frequency table for the distribution, and plot a cumulative frequency curve. [4 marks] (b) Use your cumulative frequency curve to estimate the median and interquartile range.

[Answer : 68.5 , 10]

62. [STPM ] The numbers of trees planted in 21 housing estates are as follows: 108 75 64

42 140 44

81 137 47

53 76 72

70 45 83

[5 marks]

53 41 95

44 77 40

(a) Find the median and the interquartile range of the numbers of trees planted.

[4 marks]

(b) Determine the outlier(s), it any.

[2 marks]

(c) Draw, on the graph paper provided, a box-and-whisker plot to represent the data.

[3 marks]

[Answer : (a) 70 , 36 ; (b)137 , 140]

63. [STPM ] The pH values of water in a river at 10 different locations are measured by using a pH meter. The results obtained are shown below. 6.95

7.08

6.98

6.98

6.94

6.75

6.98

6.95

6.86

6.96

Calculate the Pearson coefficient of skewness and comment on the shape of the distribution of the pH values. [6 marks] [Answer : −0.45] 127

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4.9 2.0

7.0 6.2

6.6 4.1

7.2 6.0

6.9 7.0

6.3 4.0

4.4 2.9

6.7 2.8

(a) Calculate the mean and the standard deviation of the profit earned.

7.0 5.5

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3.1 6.8

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64. [STPM ] Profits, in RM million, earned by 20 trading company in a particular year are given below.

[5 marks]

(b) Determine the Pearson coefficient of skewness, and sketch the shape of the distribution. [4 marks] [Answer : (a) 5.37 , 1.656 ; (b) −1.322]

65. [STPM ] The times, in minutes, taken to commute from home to work place for 25 workers of a company are as follows: 40 26 38

50 32 45

65 47 60

33 7 56

48 55 44

5 59

51 69

43 43

39 51

66 22

(a) Construct a stem-and-leaf diagram for this data. Comment on your diagram.

[3 marks]

(b) State, with a reason, the appropriate measure of central tendency and the measure of dispersion to summarise the data. Determine their values. [6 marks]

128

[Answer : ]

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14: Probability

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1. [STPM ]

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Probability

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14


2 1 1 Events A and B are such that P(A) = and P(B) = and P[(A ∩ B 0 ) ∪ (A0 ∩ B)] = . Find P(A ∩ B) 5 4 6 and determine if events A and B are independent. [5 marks] [Answer :

29 ] 120

2. [STPM ] A factory producing electronic equipments obtains 40% of the electronic components from supplier A, 25% from supplier B, and 35% from supplier C. The percentages of defective electronic components supplied by supplier A, B and C are respectively 5%, 2% and 1%. (a) Find the probability that an electronic component selected randomly from overall supply to the factory, is defective. [6 marks] (b) Find the probability, from two electronic components selected randomly from the overall supply as received by the factory, i. at least one electronic component is defective. ii. both the electronic components are defective and are supplied by supplier A.

[5 marks] [4 marks]

[Answer : (a) 0.0285 ; (b) (i) 0.0562 , (ii) 0.0004]

3. [STPM ] A factory producing bulbs uses three types of machines A, B and C which respectively contribute 25%, 35% and 40% of the bulbs produced by the factory. 5% of the bulbs produced by machine A, 4% by machine B, and 2% by machine C are found to be defective. Find the probability that a randomly selected bulb, produced by the factory is defective. [5 marks] [Answer : 0.0345]

4. [STPM ] A and B are two events such that P(A) = 0.5, P(B) = 0.6, and P(A ∪ B) = 0.85. Find P(A ∩ B), P(A ∩ B 0 ), P(A0 ∩ B) and P(A0 ∪ B 0 ). [6 marks] [Answer : 0.25 , 0.25 , 0.35 , 0.75]

5. [STPM ] As many as 35 out of 40 light bulbs in a box are in good conditions. Two light bulbs are randomly selected, one after another, from the box. Find the probability that only one of the two bulbs selected is in good conditions (a) if the first selected bulb is replaced in the box.

(b) if the first selected bulb is not replaced in the box.

[2 marks]

[2 marks]

[Answer : (a)

35 7 ; (b) ] 32 156

6. [STPM ] The probability that Mimi taking tuition for Statistics is 0.3. If Mimi takes tuition, the probability that she will pass the Statistics paper is 0.8. If Mimi does not take tuition, the probability that she will pass the Statistics paper is 0.6. 129

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14: Probability

(a) Find the probability that Mimi passes the Statistics paper.

[3 marks]

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(b) Find the probability that Mimi takes tuition if she passes her Statistics paper.

[3 marks]

[Answer : (a) 0.66 ; (b) 0.3636]

7. [STPM ] One in a thousand foreign workers is known to have a certain disease. Result from a routine screening of a foreign worker may be positive or negative. A positive result suggests that the worker has the disease, but the test is not perfect. If a foreign worker has the disease, the probability of a negative result is 0.02. If a foreign worker does not have the disease, the probability of a positive result is 0.01. If the result of a test on a foreign worker is positive, find the probability that this worker has the disease. Comment on the answer you obtain. [6 marks] [Answer : 0.0893]

8. [STPM ] Among 100 students in a school, 40 like lemons, 62 like mangosteens, 56 like nutmegs, 18 like lemons and nutmegs, 15 like lemons and mangosteens, 10 like all the three fruits, and 11 do not like any of the three fruits. Find the probability that (a) a student chosen at random likes only lemons,

[3 marks]

(b) a student chosen at random likes mangosteens and nutmegs but does not like lemons.

9. [STPM ]

If A and B are two events with P(A) =

[Answer : (a)

[4 marks]

17 ; (b) 0.36] 100

4 2 1 , P(B) = , P[(A ∩ B 0 ) ∪ (A0 ∩ B)] = , find P(A ∩ B). 7 5 3

[3 marks]

[Answer :

67 ] 210

10. [STPM ] A construction contractor orders sand, cement, and bricks from three different suppliers. The probabilities that the sand, cement, and bricks are sent to the project site on or before the agreed date are 0.80, 0.90, and 0.95 respectively. Find the probability that (a) all supplies delivered to the project site are not late,

(b) at most one kind of supply delivered to the project site is late.

[2 marks]

[3 marks]

[Answer : (a) 0.684 ; (b) 0.967]

11. [STPM ] Three balls are selected at random from one blue ball, three red balls and six white balls. Find the probability that all the three balls selected are of the same color. [3 marks]

130

[Answer :

7 ] 40

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14: Probability

A: A female student of the university is selected. B: A student of the university, who studies the business program is selected.

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12. [STPM ] In a university, 48% of the students are females and 17.5% of the students are taking business programs. 4.7% of the university students are female students who study business programs. A student is selected randomly. Events A and B are defined as follows:

(a) Determine whether A and B are mutually exclusive and whether A and B are independent. (b) Find P(A|B).

[3 marks] [2 marks]

[Answer : (a) Not mutually exclusive. Not independent. (b) 0.269]

13. [STPM ] A factory has 36 male workers and 64 female workers, with 10 male workers earning less than RM1000.00 a month and 17 female workers earning at least RM1000.00 a month. At the end of the year, workers earning less than RM1000.00 are given a bonus of RM1000.00 whereas the others receive a month’s salary. (a) If two workers are randomly chosen, find the probability that exactly one worker receives a bonus of one month’s salary. [3 marks] (b) If a male worker and a female worker are randomly chosen, find the probability that exactly one worker receives a bonus of one month’s salary. [3 marks] [Answer : (a) 0.495 ; (b) 0.604]

14. [STPM ] The following table, based on a survey, shows the numbers of male and female viewers who prefer either documentary or drama programmes on television.

Male Female

Documentary 96 45

Drama 24 85

A television viewer involved in the survey is selected at random. A is the event that a female viewer is selected, and B is the event that a viewer prefers documentary programmes. (a) Find P(A ∩ B) and P(A ∪ B).

[4 marks]

(b) Determine whether A and B are independent and whether A and B are mutually exclusive. [3 marks]

[Answer : (a) 0.180 , 0.904 ; (b) Not mutually exclusive. Not independent.]

15. [STPM ] Three companies X, Y , and Z offer taxi services in a town. The percentages of residents in the town using the taxi services from companies X, Y and Z are 40%, 50%, and 10% respectively. The probabilities of taxis from companies X, Y , and Z being late are 0.09, 0.06, and 0.20 respectively. A taxi is booked at random. Find the probability that (a) the taxi is from company X and is not late, 131

[4 marks]

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14: Probability [6 marks]

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(b) the taxi is from company Y given that it is late.

[Answer : (a) 0.364 ; (b) 0.349]

16. [STPM ] Two transistors are chosen at random from a batch of transistors containing ninety good and ten defective ones. (a) Find the probability that at least one out of the two transistors chosen is defective.

[3 marks]

(b) If at least one out of the two transistors chosen is defective, find the probability that both transistors are defective. [4 marks] [Answer : (a) 0.191 ; (b) 0.0476]

17. [STPM ] According to a survey conducted in a company on job satisfaction, salary and pension benefits are two important issues. It is found that 74% of the employees are of the opinion that salary is important whereas 65% think that pension benefits are important. Among those who think that pension benefits are important, 60% think that salary is also important. Determine the percentage of employees who are of the opinion that salary and pension benefits are important. [3 marks] [Answer : 36%]

18. [STPM ] The letters in the word BANANA are to be rearranged. A word can be considered formed without being meaningful. The events R, S and T are defined as follows. R: The word starts and ends with an A.

S: All the N’s in the word are kept together.

T : All the A’s in the word are kept together.

(a) Find P(R), P(S) and P(T ).

[5 marks]

(b) Find P(R ∩ S), P(R ∪ S), P(R ∩ T ) and P(R ∪ T ).

[5 marks]

[Answer : (a)

1 13 2 1 1 1 , , ; (b) , ,0, ] 5 3 5 10 30 5

19. [STPM ] A four-digit number, in the range 0000 to 9999 inclusive, is formed. Find the probability that (a) the number begins or ends with 0,

(b) the number contains exactly two non-zero, digits.

[3 marks]

[3 marks]

[Answer : (a) 0.19 ; (b) 0.0486]

20. [STPM ] Two archers A and B take turns to shot, with archer A taking the first shot. The probabilities of A 1 1 and B hitting the bull’s eye in each shot are and respectively. Show that the probability of archer 6 5 1 A hitting the bull-eye first is . [4 marks] 2 132

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21. [STPM ]

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1 The probability that it rains in a certain area is . The probability that an accident occurs at a 5 1 1 if it rains and if it does not rain. Find the probability particular corner of a road in that area is 20 50 that it rains if an accident occurs at the corner. [4 marks] [Answer :

22. [STPM ] A teacher, 3 male students and 2 female students line up for a photograph.

5 ] 13

(a) Find the number of different arrangements if the teacher stands at the end of the line. [2 marks] (b) Find the number of different arrangements if all the male students stand together.

[2 marks]

[Answer : (a) 240 ; (b) 144]

23. [STPM ] The amounts of purchase and the modes of payment of 300 customers of a supermarket are shown in the following table. Amount of purchase Less than RM50 RM50 or more

Mode of payment Cash Credit card 50 25 75 150

A customer is selected at random from this group of customers. (a) Find the probability that the payment is made by cash.

[1 marks]

(b) Find the probability that the amount of purchase is less than RM50 and the payment is made by cash. [1 marks] (c) If the amount of purchase is at least RM50, find the probability that the payment is made by cash. [1 marks] (d) Find the probability that the amount of purchase is less than RM50 given that the payment is made by cash. [1 marks] (e) State, with a reason, whether the events “the amount of purchase is less than RM50” and “the payment is made by cash” are mutually exclusive. [2 marks] (f) State, with a reason, whether the events “the amount of purchase is less than RM50” and “the payment is made by cash” are independent. [2 marks] [Answer : (a) 0.417 ; (b)

1 1 ; (c) ; (d) 0.4] 6 3

24. [STPM ] There are eight parking bays in a row at a taxi stand. If one blue taxi, two red taxis and five yellow taxis are parked there, find the probability that two red taxis are parked next to each other. [3 marks] [Assume that a taxi may be parked at any of the parking bays.]

[Answer : 133

1 ] 4

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14: Probability

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3 1 1 Two events A and B are such that P (A) = , P (B) = and P (A|B) = . 8 4 6

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25. [STPM ]

(a) Show that the events A and B are neither independent nor mutually exclusive.

[2 marks]

(b) Find the probability that at least one of the events A and B occurs.

[3 marks]

(c) Find the probability that either one of the events A and B occurs.

[4 marks]

[Answer : (b)

13 7 ; (c) ] 12 24

26. [STPM ] A study on 100 visitors to a book fair shows that 60 visitors have seen the advertisement about the fair. Out of 40 visitors who make purchases, 30 have seen the advertisement. Find the probability that a visitor who has not seen the advertisement makes a purchase. [4 marks] [Answer : 0.25]

27. [STPM ] There are 12 towels, two of which are red. If five towels are chosen at random, find the probability that at least one is red. [4 marks] [Answer :

15 ] 22

28. [STPM ] In a basket of mangoes and papayas, 70% of mangoes and 60% of papayas are ripe. If 40% of the fruits in the basket are mangoes, (a) find the percentage of the fruits which are ripe,

(b) find the percentage of the ripe fruits which are mangoes.

[3 marks]

[4 marks]

[Answer : (a) 64% ; (b) 43.75%]

29. [STPM ] Eight persons are invited to be seated in the front row of eight seats to watch a concert. If two of them have to sit next to each other, find the number of different ways in which this can be done. [3 marks] [Answer : 10080]

30. [STPM ] A student applies for two scholarships S and T to further study. The probability that he is offered scholarship S is 0.4. If he is offered scholarship S, the probability of him being offered scholarship T is 0.2. If he is not offered scholarship S, the probability of him being offered scholarship T is 0.7. (a) Find the probability that he is offered both scholarships. (b) Find the probability that he is offered only one scholarship.

[2 marks]

[5 marks]

(c) State, with a reason, whether the events ‘he is offered scholarship S’ and ‘he is offered scholarship T ’ are mutually exclusive. [2 marks] [Answer : (a) 0.08 ; (b) 0.74 ; (c) Not mutually exclusive]

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14: Probability

(a) Find the probability that three doctors are selected.

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31. [STPM ] There are 20 doctors and 15 engineers attending a conference. The number of women doctors and that of women engineers are 12 and 5 respectively. Four participants from this group are selected randomly to chair some sessions of panel discussion. [2 marks]

(b) Given that two women are selected, find the probability that both of them are doctors. [2 marks] [Answer : (a)

855 33 ; (b) ] 2618 68

32. [STPM ] A survey was carried out among mothers in Town A and Town B on the issue whether pupils should be allowed to bring handphones to school. The data obtained are shown in the table below.

Town A Town B

Agree 100 20

Neutral 50 60

Disagree 50 120

(a) Find the probability that a mother agrees if she is from Town A. (b) Determine whether

i. the events ‘Town B’ and ‘disagree’ are independent, ii. the events ‘Town A’ and ‘disagree’ are mutually exclusive.

[2 marks]

[4 marks] [1 marks]

[Answer : (a) 0.5 ; (b) (i) not independent , (b) not mutually exclusive]

33. [STPM ] The probability that it rains in a day is 0.25, and the probability that a student carries an umbrella is 0.6. The probability that it rains or the student does not carry an umbrella is 0.5. If it rains on a particular day, find the probability that the student does not carry an umbrella. [4 marks] [Answer : 0.6]

34. [STPM ] A company has 400 employees of whom 240 are females. There are 57 female employees with degree qualifications and 85 male employees with non-degree qualifications. (a) Find the probability that a randomly chosen employee has a degree qualification.

[1 marks]

(b) Find the probability that a randomly chosen employee has a degree qualification given that the employee is a female. [2 marks] (c) Determine whether the events ‘an employee has a degree qualification’ and ‘an employee is a female’ are independent. [2 marks] [Answer : (a) 0.33 ; (b) 0.2375]

35. [STPM ] Mr. Tan works five days a week and he takes either a bus, a commuter or a taxi to his office. The probability that he takes a taxi is 0.20 and the probability that he takes a commuter is 0.50. The probabilities that he will arrive late at his office if he takes a bus, a commuter, or a taxi are 0.30, 0.10 and 0.15 respectively. 135

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14: Probability

(a) Find the probability that Mr. Tan will arrive late at his office on any given day.

[3 marks]

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(b) Mr. Tan arrives late at his office on a given day. Determine the mode of transport that he is likely to take. Give a reason for your answer. [4 marks] (c) Determine the probability that in a given week, Mr. Tan arrives late at his office on alternate days. [4 marks] [Answer : (a) 0.17 ; (c) 0.0199]

36. [STPM ] The number of shirts by colour and size in a batch of 100 is shown in the table below.

Blue White

Small 11 22

Medium 20 24

Large 9 14

Two shirts are selected at random, without replacement. Calculate the probability that (a) both shirts are of the same colour, (b) at least one shirt is large.

[3 marks]

[3 marks]

[Answer : (a)

92 17 ; (b) ] 33 225

37. [STPM ] There are eight male and four female architects in a consultant company. Three architects are randomly chosen to be posted to Johor Bahru, Kuala Lumpur and Miri. Find the probability that (a) three male architects are posted to the three cities,

[2 marks]

(b) one male architect is posted to Kuala Lumpur, one female architect is posted to Johor Bahru and one female architect is posted to Miri. [2 marks] [Answer : (a)

14 4 ; (b) ] 55 55

38. [STPM ] An opinion poll on a certain political party is conducted on 1000 voters, of whom 600 are males. It is found that 250 voters are in favour of the party. It is also found that 450 male voters are not in favour of the party. (a) Construct a two-way classification table based on the above information.

[2 marks]

(b) Find the probability that a randomly chosen voter is in favour of the party if the voter is a female. [1 marks]

(c) Find the probability that a randomly chosen voter is a male or not in favour of the party.[2 marks] (d) Determine whether the events “a voter is a male” and “a voter is in favour of the party” are independent. [3 marks] [Answer : (b) 0.25 ; (c) 0.9]

39. [STPM ] There are nine boxes, three of which contain oranges and the rest apples. Three girls and six boys randomly select a box each. Find the probability that 136

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14: Probability

(a) the girls select more boxes which contain oranges than the boys,

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(b) none of the girls has a box which contain oranges.

[3 marks] [3 marks]

[Answer : (a)

5 19 ; (b) ] 84 21

40. [STPM ] There are three male and two female workers who will be assigned to four different tasks. Each worker has an equal chance of being assigned to any task and may perform at most one task. (a) Determine the number of different ways the workers can be assigned to the tasks, with two of the tasks allocated to the female workers. [2 marks] (b) Find the probability that the female workers are assigned to two of the tasks.

[2 marks]

3 5

[Answer : (a) 72 ; (b) ]

41. [STPM ] At a cineplex 70% of the moviegoers buy popcorn at the snack counter. Of those who buy popcorn, 80% of them buy drinks. Among those moviegoers who do not buy popcorn, 10% of them buy drinks. (a) Find the probability that a moviegoer buys a drink.

[4 marks]

(b) Find the probability that a moviegoer does not buy popcorn if he buys a drink.

42. [STPM ] The events A and B are such that P(A) 6= 0 and P(B) 6= 0. (a) Show that P(A0 |B) = 1−P(A|B). 0

0

(b) Show that P(A |B) =P(A ) if A and B are independent.

[2 marks]

[Answer : (a) 0.59 ; (b) 0.05085]

[2 marks] [3 marks]

43. [STPM ] Two companies X and Y operate call-taxi services in a town. The percentages of residents in the town using the taxi services from companies X and Y are 40% and 60% respectively. The probabilities of taxis from companies X and Y being late are 0.02 and 0.01 respectively. A taxi is booked at random. Find the probability that (a) the taxi is late and it is from company X, (b) the taxi is late.

[2 marks]

[3 marks]

[Answer : (a) 0.008 ; (b) 0.014]

44. [STPM ] In a woman single badminton match, the first player who wins two out of three sets wins the match. The probability of a particular player winning a set is 0.7 if she has won the previous set. The probability of the player winning a set is 0.4 if she has lost the previous set. (a) Find the probability that the player wins the match given that she has lost the first set. [2 marks] (b) If the probability that the player wins the match is 0.64, find the probability that she wins the first set. [6 marks] 137

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[Answer : (a) 0.28 ; (b) ] 45. [STPM ] It is known that of the customers buying laptops, 75% buy an antivirus software, 40% buy additional memory card and 30% buy both. (a) Find the probability that a randomly selected customer does not buy any of those items [3 marks] (b) Find the probability that a randomly selected customer buys an antivirus software given that an additional memory card is bought. [2 marks] (c) Determine whether the event “an additional memory card is bought” is independent of the event “an antivirus software is bought”. [2 marks]

46. [STPM ]

[Answer : (a) 0.15 ; (b) 0.75 ; (c) Yes]

1 1 1 Two events A and B are such that P(A) = , P(B) = and P(A ∪ B) = . Find P(B|A0 ). [5 marks] 7 5 3

47. [STPM ]

[Answer :

2 ] 9

1 1 3 Two events A and B are such that P(A ∩ B) = , P(A ∩ B 0 ) = and P(A0 ∩ B) = . 8 4 8 (a) Find P(A), P(B) and P(A ∪ B).

[5 marks]

(b) Determine whether events A and B are independent.

[2 marks]

[Answer : (a)

3 1 3 , , ; (b) Dependent. ] 8 2 4

48. [STPM ] A student goes to school by bus, taxi or motorcycle. The probability that he travels by motorcycle is 0.44 and he is equally likely to take a bus or a taxi. The probability that he is late for school if he 1 1 1 goes by bus, taxi or motorcycle is , or respectively. Calculate the probability that 5 6 10 (a) he is late for school on a randomly chosen school day,

[2 marks]

(b) he goes to school by bus if he is late for school,

[3 marks]

(c) he is not late for school if he goes to school by bus or motorcycle.

[Answer : (a)

[4 marks]

11 21 31 ; (b) ; (c) ] 75 55 36

49. [STPM ] A study of job satisfaction is conducted for accountants and lawyers, where the satisfaction score is measured on a scale of 0 to 100. The number of respondents from these two occupations for each category of job satisfaction score is shown in the table below.

Occupation of respondents Accountant Lawyer

Score < 50 5 6

138

Number of respondents Score 50 − 64 Score 65 − 70 10 30 6 21

Score > 79 5 7

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(a) Find the probability that a randomly chosen respondent has a satisfaction score of less than 65 given that the respondent is an accountant. [2 marks] (b) Find the probability that a randomly chosen respondent is an accountant and has a satisfaction score of less than 65. [2 marks] (c) Find the probability that two randomly chosen respondents each has a satisfaction score of less than 65. [3 marks] (d) Determine whether the event “the respondent is an account” and the event “the satisfaction score is less than 65” are independent. [3 marks]

50. [STPM ]

[Answer : (a)

1 39 3 ; (b) ; (c) ; (d) Independent ] 10 6 445

4 1 3 Two events X and Y are such that P(X) = , P(Y ) = and P(X|Y ) = . Find the probability that 7 2 4 (a) both events occur, (b) both events do not occur, (c) only one of the events occurs.

[2 marks]

[3 marks]

[3 marks]

[Answer : (a)

17 9 3 ; (b) ; (c) ] 8 56 28

51. [STPM ] In an undergraduate programme at a private university, there are 132 male students and 22 international students. 8 of the male students are international students. The event “a student is a male” and the event “a student is an international student” are independent. (a) Determine the total number of students in the programme.

[3 marks]

(b) Find the probability that a student in the programme selected at random is a male or an international student. [3 marks] [Answer : (a) 363 ; (b)

146 ] 363

52. [STPM ] A box contains 3 blue balls, 4 red balls and 5 green balls. Six balls are taken out at random from the box without replacement. Calculate the probability that (a) at least one ball is green,

(b) exactly four are red balls, given that at least one ball is green.

[3 marks]

[4 marks]

[Answer : (a)

131 25 ; (b) ] 132 917

53. [STPM ] Based on a market survey, 70% of respondents in a certain area own smartphones. Among those who own smartphones, 45% of them own tablet computers. It is also found that 80% of the respondents own smartphones or tablet computers. (a) Find the probability that a respondent selected at random owns i. a smartphone and a tablet computer. 139

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14: Probability [2 marks]

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ii. a tablet computer. iii. a smartphone but not a tablet computer.

[2 marks]

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(b) Determine whether the events ”a respondent owns a smartphone” and ”a respondent owns a tablet computer” are independent. [2 marks] [Answer : (a)(i) 0.315 , (ii) 0.415 , (iii) 0.385 ; (b) Dependent. ]

54. [STPM ] The frequency distribution of blood types of 100 residents in a village is given below. Blood type Frequency

A 20

B 18

AB 4

O 58

Four residents are chosen at random. Find the probability that (a) all the four residents are of different blood types.

[3 marks]

(b) at least one of the four residents is of blood type AB, given that all of them are not of blood type A. [3 marks] [Answer : (a)

5921 5568 ; (b) ] 261415 316316

55. [STPM ] An office security system has a four digit personal identification number (PIN). The door will not open alter three unsuccessful attempts to enter the PIN. Assume that a worker does not try the same wrong PIN twice. (a) If a worker remembers the first two digits and knows that the last digit is either 8 or 9. find the probability that he will guess the correct PIN i. in the first attempt, ii. in the second attempt.

[2 marks] [2 marks]

(b) If a worker remembers the first two digits only, find the probability that i. all his three attempts are unsuccessful, ii. he will guess the correct PIN within three attempts. [Answer : (a) (i)

[3 marks] [2 marks]

1 1 97 3 , (ii) ; (b) (i) , (ii) ] 20 20 100 100

56. [STPM ] The participants in a competition have to choose five important features from 10 possible features of a product and list them in order of preference. Any entry that matches the predetermined order of preference will win a luxurious holiday. Assume that each feature is equally likely to be selected. (a) Find the probability that a participant who submits one entry will win a luxurious holiday.[2 marks] (b) Determine the minimum number of different entries that a participant needs to submit in order that the probability of winning a luxurious holiday is at least 0.25. [3 marks]

140

[Answer : (a)

1 ; (b) 7560 ] 30240

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Probability Distributions

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1. [STPM ] At a factory that employs many workers, 30% of the workers are members of the trade union A, 60% are members of the trade union B, and 10% are not members of any trade union. A sample of five workers are selected from the workers of the factory. (a) Determine the mean and standard deviation for the number of workers in the sample who are members of A. (b) Find, correct to three significant figures, the probability that the sample contains not more than 2 members of A. (c) Given that exactly 4 workers in the sample are members of trade unions, find the probability that the majority of the workers in the sample are members of A. 2 9

[Answer : (a) 1.5 , 1.025 ; (b) 0.837 ; (c) ]

2. [STPM ] A continuous random variable X can takes the values in the range 0 to 4. The probability that X takes a value larger than x is equal to αx2 + β, 0 ≤ x ≤ 4. (a) Determine the values of α and β.

(b) Find f (x), the probability density function of X. 8 (c) Show that the value µ, expectation for X is . 3 2√ 2. (d) Show that the standard deviation σ for X is 3

(e) Show that the probability (µ − σ) ≤ x ≤ (µ + σ) is

4√ 2. 9

[Answer : (a) α = −

1 , β = 1] 16

3. [STPM ] Each Supabrek packet contains a card from a set of n picture cards. A packet selected at random has the same probability of containing any one of the a cards. The Smith family already possess k (< n) different picture cards. They purchase a few more packets, selected one by one randomly, until they find a card that is different from the k cards they already have. If this happens when they have bought R packets, show that

Given the result

∞ X r=1

P (R = r) =

n−k n

r−1 k . n

rxr−1 = (1 − x)−2 for 0 < x < 1, show that E(R) =

n . n−k

For the case n = 4, deduce that the expected total number of packets that a family must buy if they 1 do not possess any yet, and picks the packets one by one until they have a complete set is 8 . 3

141

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4. [STPM ] Two fair dice are rolled and the numbers obtained are D1 , and D2 . In a game, the score S is the smaller of the two numbers and D1 and D2 , unless D1 = D2 when the score would be 0. 35 Obtain the probability distribution for S. Show that the expected value of S is . Find the variance 18 for S. Find P(D1 > S). The total score, T , obtained by a player is given by T = S1 + 2S2 + 3S3 where S1 , S2 and S3 are the scores obtained as described above, for three successive throws of the two dice. Determine the 4655 expected value of T , and show that the variance of T is . 162 [Answer :

665 7 35 , , ] 324 12 3

5. [STPM ] A box contains ten balls; six of the balls are numbered 1 and the other four are numbered 2. A sample of two balls are drawn at random from the box, replacing the first ball into the box before the second ball is drawn. The sum of the numbers on the two balls is denoted by R. The second ball is returned to the box. Another sample of two balls are drawn at random from the box, this time without replacement. The sum of the numbers on these two balls is denoted by W. Determine the probability distribution for R and for W . (a) Show that E(R)=E(W ). (b) Show that 8 Var(R)=9Var(W ).

6. [STPM ] A continuous random variable X takes the values in the range 1 ≤ x ≤ 4. The cumulative distribution function F (x), for X is a linear function that differ for the ranges 0 ≤ x ≤ 3 and 3 ≤ x ≤ 4, and takes the values 0, 0.5, and 1 corresponding with x = 1, 3 and 4, as shown in the following diagram.

(a) Sketch the graph for the probability density function of X. 11 (b) Show that the expectation value, µ, of X is equal to . 4 (c) Find the standard deviation of X, leaving your answer in the form of surds. (d) Find the probability that |X − µ| exceeds 1.

1 [Answer : (c) 4

r

37 5 ; (d) ] 3 16

7. [STPM ] The assembly of the students of a school begins 15 minutes after 8 oclock every school day. A male 142

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student changes his shoes as soon as he arrives at school. The time taken in minutes, from the moment he arrives at school, to change his shoes, can be assumed to have a normal distribution with mean 2 and standard deviation 1.5. The time after eight oclock, in minutes, the student arrives at school can be assumed to be normally distributed with mean 10 and standard deviation 2. The time taken to change his shoes can be assumed to be independent of the time he arrives at the school. (a) Show that, on a day selected at random, the probability that the student has changed his shoes before assembly begins is 0.885 (correct to three significant figures). (b) Find the probability that, for 10 days selected at random, the student has changed his socks before assembly starts on less than 9 days out of the 10 days. (c) By using a suitable approximation, find the probability that, for 100 days selected randomly, the student has changed his socks before assembly starts on less than 95 days out of the 100 days. [Answer : (b) 0.322 ; (c) 0.970]

8. [STPM ] For each of the functions f , g, h defined below, state if it is a probability density function, giving a reason for your answer for each case. ( 4x − 5, 1 ≤ x ≤ 2; (a) f (x) = 0, otherwise   3 x2 , 1 ≤ x ≤ 2; (b) g(x) = 7 0, otherwise   1 (3 − x), 1 ≤ x ≤ 2; (c) h(x) = 2 0, otherwise

9. [STPM ]

[Answer : (a) No ; (b) Yes ; (c) No]

(a) A drawer contains 2 black socks, 2 white socks and 2 green socks, The socks are removed from the drawer randomly, one at a time, until 2 socks of the same colour are removed, The random 2 variable X is the number of socks removed from the drawer. Show that P(X = 3)= . Find the 5 expected value and variance of X. (b) A 6-sided fair dice is tossed repeatedly until the total score is 3 or more. The random variable Y is the number of throws needed. Find the expected value of Y . [Answer : (a) 3.2 , 0.56 ; (b)

49 ] 36

10. [STPM ] A fair dice is rolled once. If the score is 3 or more, then the result X is the score. If the score is 1 or 2, then the dice is rolled once again and the result X is the sum of the scores of the two rolls. Show 2 1 that P(X = 6)= dan P(X = 7)= . 9 18 14 Show that E(X)= . 3 The procedure is repeated 6 times and N is the number of times the result 6 or 7 is obtained. Find E(N ) and Var(N ). Find P(N = 4), giving your answer correct to 3 decimal places. 143

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15: Probability Distributions 5 65 , , 0.047] 3 54

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[Answer :

11. [STPM ] The number of theft cases reported every day at a small police station has a Poisson distribution with mean 2. (a) Show that the probability that, on a particular day, at least one theft is reported, is 0.865, correct to 3 decimal places. (b) Find the probability that the total number of theft cases reported on 3 particular days is at least 3. (c) Find the probability that, for 5 particular days, no theft cases were reported on 2 days.

(d) Find the probability that, for 100 particular days, at least one theft is reported on less than 80 days. [Answer : (b) 0.9380 ; (c) 0.1179 ; (d) 0.0203]

12. [STPM ] The marks obtained by the candidates for a paper in an examination are distributed normally with mean 50 and standard deviation 10. (a) If a candidate must obtain 70 marks to get a distinction for that paper, find the percentage of the candidates who score distinctions for this paper in the examination. (b) If 70% of the candidates pass the paper in this examination, find the minimum marks required for a passing grade for that paper. [Answer : (a) 2.28% ; (b) 45]

13. [STPM ] An institute of higher learning is supplied with 500 microcomputer discs. To determine if the discs supplied are in good condition, 10 discs are selected at random and inspected. If none or only one of the discs is defective, then the supply of 500 discs will be accepted. Suppose that the entire supply of 500 discs contains 25 defective discs. Find (a) the probability that the supply will be accepted entirely,

(b) the probability that exactly one disc will be found to be defective if it is known that the entire supply has been accepted. [Answer : (a) 0.914 ; (b) 0.345]

14. [STPM ] A committee consisting of 3 people will be chosen at random from 6 men and 3 women. If X represents the number of men selected, (a) tabulate the probability distribution P(X = x). (b) calculate the mean and variance of X.

(c) write down the distribution function F (x) = P (X ≤ x).

If Y represents the number of women selected, determine the mean and variance for Y , without calculating the probability distribution P (Y = y). 144

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15. [STPM ] A continuous random variable X has the probability density function ( ax + b, −1 ≤ x ≤ c f (x) = 0, otherwise 1 7 1 1 = = . with P X ≤ − and P X ≥ 2 16 2 16 Find the values of a, b, and c. Calculate the median and mean of X.

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[Answer : (b) 2 , 0.5 ; 1 , 0.5]

[Answer :

1 1 1 ; ; 1 ; 0.414 ; ] 2 2 3

16. [STPM ] A continuous random variable X which represents the amount of sugar (in kg) used by a family per week, has the probability density function ( c(x − 1)(2 − x), 1 ≤ x ≤ 2, f (x) = 0, otherwise. (a) Determine the value of the constant c.

(b) Obtain the cumulative distribution function F (x) = P (X ≤ x). (c) Calculate the mean and variance of X.

(d) Find the probability that the family uses less than 1.8 kg sugar but more than 1.3 kg in a week. [Answer : (a) 6 ; (c) 1.5 , 0.05 ; (d) 0.68]

17. [STPM ] Five patients need to undergo an operation that has a success rate of 0.8. Assuming that each operation is independent of each other, find (a) the probability that none of the operations fail,

(b) the probability that at least four of the patients will undergo the operation successfully.

[Answer : (a) 0.328 ; (b) 0.737]

18. [STPM ] A bank manager starts work at 0900 hours. He notices that his journey from his house to his office follows a normal distribution with a mean of 38 minutes and a standard deviation of 5 minutes. (a) If the manager leaves his house at 0820 hours every day, find the probability that he arrives late for work on a day selected normally. Find also, by using a suitable approximation, the probability that the manager will be late for work for, at the most, 33 out of 100 working days. (b) Determine at what time, correct to the nearest minute, the manager should leave his house, if the probability that he will be late is 0.1. [Answer : (a) 0.3446 , 0.4199 ; (b) 0816 hours]

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(a) the probability that no accidents occur one week.

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19. [STPM ] At a busy road junction, the numbers of accidents that occur are distributed according to the Poisson distribution. If the mean number of accidents that occur is 1.8 per week, find

(b) the smallest integer n such that the probability that n accidents occur in one week is less than 0.02. (c) the probability that no accidents occur in a period of 2 weeks.

(d) the largest integer k such that the probability that no accidents occur in a period of k successive weeks exceeds 0.0001. [Answer : (a) 0.1653 ; (b) 6 ; (c) 0.0273 ; (d) 5]

20. [STPM ] An electrical company sells X television sets a day with the probabilities P(X = x) = a(5 − x), where a is a constant, for x = 0, 1, 2, 3, 4, and zero for any other value of x. (a) Determine the value of a.

(b) Write down the probability distribution function of X. (c) Find P( |X − E(X) | < 1).

None of the customers will buy only the aerial, and the probability that a customer who buys a 2 television set also buys an aerial is . Find the probability that the number of aerials sold one day 5 is 3, and find also the probability that the number of television sets sold is 4 if it is known that the number of aerials sold is 3. [Answer : (a)

21. [STPM ] A continuous random variable X has the probability density function   2 (x − 2)(5 − x), 2 ≤ x ≤ 5, f (x) = 9 0, otherwise. (a) Sketch the graph f (x), and determine the mode of X. (b) Find the mean and variance of X.

(c) Find the cumulative distribution function of X, and P(X > 3).

[Answer : (a) 3.5 ; (b) 3.5 , 0.45 ; (c)

22. [STPM ] The discrete random variable X takes the value k with the probability

where c is a constant.

3 P (X = k) = c| − k|, k = 0, 1, 2, 2

(a) Determine the value of c. 146

7 1 ; (c) ; 0.0188 ; 0.5456] 15 15

20 ] 27

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(c) Calculate the mean and variance of X.

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(b) Tabulate the values of P (X = k), k = 0, 1, 2. (d) Write down the cumulative distribution function of X, and sketch the graph of this function. [Answer : (a)

23. [STPM ] A continuous random variable Y has the probability density function ( y(a − by), 0 ≤ y ≤ 2, f (y) = 0, otherwise. with E(Y ) = 1.

(a) Find the values of a and b. (b) Determine the variance of Y .

(c) Find the cumulative distribution function of Y . 3 1 ≤Y ≤ (d) Find P 2 2

[Answer : (a)

2 3 16 ; (c) , ] 5 5 25

11 3 3 , ; (b) 0.2 ; (d) ] 2 4 16

24. [STPM ] A chocolate manufacturer labels each piece of chocolate it produces as having a mass of 200 g. It is known that the actual mass of the pieces of chocolates are distributed normally with mean 204 g and standard deviation 2 g. (a) Find the probability that the mass of a piece of chocolate selected at random is less than 200 g. (b) By using a Poisson approximation, estimate the probability that at the most, a piece of chocolate has a mass less than 200 g in a box containing 100 pieces of chocolate. [Answer : (a) 0.0228 ; (b) 0.3355]

25. [STPM ] At a flower pot kiln, 20% of the pots produced are cracked. For a sample of five pots selected at random, find the probability that (a) exactly one pot is cracked, (b) at least one pot is cracked.

26. [STPM ] A continuous random variable X has the probability density function. ( cx2 (1 − x), 0 < x < 1, f (x) = 0, otherwise. where c is a constant. 147

[Answer : (a) 0.410 ; (b) 0.672]

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(c) Plot the graph y = F (x).

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(b) Find the cumulative distribution function, F , of X.

(d) Estimate, from your graph, the median and P(0.39 < X < 0.61).

[Answer : (a) 12 ; (d) 0.61 , 0.32]

27. [STPM ] The discrete random variable X has the probability.   k(x + 2) , x = −1, 0, 1, 2 P (X = x) = 8 0, otherwise (a) Determine the value of k. (b) Calculate P (−1 < X < 1.5).

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(a) Find the value of c.

[Answer : (a)

1 4 ; (b) ] 5 2

28. [STPM ] From a study, it was found that 75% of the cars that stop at a petrol station are Proton Saga. Of these Proton Saga cars, 20% have not been installed with the third brake light. (a) In a particular period of time, 10 cars stop at the station.

i. Find the probability that 8 of the cars are Proton Saga. ii. Find the probability that 2 of the cars are Proton Saga that have the third brake light installed.

(b) On a particular day, 500 cars stop at the station. By using a suitable approximation,

i. find the probability that 95 to 150 of the cars are not Proton Saga. ii. find the probability that less than 60 Proton Saga cars do not have the third brake light.

[Answer : (a) (i) 0.282 ; (ii) 0.0106 ; (b) (i) 0.99496 ; (ii) 0.02621]

29. [STPM ] A continuous random variable X represents the period, in minutes, of a telephone call at an office. The cumulative distribution function of X is  2  x , 0 ≤ x ≤ 2. 8 F (x) = 4  1 − , x > 2 x3 (a) Sketch the graph F (x).

(b) Find the probability density function for X.

(c) Determine the mean and standard deviation of X.

(d) Calculate the probability that a telephone call in the office is more than 1 minute long.

(e) Calculate the probability that a telephone call in the office is between 1 to 4 minutes long. [Answer : (c) 148

13 7 13 , 1.518 ; (d) ; (d) ] 6 8 16



(a) Determine the value of a. (b) Sketch the graph of f (x).

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30. [STPM ] The probability density function of the continuous random variable x is   0 ≤ x < a, 2x, f (x) = 6 − 6x, a ≤ x ≤ 1,   0, otherwise.

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1 2

[Answer : (a) ]

31. [STPM ] A random variable X is distributed normally with mean µ and standard deviation σ. If P(X < 32) = 0.0228 and P(X > 45) = 0.1056, find the values of µ and σ.

32. [STPM ] The cumulative distribution function of the discrete  0,      2, F (x) = 15 5   ,     15 1,

[Answer : 40 , 4]

random variable X is x < 1; 1 ≤ x < 2; 2 ≤ x < 9;

x ≥ 9.

(a) Find the probability distribution function of X. (b) Find E(15X + 2).

  x + 1 , x = 1, 2, 9 ; (b) 100] [Answer : (a) P (X = x) = 15 0, otherwise

33. [STPM ] The continuous random variable X is distributed uniformly in the interval [a, b]. Write down the probability density function of X. Hence, find the mean and variance of X in terms of a and b. 3 3 If the mean and variance of X respectively are and , find the values of a and b. 2 4 Hence, show the the probability density function of X is   1 , 0 ≤ x ≤ 3, f (x) = 3 0, otherwise. (a) Find the cumulative distribution function of X, and sketch its graph. 1 (b) Find P |X − 1| > 2 [Answer : 149

a+b 1 2 , (a − b)2 , a = 0 b = 3 ; (b) ] 2 12 3



(a) (b) (c) (d)

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34. [STPM ] A continuous random variable X has the cumulative distribution function   0, x < 0,    2   ax , 0 ≤ x < 1,   1 1 1 ≤ x < 2, F (x) = 2 x − 4 ,   1 5   − x2 + bx − , 2 ≤ x < 3,   4 4  1, x ≥ 3. Determine the values of a and b. Calculate P(0.5 ≤ X < 1.5). Find the probability density function of X. Find the mean and variance of X.

[Answer : (a)

35. [STPM ] Determine if each of the following is a probability density function.   1 + x , x = −1, 0, 1; (a) P (X = x) = 2 0, otherwise.   1 + x , −1 ≤ x ≤ 1; (b) f (x) = 2 0, otherwise

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3 5 1 3 , ; (b) 0.4375 ; (d) , ] 4 2 2 12

[Answer : (a) No ; (b) Yes]

36. [STPM ] During an election, a state constituency has 20 000 male residents and 30 000 female residents. 60% of the male residents are qualified to vote and 75% of them are registered voters. 50% of the female residents are qualified to vote and 80% of them are registered voters. Only residents registered as voters are allowed to cast their votes, and the percentage of male residents and female residents who voted were 80% and 65% respectively. (a) Find the probability that a male resident in that constituency who is a registered voter, voted in the election. (b) Find the probability that a resident in that constituency picked at random, voted in the election. (c) If 10 residents in that constituency are selected at random from the male residents who are registered voters, find the probability that at least 8 of them voted in the election. (d) If 100 residents in that constituency are selected at random, find the probability that more than 20 of them voted in the election. [Answer : (a) 0.8 ; (b) 0.3 ; (c) 0.6778 ; (d) 0.9809]

37. [STPM ] A factory uses cables of radius 15 mm to 25 mm. The supply of cables are obtained from Syarikat Utama Berhad and Syankat Wawasan Berhad respectively in the ratio 2:3. The radii of the cables supplied by Syarikat Utama Berhad are distributed normally with mean 18 mm and standard deviation 3 mm, while the radii of the cables supplied by Syarikat Wawasan Berhad are distributed uniformly between 12 to 26 mm. 150

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(a) Find the probability of a cable produced by Syarikat Utama Berhad, that is selected at random, can be used by the factory. (b) Find the probability of a cable produced by Syarikat Wawasan Berhad, that is selected at random, can be used by the factory. (c) Find the probability of a cable obtained by the factory that is selected at random has a radius of less than 16 mm. [Answer : (a) 0.8315 ; (b) 0.7143 ; (c) 0.2723]

38. [STPM ] The discrete random variable X can only take the values of 1, 2, 3, and 4, with the probabilities P (X = 1) = P (X = 2) and P (X = 3) = P (X = 4) = 2P (X = 1). If A = {1, 3}, find P(A). [Answer : 0.5]

39. [STPM ] The probability distribution of the random variable X is as follows X=x P(X = x)

(a) Determine of the value of c. 1 (b) Find E (X − 2) . 2

1 0.40

2 c

3 0.10

4 0.09

5 0.06

[Answer : (a) 0.35 ; (b) 0.03]

40. [STPM ] The mathematics paper in an examination is scheduled from 0900 to 1200 hours. From the experience of the years before, the time taken by the candidates to complete the paper have a mean of 150 minutes and a standard deviation of 25 minutes. By assuming that the time needed to complete the paper is distributed normally, (a) find the probability that a candidate can complete the paper in the time given.

(b) find the probability that a candidate can complete the paper in the time given if he enters the examination hall at 0945.

41. [STPM ] A continuous random variable X has the probability density function  2  kx , 0 < x < 1, f (x) = 3 0, otherwise (a) Determine the value of k.

(b) Find the cumulative distribution function of X. (c) Find the mean and median of X. (d) Find variance of X. 151

[Answer : (a) 0.8849 , (b) 0.2743]

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(e) If Y = 2X, calculate P(1 < Y ≤ 4).   x ≤ 0, 0, [Answer : (a) 9 ; (b) F (x) = x3 , 0 < x < 1, ;   1, x ≥ 1. 7 (c) 0.75 , 0.7937 ; (d) 0.0375 ; (e) ] 8

42. [STPM ] A computer program has an error that causes the program not to function perfectly. n programmers are assigned separately to detect the error. The probability that each programmer will detect the error is 0.875. Determine the value of n if the probability that at least one programmer detects the error is 0.998. [5 marks] [Answer : 3]

43. [STPM ] The amount of petrol X, in thousand litres, sold in a day by a petrol station is a random variable with probability density function f (x) =

1 (x2 − 18x + 81), 0 ≤ x ≤ 9. 243

Find the expected amount of petrol sold by the petrol station in 30 days.

[5 marks]

[Answer : 67500 litres]

44. [STPM ] The number of trees of a certain species of tree in a forest is believed to have a Poisson distribution with a tree per hectare. Find the probability there are at least 3 trees of that species in a circular area of radius 100 m in that forest. Give your answer correct to three decimal places. [1 hectare = 104 m2 ]

45. [STPM ] Discrete random variable X has the following probability function:   1 x(9 − x)n , x = 5, 7, 8 p(x) = 116 0, otherwise where n is a constant. (a) Determine the value of n.

(b) Sketch the graph of the probability function of X. (c) Calculate the mean and variance of X.

[5 marks]

[Answer : 0.608]

[6 marks]

[4 marks]

[5 marks]

[Answer : (a) 2 ; (c)

165 934 , ] 29 841

46. [STPM ] A sugar factory has two packaging machines which operate simultaneously and independently. Machine A is two times faster than machine B. Mass of packets of sugar packed by machine A are distributed normally with mean 1 000 g and standard deviation 4 g, whereas mass of packets of sugar packed by machine B are distributed normally with mean 995 g and standard deviation 5 g. 152

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(a) Show that the probability of the mass of a packet of sugar that is selected randomly is less than 997 g is 0.370. [8 marks] (b) If a packet of sugar selected randomly has a mass less than 997 g, find the probability that the packet is packed by machine B. [3 marks] (c) Find the probability that from 5 packets of sugar selected randomly, at least 2 packets have masses less than 997 g. [4 marks] [Answer : (b) 0.591 ; (c) 0.609]

47. [STPM ] The number of computers, X, sold in a day by a computer shop has the probability distribution as follows:   0.25 − 0.05x, x = 0, 1, 2, 3; P (X = x) = a, x = 4, 5   0, otherwise. (a) Determine the value of the constant a,

(b) Find the probability that 3 or 4 computers have been sold in a day.

[2 marks]

[2 marks]

[Answer : (a) 0.15 ; (b) 0.25]

48. [STPM ] The continuous random variable X is normally distributed with P(X < 17.8) = 0.75 and P(X > 19.2) = 0.15. Find the mean and standard deviation of X. [5 marks] [Answer : 15.19 , 3.87]

49. [STPM ] The annual revenue of a company is distributed normally with mean RM2.5 million and standard deviation RM0.3 million. The annual expenditure of the company consists of two parts, namely fixed expenditure of RM 1.8 million and variable expenditure which is of the annual income. (a) If Y is the annual profit of the company when its annual revenue is X, show that Y = 0.95X −1.8. Hence, find the expected annual profit and its standard deviation. [9 marks] (b) The company management promises to pay bonus to its workers if the annual profit is more than 30% of the annual revenue. Find the probability that bonus will not be paid to its workers. [6 marks]

[Answer : (a) RM 0.575 million , RM 0.285 million ; (b) 0.721]

50. [STPM ] X and Y are two independent random variables with E(X)=7, E(X 2 )=51, E(Y )=1, and E(Y 2 )=103. Find (a) E(5X + Y − 2). (b) Var(5X + Y − 2).

[2 marks] [4 marks]

[Answer : (a) 34 ; (b) 152]

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(a) the probability that one randomly chosen year is free from accident.

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51. [STPM ] The number of accidents that occur on an assembly line has a Poisson distribution with a mean of 3 accidents a year. Find, correct to three significant figures, [2 marks]

(b) the probability that two years out of five years chosen at random are free from accidents.[3 marks] [Answer : (a) 0.0498 ; (b) 0.0213]

52. [STPM ] Continuous random variable X has the following probability density function:   1 , 0 ≤ x ≤ 9; f (x) = 8 0, otherwise. Continuous random variable Y is defined by Y = ln X. (a) Show that,

1 P (Y ≤ y) = (ey − 1), 0 ≤ y ≤ ln 9. 8

[4 marks]

(b) Find and sketch the cumulative distribution function of Y .

[4 marks]

(c) Find the probability density function of Y . 9 (d) Show that E(Y )= ln 3 − 1. 4 53. [STPM ]

[3 marks] [4 marks]

(a) The probability that a pair of brand A shoes will be spoilt within the period of 6 months of usage is 0.2. By using a suitable approximation, find the probability that there are 75 to 100 pairs of shoes from 500 randomly chosen pairs which will be spoilt within the period of 6 months.[6 marks] (b) A large quantity of shoes from brand B is inspected before been distributed to sales out lets. It is found that 1% of the left side shoes are defective while 2% of the right side shoes are defective. By assuming that the defects occur independently and using suitable approximation, calculate the probability that in a random sample of 50 pairs of shoes, i. there are two pairs of shoes with defects on their left side. ii. there is a pair of shoes with defects on both sides.

[4 marks] [5 marks]

[Answer : (a) 0.520 ; (b)(i) 0.076 ; (ii) 0.251]

54. [STPM ] The profit for one lot of shares of BOD and its probability distribution are shown in the table below. Profits (RM) Probability

100 0.20

210 0.25

315 0.30

(a) Calculate the expected profit for one lot of the shares.

525 0.15

1050 0.10

[2 marks]

(b) An investor will buy the shares if the expected profit is more than 10% of the share price. If the price of one lot of the shares is RM2200, determine whether the investor will buy the shares. [2 marks]

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[Answer : (a) RM350.75 ; (b) Yes.] 55. [STPM ] Discrete random variable X represents the number of radios sold by a company in a week. X has a Poisson distribution with mean µ and E(X 2 )=6. (a) Find the value of µ.

[3 marks]

(b) Find the probability that five radios are sold by the company in four weeks.

[5 marks]

[Answer : (a) 2 ; (b) 0.0916]

56. [STPM ] A survey shows that 60% of the housewives have seen the advertisement of a new product in televisions. The probability that a housewife who has seen the advertisement buys the new product is 0.9 while the probability that a housewife has not seen the advertisement buys the new product is 0.3. (a) Find the probability that a housewife buys the new product.

[6 marks]

(b) Find the probability that a housewife who buys the new product has seen the advertisement. [3 marks]

(c) Using appropriate approximation, find the probability of less than 10 of 50 housewives who buy the new product, have not seen the advertisement. [8 marks] [Answer : (a) 0.66 ; (b) 0.8182 ; (c) 0.541]

57. [STPM ] The cumulative distribution function of the time T , in minutes, of a student spends every morning waiting for bus to school is given by  0, t < 0;     1   t2 , 0 ≤ t < 5; F (t) = 751 1 1 2   t , 5 ≤ t < 15; − + t−   2 5 150   1, t ≥ 15. (a) Find the probability density function of T and sketch its graph. (b) Calculate the mean waiting time of the student.

[5 marks]

[4 marks]

(c) Find the probability that the student waited less than 10 minutes for at least 3 days out of 5 schooling days. [6 marks] [Answer : (b) 6.67 ; (c) 0.9645]

58. [STPM ] Continuous random variable X has the following cumulative distribution function.  0, x ≤ 0;     1   0 < x ≤ 1;  x, 2 F (x) = 3  x − a, 1 < x <≤ ;   2   3  b, x> . 2 155



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1 (a) Show that a = and state the value of b. 2 5 1 . (b) Find P ≤X≤ 2 4

[2 marks]

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[3 marks]

1 2

[Answer : (a) B = 1 ; (B) ]

59. [STPM ] On average, the number of cars stopping over at a petrol station is one car per minute. Using Poisson distribution, find, correct to three decimal places, the probability (a) that exactly three cars stop over at the petrol station in a period of one minute,

[2 marks]

(b) that more than two cars stop over at the petrol station in a period of five minutes.

[4 marks]

[Answer : (a) 0.061 ; (b) 0.875]

60. [STPM ] Random variable T , in hours, represents the lifespan of a system of heat detector. The probability that the system do not function at time t hour is given by 21

P (T < t) = 1 − e− 5500 t

(a) Find the probability that the system functions for at least 250 hours.

[4 marks]

(b) Find the probability distribution function of the lifespan of the system and sketch its graph. (c) Calculate the expected lifespan of the system.

[5 marks]

[6 marks]

[Answer : (a) 0.385 ; (c) 261.9 hours]

61. [STPM ] A company sells two types of rice, brand A and brand B, in a 10 kg bag. The mass of brand A rice in each bag is normally distributed with a mean of 10.05 kg and a standard deviation of 0.2 kg, whereas the mass of brand B rice in each bag is normally distributed with a mean of 10.05 kg and a standard deviation of 0.05 kg. (a) Find the probability that a bag of brand A rice and a bag of brand B rice selected at random both are more than 10 kg. [6 marks] (b) If T represents the total mass of 2 bags of brand A rice and 6 bags of brand B rice, find the mean and variance of T . Hence or otherwise, find the probability that the total mass of the 8 bags of rice is less than 80 kg. [9 marks] [Answer : (a) 0.5037 ; (b) mean=80.4 , variance=0.095 , 0.097]

62. [STPM ] The probability distribution of the number of typing errors in a page of document in an office is given in the following table.

156

Number of errors 0 1 2 3 4 5

Probability 0.01 0.10 0.25 0.25 0.30 0.09


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(a) Find the probability that at least two errors happen in one page of document.

[2 marks]

(b) Find the mean number of typing errors in one page of document.

[2 marks]

[Answer : (a) 0.89 ; (b) 3]

63. [STPM ] The mass of an empty cereal box is normally distributed with a mean of 7 g and a standard deviation of 0.05 g, whereas the mass of a box filled with cereal is also normally distributed but with a mean of 230 g and a standard deviation of 2 g. Find the probability that the net weight of a box of cereal is less than 220 g. [5 marks] [Answer : 0.0668]

64. [STPM ] The following is the graph of the probability density function of continuous random variable X which only takes values of -2 to 4 inclusive.

1 (a) Show that a = . 3 (b) Find the probability density function of X. (c) Calculate the mean and variance of X. (d) Find P[|X + 3E(X)| < 4].

[2 marks]

[4 marks]

[6 marks] [3 marks]

[Answer : (c)

1 4 14 , ; (d) ] 3 9 6

65. [STPM ] In a production process of flower vases, is known that on average 45 flower vases have racks in every 1000 flower vases produced. (a) Find the probability that there are at most two flower vases having cracks in a random sample of size 5. [4 marks] (b) If the probability that no flower vases, in n flower vases selected at random, having cracks is 0.1, determine the value of n. [5 marks] (c) By using suitable approximation, find the probability that more than three flower vases have cracks in a random sample of size 100. [6 marks] 157

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[Answer : (a) 0.9992 ; (b) 50 ; (c) 0.658] 66. [STPM ] A random variable X has a Poisson distribution with P(X = 0)=P(X = 1). Find E(X 2 ).

[5 marks]

[Answer : 2]

67. [STPM ] The distance travelled by a newspaper vendor in a residential district for each weekday (Monday to Friday) has mean 13 km and standard deviation 0.8 km. For Saturdays and Sundays, the daily distance travelled has mean 11 km and standard deviation 0.7 km. The distances travelled on different days may be assumed to be independent. If D is the average daily distance travelled by the newspaper vendor in a week, find E(D) and Var(D). Assuming that D has a normal distribution, find the probability that, in a randomly chosen week, the mean daily distance travelled by the newspaper vendor is less than 12 km. [7 marks] [Answer : 12.429 , 0.0853 , 0.0708]

68. [STPM ] A manufacturer produces a type of car battery with a lifetime of X years which is a random variable having the probability density function  ax2 + bx − 9 , 1 ≤ x ≤ 3; f (x) = 4 0, otherwise, where a and b are constants.

(a) It is found that 50 out of 100 batteries produced by the manufacturer have lifetimes of less than 2 years. Determine the values of a and b. [7 marks] (b) Find the probability that a battery produced by the manufacturer lasts more than 2 years and 4 months. [3 marks] 3 4

[Answer : (a) a = − , b = 3 ; (b)

7 ] 27

69. [STPM ] A discrete random variable X can take only the values 1, 2, 3, and 4, with P(X = 1)=P(X = 2) and P(X = 3)=P(X = 4)=2P(X = 1). If A = {1, 3}, find P(A). [5 marks] [Answer : 0.5]

70. [STPM ] The marks of a test are normally distributed with mean 55 and standard deviation 12. A student who obtains 75 marks or more is given grade A, and a student who obtains 85 marks or more is awarded a certificate of excellence. Find the probability that (a) a student receives grade A,

[2 marks]

(b) a student who receives grade A is also awarded a certificate of excellence.

[4 marks]

[Answer : (a) 0.0478 ; (b) 0.1299]

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71. [STPM ] Two percent of the bulb produced by a factory are not usable. Find the smallest number of bulbs that must be examined so that the probability of obtaining at least one non-usable bulb exceeds 0.5. [6 marks]

[Answer : 35]

72. [STPM ] The mass of yellow water melon produced by a farmer is normally distributed with a mean of 4 kg and a standard deviation of 800 g. The mass of red water melon produced by the farmer is normally distributed with a mean of 6 kg and a standard deviation of 1 kg. (a) Find the probability that the mass of a red water melon, selected at random, is less than 5 kg. Hence, find the probability that a red water melon with mass less than 5 kg has mass less than 4 kg. [5 marks] (b) If Y = M − 2K, where M represents the mass of a red water melon and K the mass of a yellow water melon, determine the mean and variance of Y . Assuming that Y is normally distributed, find the probability that the mass of a red water melon selected at random is more than twice the mass of yellow water melon selected at random.[6 marks] [Answer : (a) 0.1587 , 0.144 ; (b) -2 , 3.56 ; 0.1446]

73. [STPM ] Continuous random variable X is defined in the interval 0 to 4, with  1 − ax, 0 ≤ x ≤ 3 P (X > x) = 1 b − x, 3 < x ≤ 4 2 with a and b as constants,

1 and b = 2, 6 (b) Find the cumulative distribution function of X and sketch its graph. (a) Show that a =

(c) Find the probability density function of X.

(d) Calculate the mean and standard deviation of X.

[3 marks]

[4 marks]

[2 marks]

[6 marks]

[Answer : (d)

5 , 1.190] 2

74. [STPM ] The discrete random variable X can only take the values 1, 3, 5 and 9, with probabilities: P(X=1)=0.2, P(X=3)=0.3, P(X=5)=0.4, and P(X=9)=0.1. Find E(X) and Var(X). [4 marks] [Answer : 4.0 , 5]

75. [STPM ] The probability of a person allergic to a type of anaesthetic is 0.002. A total of 2000 persons are injected with the anaesthetic. Using a suitable approximate distribution, calculate the probability that more than two persons are allergic to the anaesthetic. [5 marks] [Answer : 0.7619] 159

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76. [STPM ] Tea bags are labelled as containing 2 g of tea powder. In actual face, the mass of tea powder per bag has mean 2.05 g and standard deviation 0.05 g. Assuming that the mass of tea powder of each bag is normally distributed, calculate the expected number of tea bags which contain 1.95 g to 2.10 g of tea powder in a box of 100 tea bags. [5 marks] [Answer : 82]

77. [STPM ] The lifespan of a species of plant is a random variable T (tens of days). The probability density function is given by   1 e− 18 t , t > 0 f (t) = 8 0, otherwise (a) Find the cumulative distribution function of T and sketch its graph.

[6 marks]

(b) Find the probability, to three decimal places, that a plant of that species randomly chosen has a lifespan of more than 20 days. [3 marks] (c) Calculate the expected lifespan of that species of plant.

[5 marks]

[Answer : (b) 0.779 ; (c) 80 days]

78. [STPM ] The probability distribution of a random variable X is given by P(X = 0) = P(X = 2) = 3k, P(X= l)=P(X=3)=2k, and P(X ≥ 4)=0. (a) Find the value of k.

[2 marks]

(b) If Y = 2X + 3, find the probability distribution of Y , and hence find the expected value of Y . [4 marks]

[Answer : (a) k =

1 ; (b) 5.8] 10

79. [STPM ] The time taken by the customers of a company to settle invoices is normally distributed with mean 20 days and standard deviation 5 days. A discount is given for every invoice which is settled in less than 12 days. (a) Find the probability that an invoice is settled in less than 12 days. (b) Find the probability that an invoice is settled in 18 to 26 days.

[2 marks]

[3 marks]

(c) Determine, out of 200 invoices, the expected number of invoices which are given discounts.[2 marks] (d) Find the probability that at most 2 out of 10 invoices are given discounts.

[4 marks]

[Answer : (a) 0.0548 ; (b) 0.5403 ; (c) 11 ; (d) 0.9853]

80. [STPM ] A type of seed is sold in packets which contain ten seeds each. On the average, it is found that a seed per packet does not germinate. Find the probability that a packet chosen at random contains less than two seeds which do not germinate. [4 marks] [Answer : 0.7361] 160



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81. [STPM ] The continuous random variable X has the probability density function   4 x2 (3 − x), 0 < x < 3, f (x) = 27 0, otherwise. 3 . (a) Calculate P X < 2 (b) Find the cumulative distribution function of X.

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[3 marks]

[3 marks]

[Answer : (a)

5 ] 16

82. [STPM ] The lifespan of an electrical instrument produced by a manufacturer is normally distributed with a mean of 72 months and a standard deviation of 15 months. (a) If the manufacturer guarantees that the lifespan of an electrical instrument is at least 36 months, calculate the percentage of the electrical instruments which have to be replaced free of charge. [4 marks]

(b) If the manufacturer specifies that less than 0.1% of the electrical instruments have to be replaced free of charge, determine the greatest length of the guarantee period correct to the nearest month. [5 marks]

[Answer : (a) 0.82% ; (b) 25 months]

83. [STPM ] The discrete random variable X has the probability function ( k(4 − x)2 , x = 1, 2, 3, P (X = x) = 0, otherwise where k is a constant.

(a) Determine the value of k and tabulate the probability distribution of X. (b) Find E(7X − 1) and Var(7X − 1).

[7 marks]

[Answer : (a) k =

84. [STPM ] The discrete random variable X has the probability function ( k(4 − x)2 , x = 1, 2, 3, P (X = x) = 0, otherwise where k is a constant.

(a) Determine the value of k and tabulate the probability distribution of X. (b) Find E(7X − 1) and Var(7X − 1).

[3 marks]

10 1 ; (b) , 19] 14 7

[3 marks] [7 marks]

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15: Probability Distributions 1 ; (b) 9 , 19] 14

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[Answer : (a)

85. [STPM ] A courier service company claims that 95% of the letters sent using its service reach their destinations within a day. If six letters are randomly chosen, find (a) the probability that at least two letters take more than a day to reach their destinations,[4 marks] (b) the mean and variance of the number of letters that reach their destinations within a day.[2 marks] [Answer : (a) 0.0328 ; (b) 5.7 , 0.285]

86. [STPM ] The mass of a small loaf of bread produced in a bakery may be modelled by a normal random variable with mean 303 g and standard deviation 4 g. Find the probability that a randomly chosen loaf has a mass between 295 g and 305 g. [3 marks] [Answer : 0.6687]

87. [STPM ] A computer accessories distributor obtains its supply of diskettes from manufacturers A and B, with 60% of the diskettes from manufacturer A. The diskettes are packed by the manufacturers in packets of tens. The probability that a diskette produced by manufacturer A is defective is 0.05 whereas the probability that a diskette produced by manufacturer B is defective is 0.02. Find the probability that a randomly chosen packet contains exactly one defective diskette. [7 marks]

88. [STPM ] The continuous random variable X has probability density function  1 1    25 (1 − 2x), −2 ≤ x ≤ 2  1 3 f (x) = (2x − 1), ≤x≤3   25 2   0, otherwise. (a) Sketch the graph of y = f (x). 13 (b) Given that P (0 ≤ X ≤ k) = , determine the value of k. 100

89. [STPM ] The probability distribution function of the discrete random variable Y is P (Y = y) =

y , y = 1, 2, 3, . . . , 100 5050

(a) Show that E(Y ) = 67 and find Var(Y ). (b) Find P (|Y − E(Y )| ≤ 30).

[Answer : 0.2558]

[2 marks]

[6 marks]

[5 marks] [4 marks]

[Answer : (a) 561 ; (b) 162

3 2

[Answer : k = ]

4087 ] 5050

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90. [STPM ] The time taken by year 1 pupils of a school to complete a task is normally distributed with mean µ and standard deviation 5.1 minutes. (a) Given that 97.5% of the pupils require less than 40 minutes to complete the task, find the value of µ. [3 marks] (b) Find the probability that a pupil chosen at random takes more than 35 minutes to complete the task. [2 marks] [Answer : (a) 30 ; (b) 0.1635]

91. [STPM ] The number of requests, X, received by a company to deliver pianos in a day is a discrete random variable having probability distribution function  2  2k , x = 0, 3, P(X = x) = kx, x = 1, 2,   0, otherwise. (a) Determine the value of the constant k and construct a probability distribution table for X.[4 marks] (b) Find the probability that the company receives at least two requests in a day.

[2 marks]

(c) Find the expected number of requests per day.

[2 marks]

[Answer : (a) k =

1 5 13 ; (b) ; (c) ] 4 8 8

92. [STPM ] The probability that an employee of a company is late for work is 0.15 in any working day and 0.35 if it rains. The probability that it rains is 0.24. Calculate (a) the probability that it rains and the employee is late,

[2 marks]

(b) the probability that it rains if the employee is late,

[2 marks]

(c) the probability that the employee is late on at least 2 out of 5 consecutive working days.[4 marks] [Answer : (a) 0.084 ; (b) 0.56 ; (c) 0.1648]

93. [STPM ] The independent Poisson random variables X and Y have parameters 0.5 and 3.5 respectively. The random variable W is defined by W = X − Y . (a) Find E(W ) and Var(W ).

(b) Give one reason why W is not a Poisson random variable.

94. [STPM ] The probability that a heart patient survives after surgery in a country is 0.85.

[4 marks]

[1 marks]

[Answer : (a) -2 , 4]

(a) Find the probability that, out of five randomly chosen heard patients undergoing surgery, four survive. [3 marks] 163

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[Answer : (a) 0.3915 ; (b) 0.97]

95. [STPM ] The continuous random variable X has probability density function r  x−1 , 1≤x≤b f (x) = 12  0, otherwise where b is a constant. (a) Determine the value of b.

[4 marks]

(b) Find the cumulative distribution function of X and sketch its graph. (c) Calculate E(X).

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(b) Using a suitable approximate distribution, find the probability that more than 160 survive after surgery in a random sample of 200 heart patients. [6 marks]

[5 marks] [6 marks]

[Answer : (a) b = 4 ; (c)

14 ] 5

96. [STPM ] The time taken by a manager to travel from his home to his workplace is normally distributed with mean 45 minutes and standard deviation 3 minutes. Determine the time when the manager has to leave his house so that he is 95% confident of arriving at the workplace by 8:00 am. [5 marks] [Answer : 7:10 am]

97. [STPM ] The probability distribution of a random variable X is as shown in the table below.

(a) Determine the value of p.

x

P(X = x)

0

p

1 5 21

2 10 21

3 5 21

4 p

(b) Calculate the mean and variance of X.

[2 marks]

[5 marks]

[Answer : (a)

2 1 ; (b) 2 , ] 42 3

98. [STPM ] The random variable X is normally distributed with mean µ and standard deviation 100. It is known that P (X > 1169) ≤ 0.117 and P (X > 879) ≥ 0.877. Determine the range of the values of µ. [7 marks]

99. [STPM ] The probability that a lemon sold in a fruit store is rotten is 0.02.

[Answer : {µ : 995 ≤ µ ≤ 1050}]

(a) If the lemons in the fruit store are packed in packets, determine the maximum number of lemons per packet so that the probability that a packet chosen at random does not contain rotten lemons is more than 0.85. [5 marks] 164

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(b) If the lemons in the fruit store are packed in boxed each containing 60 lemons, find using a suitable approximation, the probability that a box chosen at random contains less than three rotten lemons. [5 marks] [Answer : (a) 8 ; (b) 0.8795]

100. [STPM ] In a country, one person in 20 is left-handed. Find the probability that, in a random sample of 20 persons, at least three will be left-handed. [5 marks] [Answer : 0.0755]

101. [STPM ] A random variable X is normally distributed with mean µ and variance 25. Find the least value of µ for which P (X ≥ 500) > 0.9. [5 marks]

102. [STPM ]

[Answer : 506.41]

1 The random variable X has a binomial distribution with parameters n = 500 and p = . 2 Using a suitable approximate distribution, find P (|X − E(X)| ≤ 25).

103. [STPM ] The continuous random variable X has probability density function   0, x < 0,   5 f (x) = 4 − x, 0 ≤ x < 1,     1 , x ≥ 1. 4x2 (a) Find the cumulative distribution function of X.

[6 marks]

[Answer : 0.9774]

[7 marks]

(b) Calculate the probability that at least one of two independent observed values of X is greater than three. [4 marks] [Answer : (b)

23 ] 144

104. [STPM ] A car rental shop has four cars to be rented out on a daily basis at RM50.00 per car. The average daily demand for cars is four. (a) Find the probability that, on a particular day,

i. no cars are requested, ii. at least four requests for cars are received.

(b) Calculate the expected daily income received from the rentals.

[2 marks] [2 marks]

[5 marks]

(c) If the shop wishes to have one more car, the additional cost incurred is RM20.00 per day. Determine whether the shop should buy another car for rental. [5 marks] [Answer : (a) (i) 0.0183 , (b) 0.5665 ; (b) 160.93 ; (c) No] 165



(a) Given that E(X) = 6 and Var(X) = (b) Calculate P(X = 5).

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105. [STPM ] A random variable X has a binomial distribution with parameters n and p. 12 , determine the values of n and p. 5

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[4 marks] [3 marks]

[Answer : (a) p = 0.6 , n = 10 ; (b) 0.201]

106. [STPM ] A discrete random variable X takes the values of 0, 1 and 2 with the probabilities of a, b and c 5 4 [6 marks] respectively. Given that E(X) = and Var(X) = , find the values of a, b and c. 3 9 [Answer : a =

1 1 1 ,b= , ] 6 3 2

107. [STPM ] The independent random variable Yi , where i = 1, 2, . . . , n, takes the values of 0 and 1 with the probabilities of q and p respectively, where q = 1 − p. (a) Show that E(Yi ) = p and Var(Yi ) = pq.

[3 marks]

(b) If X = Y1 + Y2 + . . . + Yn , determine E(X) and Var(X). Comment on the distribution of X. [5 marks]

[Answer : (b) np, npq ; binomial distribution]

108. [STPM ] The number of hours spent in a library per week by arts and science students in a college is normally distributed with mean 12 hours and standard deviation 5 hours for arts students, and mean 15 hours and standard deviation 4 hours for science students. A random sample of four arts students and six science students is chosen. Assuming that X is the mean number of hours spent by these 10 students in a week, (a) calculate E(X) and Var(X),

[7 marks]

(b) find the probability that in a given week, the mean number of hours spent by this sample of students is between 11 hours and 15 hours. [3 marks] [Answer : (a) 13.8 , 1.96; (b) 0.0.7814]

109. [STPM ] The time to repair a certain type of machine is a random variable X (in hours). The probability density function is given by   0.01x − p, 10 ≤ x < 20, f (x) = q − 0.01x, 20 ≤ x ≤ 30,   0, otherwise, where p and q are constants. (a) Show that p = 0.1 and q = 0.3.

[6 marks]

(b) Find the probability that the repair work takes at least 15 hours.

[4 marks]

(c) Determine the expected value of X.

[4 marks]

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[Answer : (b)

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(d) If the total cost of repair of the machine comprises a surcharge of RM500 and an hourly rate of RM100, express the total cost of repair in terms of X, and determine the expected total cost of repair. [3 marks] 7 ; (c) 20 hours ; (d) RM 2500] 8

110. [STPM ] A rice dispenser is capable of filling up a cup of rice with an average weight of 130 g. Given that the weight of the rice is normally distributed with a standard deviation of 6 g. (a) Find the percentage number of cups of rice that weigh more than 140g.

[3 marks]

(b) If a cup can hold a maximum of 150 g of rice, find the probability that an overflow occurs.[3 marks] [Answer : (a) 4.78% ; (b) 0.000430 or 0.000429]

111. [STPM ] The probability distribution of the number of cars owned by households in a city is as follows: x P(X = x)

0 0.02

1 m

2 n

3 0.07

Given that the mean number of cars owned is 1.38. (a) Determine the values of m and n. (b) Calculate the variance of X.

(c) Find the probability that a randomly chosen household owns at least two cars.

[4 marks]

[3 marks]

[2 marks]

[Answer : (a) m = 0.65 , n = 0.26 ; (b) 0.4156 ; (c) 0.33]

112. [STPM ] The random variable X is normally distributed with mean 48 and standard deviation 10. Find the least integer k such that P(|X − 48| > k) < 0.3. [5 marks]

113. [STPM ] A discrete random variable X has cumulative distribution function   0, x < 1,    0.6, 1 ≤ x < 3, F (x) =  0.9, 3 ≤ x < 5,    1, x ≥ 5. (a) Construct the probability distribution table for X. (b) Find the mean and variance of X.

167

[Answer : 11]

[2 marks]

[4 marks]

[Answer : 2,1.8]

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114. [STPM ] The lifespan, in months, of a type of bulb is a random variable X. The probability density function is given by   1 xe− x3 , x ≥ 0, f (x) = 9 0, x < 0. (a) Find the cumulative distribution function of X, and hence, sketch the graph.

[6 marks]

(b) Determine the probability that a randomly chosen bulb has a lifespan of more than 9 months. [3 marks]

[Answer : (b) 0.199]

115. [STPM ] The probability that a chicken egg placed in an incubator fails to hatch is 0.01.

(a) Determine the maximum number of eggs that may be placed in the incubator so that the probability that all the eggs hatch is more than 0.75. [6 marks] (b) Using a suitable approximate distribution, find the probability that more than three out of 200 eggs fail to hatch. [5 marks] [Answer : (a) 28 ; (b) 0.1429]

116. [STPM ] Five per cent of credit card holders of a bank do not pay their monthly bills on time. A random sample of 10 credit card holders is taken. (a) Find the probability that at least one card holder do not pay his/her bills on time.

[4 marks]

(b) State the modal number of card holders who do not pay their monthly bills on time. Give a reason for your answer. [2 marks] [Answer : (a) 0.401 ; (b) X = 0]

117. [STPM ] A supervisor claims that five in every 1000 compact discs produced by his factory are defective. The compact discs are packed into boxes containing 200 compact discs in each box. (a) Using a suitable approximate distribution, find the probability that a randomly chosen box i. does not contain any defective compact disc, ii. contains at least four defective compact discs.

[2 marks] [2 marks]

(b) Two boxes are chosen randomly. If X and Y represent the number of defective compact discs in the first and second boxes respectively, calculate P(X + Y = 2) using a suitable approximate distribution. [3 marks] [Answer : (a)(i) 0.3679 , (ii) 0.01899 ; (b) 0.2707]

118. [STPM ] The continuous random variable X has the probability density function   √ 1 , 1 ≤ x ≤ 5, f (x) = 2 2x − 1  0, otherwise. 168

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13 and deduce E(X). 3 (b) Given that E[(2X − 1)2 ] = 24.2, find Var(X).

[4 marks]

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(a) Show that E(2X − 1) =

[3 marks]

[Answer : (a)

61 8 ; (b) ] 3 45

119. [STPM ] There are two boxes which contains four marbles each numbered 1, 2, 3 and 4. One marble is drawn from each box. Let X denotes the difference of the numbers on the marbles. (a) Construct a probability distribution table of X. (b) Calculate E(X). (c) Calculate P(|X − E(X)| ≤ 1).


5 8

[Answer : (b) 1.25 ; (c) ]

120. [STPM ] The mass of a durian from a local farm is normally distributed with a mean of 1.2 kg and a standard deviation of 0.32 kg. The durians are classified as small, medium, or large. The durian is small if its mass is less than 1.0 kg and large if its mass is more than 1.6 kg. (a) Find the percentages of small, medium and large durians in the farm. (b) Calculate the probability that the mass of a small durian is at least 0.7 kg.

(c) Calculate the probability that two randomly selected durians are both small.

[4 marks]

[4 marks]

[2 marks]

(d) Calculate the probability that the total mass of two randomly selected durians is at least 1.7 kg. [4 marks]

[Answer : (a) 26.6%, 62.8%, 10.6% ; (b) 0.778 ; (c) 0.0708 ; (d) 0.939]

121. [STPM ] A discrete random variable X has probability distribution function  x 1     3 , x = 1, 2, 3, 4, P(x) = k, x = 5, 6,    0, otherwise. (a) Determine the value of the constant k.

[2 marks]

(b) State the mode, and calculate the mean of X.

[3 marks]

[Answer : (a)

122. [STPM ] Experience shows that 40% of the throws of a bowler result in strikes.

41 1 ; (b) mode=1 , mean=3 ] 162 2

(a) Find the probability that, out of ten throws made by the bowler, at least two throws result in strikes. [4 marks]

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(b) Find the probability that at most five throws need to be made by the bowler so that four throws result in strikes. [5 marks] [Answer : (a) 0.954 ; (b) 0.08704]

123. [STPM ] The lifespan of a certain torch battery is normally distributed with a mean of 150 hours and a standard deviation of 10 hours. (a) Find the probability that a randomly chosen torch battery has a lifespan of not less than 160 hours. [2 marks] (b) Determine the value of t, to the nearest integer, if it is found that 5% of the batteries are exhausted after t hours. [4 marks] [Answer : (a) 0.1587 ; (b) 134]

124. [STPM ] A discrete random variable X which takes the values of 1, 4, 9, 16 and 25 has cumulative distribution function   0, x < 1,   1 F (x) = n, n2 ≤ x < (n + 1)2 , where n = 1, 2, 3 and 4,  5   1, x ≥ 25. (a) Tabulate the probability distribution of X. (b) Find P(X ≥ 9). √ √ (c) Calculate E( X) and Var( X).


[Answer : (b)

3 ; (c) 3 , 2] 5

125. [STPM ] The result of a diagnostic test for a certain infection may be negative or positive, but the test is not completely reliable. If an individual has the infection, the probability that the result will be negative is 0.01. If an individual does not have the infection, the probability that the result will be negative is 0.95. In a certain population, the percentage of the population affected by the infection is 8%. (a) An individual is chosen at random and tested.

i. Find the probability that the result of the test is negative. [4 marks] ii. If the result of the test is negative, show that the probability of the individual not infected is 0.999, by giving your answer correct to three decimal places. [2 marks]

(b) Estimate the percentage of the population which shows positive results for two independent diagnostic tests. [3 marks] (c) Find the probability that in a sample of ten independent diagnostic tests, more than two results are positive. [4 marks] [Answer : (a) (i) 0.8748 ; (b) 1.57% ; (c) 0.120]

126. [STPM ] There are two boxes which each contain three golf balls numbered 1, 2 and 3. A player draws one ball at random from each box, and the score X of the player is the sum of the numbers on the two balls. 170

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15: Probability Distributions [3 marks]

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(b) Find E(X) and Var(X).

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(a) Determine the probability distribution function of X.

[4 marks]

1 3

[Answer : (b) 4 , 1 ]

127. [STPM ] The amount of time devoted to studying Economics each week by students who achieve a grade A in an examination is normally distributed with a mean of 8.0 hours. (a) Given that 14% of the grade A students study more than 10.7 hours weekly, show that its standard deviation is 2.5 hours. [3 marks] (b) Find the percentage of the grade A students who study less than 5 hours weekly.

[3 marks]

(c) Find the probability that all three randomly selected grade A students study less than 5 hours weekly. [2 marks] [Answer : (b) 11.5% ; (c) 0.00152]

128. [STPM ] The number of defective electrical components per 1000 components manufactured on a machine may be modelled by a Poisson distribution with a mean of 4. (a) Calculate the probability that there are at most 3 defective electrical components in the next 100 components manufactured on the machine. [3 marks] (b) State the assumptions that need to be made about the defective electrical components in order that the Poisson distribution is a suitable model. [2 marks] [Answer : (a) 0.999]

129. [STPM ] A random variable T , in hours, represents the life-span of a thermal detection system. The probability that the system fails to work at time t hour is given by 21

P (T < t) = 1 − e− 5500 t .

(a) Find the probability that the system works continuously for at least 250 hours.

[3 marks]

(b) Calculate the median life-span of the system.

[3 marks]

(c) Find the probability density function of the life-span of the system and sketch its graph.[4 marks] (d) Calculate the expected life-span of the system.

[5 marks]

[Answer : (a) 0.385 ; (b) 181.54 ; (d) 261.9]

130. [STPM ] The probability density function of a continuous random variable X is given by ( ax(b − x), 0 ≤ x ≤ b, f (x) = 0, otherwise, where a and b are positive constants. It has a value of 171

1 at the mean value of X. 2

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(a) Determine the values of a and b.

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(b) Sketch the graph of the probability density function.

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[7 marks] [2 marks]

[Answer : (a) a = 6, b = 1]

131. [STPM ] An insurance company finds that 8% of the people who are insured with the company against a certain risk make claims within a year. (a) Calculate the probability that at least one out of two randomly selected people who are insured against the risk makes a claim within a year. [3 marks] (b) Calculate the probability that at least 3 out of 60 randomly selected people who are insured against the risk make claims within a year. Comment on the validity of the probability model you use. [6 marks] (c) Using an approximate probability distribution, estimate the probability that at least 12 out of 100 randomly selected people who are insured against the risk make claims within a year. Justify your use of this approximation. [6 marks] [Answer : (a) 0.1536 ; (b) 0.8683 ; (c) 0.0985]

132. [STPM ] The continuous random variable X has the probability density function   k, 0 ≤ x < 1,   k f (x) = , 1 ≤ x < 4, 2  x   0, otherwise. 4 (a) Show that k = . 7 (b) Calculate the mean and Var(X).

[3 marks]

[7 marks]

[Answer : (b) 1.08 , 0.743]

133. [STPM ] Studies conducted in the past show that the mean number of serious accidents in a certain factory is one per year. The factory wishes to find the probability of the number of serious accidents occurring on a yearly basis for the purpose of revising the insurance premium of its employees. The factory is required to increase the insurance premium if the probability of having less than two serious accident-free years out of five years is more than 0.35. (a) Define the relevant random variable, and identify its distribution. (b) Find the probability that

i. there is no serious accident occuring in the next one year, ii. there are at least three serious accidents occuring in the next two years. iii. there are less than two accident-free years occurring out of five years.

[3 marks]

[2 marks] [4 marks]

[5 marks]

(c) Based on the result in (b)(iii), state whether the factory is required to increase the insurance premium or not. [1 marks] [Answer : (b) (i) 0.368 ; (ii) 0.323 ; (iii) 0.394] 172

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(a) exactly six calls are received in one hour,

[3 marks]

(b) at least three calls are received in half an hour.

135. [STPM ]

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134. [STPM ] On the average, the number of telephone calls received by the staff at the reservation desk of an airline company is six calls per hour. Assuming that the number of calls are random and independent, find the probability that

[4 marks]

[Answer : (a) 0.160 ; (b) 0.577]

(a) State the parameters of a binomial distribution and the conditions under which a binomial distribution can be approximated by a normal distribution. Determine the parameters of the approximate distribution. [4 marks] (b) An article in an IT magazine states that 25% of hand phones are installed with a certain application software. i. Calculate the probability that exact 3 out of 10 randomly selected hand phones are installed with the application software. [3 marks] ii. Calculate the probability that at most 3 out of 10 randomly selected hand phones are installed with the application software. [3 marks] iii. Calculate the probability that at least 60 out of 200 randomly selected hand phones are installed with the application software. [5 marks] [Answer : (b) (i) 0.2503 ; (ii) 0.776 ; (iii) 0.0605]

136. [STPM ] The discrete random variable X has the probability distribution as follows: x P(X = x)

It is given that P(X ≤ 20) = 0.25. (a) Determine the values of p and q. 2

(b) Find E(X) and E{[X − E(X)] }.

137. [STPM ]

−60 p

20 0.10

24 q

60 0.25

[2 marks] [5 marks]

[Answer : (a) p = 0.15, q = 0.5 ; (b) 20 , 1368]

(a) Describe briefly the standard normal random variable.

[2 marks]

(b) The life span of a certain light bulb is a normal random variable with a mean of 950 hours and a standard deviation of 50 hours. i. Find the probability that a randomly chosen light bulb has a life span of more than 1000 hours. [3 marks]

173

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ii. Determine the value of h such that 99% of the light bulbs have life span between (950 − h) hours and (950 + h) hours. [5 marks] iii. Find the probability that at most one out of eight independently selected light bulbs has a life span of more than 1000 hours. [5 marks] [Answer : (a) 0.1587 ; (b) 128.8 ; (c) 0.630]

138. [STPM ] The number of new members in a fitness centre recruited by an agent in a month has a Poisson distribution with a mean of 2. The fitness centre offers an incentive of RM 1000 if an agent recruits more than five new members in two months. (a) Find the probability that an agent fails to recruit any new members in a month period. [2 marks] (b) Find the probability that an agent receives an incentive in a two-month period.

[4 marks]

(c) Determine the expected amount of incentive that an agent receives in a two-month period.[2 marks] [Answer : (a) 0.135 ; (b) 0.215 ; (c) RM214.87]

139. [STPM ] A stall sells a type of durian either in packets at RM15 per packet or as whole durians at RM8 per kg. On any day, the number of packets sold is a discrete random variable X which has a binomial distribution with parameters n = 10 and p = 0.6, and the weight, in kilograms. of whole durians sold is a continuous random variable Y which has probability density function ( k(y + 1), 0 ≤ y ≤ 25, f (y) = 0, otherwise, where k is a constant.

(a) Find the probability that, on any day, at least 8 packets of durians are sold. [3 marks] 2 (b) Show that k = and calculate the probability that, on any day, the weight of whole durians 675 sold is between 15 kg and 20 kg. [5 marks] (c) Determine the expected revenue of the sale of durians on any day.

[Answer : (a) 0.167 ; (b)

140. [STPM ] The continuous random variable X has probability density function   x − 1, 1 < x ≤ 2, f (x) = k, 2 < x ≤ 3,   0, otherwise. 1 (a) Show that k = . 2 (b) Find the cumulative distribution function.

(c) Sketch the graph of the cumulative distribution function. 5 3 ≤X≤ . (d) Find P 2 2 174

[7 marks]

37 ; (c) RM 220.86] 135

[2 marks]

[4 marks]

[2 marks] [2 marks]

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15: Probability Distributions 5 8

141. [STPM ] A discrete random variable X has cumulative distribution function given by   0, x<0      0.3, 0 ≤ x < 1,    0.6, 1 ≤ x < 2, P(X ≤ x) =  0.8, 2 ≤ x < 3,      0.9, 3 ≤ x < 4,    1, x ≥ 4. (a) Find P(X > 3). (b) Show that E(X) = 1.4. (c) Find P[|X − E(X)| < 1].

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[Answer : (d) ]


[Answer : (a) 0.1 ; (c) 0.5]

142. [STPM ] On average, 48 cars enter a car park of a shopping complex per hour. The number of cars which enter the car park has a Poisson distribution. (a) Find the probability that at most two cars enter the car park in a five-minute interval. [3 marks] (b) Determine time interval, in minutes, if the probability that at least one car enters the car park in this interval is 0.85. [4 marks]

143. [STPM ] A continuous random variable X has probability density function   0 ≤ x ≤ 2,  k, m , 2 < x ≤ 5, f (x) = x2   0, otherwise. It is given that 3P(X ≤ 2)=2P(X > 2).

1 and find the value of m. 5 (b) Find the cumulative distribution function of X. (a) Show that k =

(c) Determine P(1 ≤ X ≤ 3). (d) Calculate the median value of X.

[Answer : (a) 0.238 ; (b) 2.37]

[5 marks]

[5 marks] [2 marks]

[3 marks]

[Answer : (a) m = 2 ; (c)

8 20 ; (d) ] 15 9

144. [STPM ] The average number of electrical appliances returned by customers to hypermarket H in a week is 2. 175

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(a) State an appropriate probability distribution model for the random variable involved and the conditions for this model. [3 marks] (b) Calculate the probability that at least three electrical appliances are returned in a particular week. [3 marks] (c) Calculate the probability that no electrical appliances are returned on a particular Sunday.[3 marks] [Answer : (b) 0.323 ; (c) 0.751]

145. [STPM ] An office sends a number of faxes on every working day. The probability of a fax is successfully sent from the office is 0.8. (a) Find the smallest number of faxes to be sent so that the probability that at least a fax is successfully sent is more than 0.99. [4 marks] (b) Calculate the probability that, out of 15 faxes, the number of faxes successfully sent is more than the expected number. [4 marks] (c) Using a suitable approximation, calculate the probability that, out of 50 faxes, between 38 and 45 faxes are successfully sent. Justify your choice of the approximate distribution. [7 marks] [Answer : (a) 3 ; (b) 0.398 ; (c) 0.6461]

146. [STPM ] The number of car accidents at a particular junction is not more than four accidents in a day. The probability that there is no car accidents in a day is 0.4 and the probabilities that there are at most one, two and three car accidents in a day are 0.7, 0.85 and 0.95 respectively. (a) Construct a probability distribution table for the number of car accidents at the junction in a day. [3 marks] (b) Calculate the mean of the number of car accidents at the junction in a day.

[2 marks]

[Answer : ]

147. [STPM ] In a given population, 25% of the population choose jogging as their leisure activity. (a) Find the probability that, of the 20 people selected at random, i. exactly five people choose jogging as their leisure activity, ii. at most 17 people do not choose jogging as their leisure activity.

[3 marks] [4 marks]

(b) Use a suitable approximation to find the probability that, out of 60 people selected at random, 12, 13, 14 or 15 choose jogging as their leisure activity. Justify the choice of your approximation. [8 marks]

[Answer : ]

148. [STPM ] A binomial random variable X has mean 1 and variance 0.75. Find the parameters and state the probability functions of X. [5 marks] 176

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(a) Find the cumulative distribution function of T .

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149. [STPM ] The probability that the departure time ofa flight, randomly chosen at an airport, is delayed by more 1 than t hours is (t − 2)2 where 0 ≤ t ≤ 2. No flights leave earlier than scheduled and none is delayed 4 for more than 2 hours. The delay time, in hours, for a randomly chosen flight is denoted by T . [3 marks]

(b) Find the probability density function of T and sketch its graph.

[3 marks]

(c) Show that the expected delay time is 40 minutes.

[4 marks]

(d) Find the probability that a flight from the airport is delayed less than 40 minutes. 1 (e) Determine the value of m such that P( T < m)= . 2

[Answer : (d)

[2 marks] [3 marks]

5 ; (e) 0.586] 9

150. [STPM ] A laboratory equipment shop gets 45% of the electronic weighing scale supplies from country Q. Five electronic weighing scales are selected at random. (a) Determine the mean and standard deviation for the number of the electronic weighing scales supplied by country Q. [4 marks] (b) Find the probability that less than three electronic weighing scales are supplied by country Q. [4 marks]

[Answer : ]

151. [STPM ] An investor makes an average of 18 stock transactions in a year. Assume that the probability of transaction is the same for any month and that transactions in one month are independent of transactions in any other month. Find the probability that he makes more than one transaction in a month.[5 marks]

152. [STPM ] The continuous random variable X has probability density function f given by ( cx2 (1 − x), 0 ≤ x ≤ 1, f (x) = 0, otherwise, where c is a constant. (a) Show that c = 12.

(b) Calculate the mean and variance of X. (c) Find P(X >E(X)).

[Answer : ]

[3 marks]

[6 marks] [3 marks]

(d) Find the probability that, out of three independent observed values of X, exactly two are greater than E(X). [3 marks]

177

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[Answer : ]

153. A crossword puzzle is published in a newspaper every day except on Sundays. A man can solve, on average, 8 of the 10 puzzles. (a) Find the expected value and standard deviation of the number of puzzles solved in a week.

(b) Show that the probability that the man solves at least 5 puzzles in a week is 0.655 (correct to 3 significant figures). (c) Given that he solves the puzzle on Monday, find to 3 significant figures, the probability that he can solve at least 4 puzzles in the remaining days of the week. (d) Find, to 3 significant figures, the probability that in 4 weeks, he solves 4 or less puzzles in only one of the 4 weeks.

178

[Answer : (a) 4.8 , 0.98 ; (c) 0.737 ; (d) 0.388]

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16: Sampling and Estimation

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1. [STPM ]

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Sampling and Estimation

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16


(a) All the Upper Six students in a state sat for a special English test. The mean and standard deviation of the marks obtained by the students are 100 and 25 respectively. i. Find the number of students from a random sample of 200 students who are expected to obtain not less than 80 marks but not more than 120 marks. ii. If the 50 best students from the sample will be awarded prizes, find the minimum marks obtained by a student to be awarded a prize.

(b) The monthly wages of the workers at a factory are distributed normally with mean RM500 and standard deviation RM100. i. Find the probability that the mean monthly wage of a sample of 25 workers selected randomly is RM450. ii. Find the size of the random sample required so that the mean of the sample is within a range of RM10 from the mean of the population with a probability of 0.9. [Answer : (a) (i) 118 ; (ii) 117 ; (b) (i) 0.00621 ; (ii) 271]

2. [STPM ] A supermarket reports that its daily sale is distributed normally with mean RM20 000 and standard deviation RM4000. (a) Estimate the number of days, in a period of 25 days selected at random, when the daily sale of the supermarket is less than RM19 000. (b) Calculate the probability that the mean daily sale of the supermarket in a period of 25 days selected at random, is within a range of RM 1000 from RM20 000. (c) Find the maximum estimation error that occurs with a probability of 0.9, for the daily sale of the supermarket in the period of 25 days selected at random. Explain your answer. [Answer : (a) 8 ; (b) 0.789 ; (c) 1316]

3. [STPM ] A drink manufacturer intends to estimate the percentage of Malaysians who likes the flavour of its new drink. In a market survey, 224 people from 400 people who tasted the drink say that they like the flavour of the drink. Obtain a 95% confidence interval for the percentage of Malaysians whose likes the flavour of the new drink. [5 marks] [Answer : (51.14%, 60.86%)]

4. [STPM ] The mass of a pill made by a factory is distributed normally with mean p and standard deviation 1.5 g. (a) If µ = 7 g, find the probability that the mean mass of a random sample of 9 pills is less than 6 g. [4 marks] (b) If the mean mass of a random sample of 9 pills is 6.8 g, obtain a 95% confidence interval for µ. State, giving reasons, whether the managements claim that µ = 7.5 g is true or false. [6 marks]

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(c) Determine the smallest sample size needed so that the value of µ obtained is less than 0.5 g at a 95% confidence level. [5 marks] [Answer : (a) 0.0228 ; (b) (5.82,7.78) ; (c) 35]

5. [STPM ] In a certain bank, the service time for each customer is normally distributed with a mean of 2.5 minutes and a standard deviation of 0.9 minutes. Find the probability that the mean service time for a random sample of 49 customers is at least 2.5 minutes but not more than 2.7 minutes. [4 marks] [Answer : 0.4401]

6. [STPM ] A company sells two brands of batteries A and B. For a random sample of 50 brand A batteries, its 50 50 X X lifespan (in months), are summarised by xi = 1600 and x2i = 51641. For a random sample of i=1

i=1

40 brand B batteries, its lifespan (in months), are summarised by

40 X i=1

yi = 1240 and

40 X

yi 2 = 39064.

i=1

(a) Calculate the unbiased estimate of the mean and variance of the lifespan for brand A battery and brand B battery. Explain what is meant by unbiased estimate. [7 marks] [Answer : (a) 32 , 9 ; 31 , 16]

7. [STPM ] An independent random sample is taken from a normally distributed population with mean 80 and variance 25. Determine the smallest sample size so that the sample mean exceeds the population mean by at least 2 with a probability not exceeding 0.01. [5 marks] [Answer : 34]

8. [STPM ] In a survey of 400 supermarkets throughout the country, it is found that 136 of them sell a daily essential product which contains a certain chemical exceeding the government-approved level. (a) Estimate the percentage of supermarkets in the country which sell the product.

[3 marks]

(b) Obtain a 95% confidence interval for the percentage of supermarkets which sell the product. Give an interpretation of the confidence interval you obtain. [6 marks] (c) Determine the smallest sample size of supermarkets which should be surveyed so that there is a probability of 0.95 that the percentage of supermarkets which sell the product can be estimated with an error of less than 2%. [6 marks] [Answer : (a) 34% ; (b) (29.4%, 38.6%) ; (c) 2156]

9. [STPM ] In a sample of 50 moths from the National Park, there are 27 female moths. Obtain a 95% confidence interval for the proportion of female moths in the National Park. [5 marks] [Answer : (0.4019,0.6781)]

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10. [STPM ] A normal population has a mean of 75 and a variance of 25. Obtain the smallest sample size needed so that the probability that the error of estimation is less than 1 is at least 0.90. [5 marks] [Answer : 68]

11. [STPM ] The height of a certain type of mustard is distributed normally with mean 21.5 cm and variance 90 cm2 . A random sample of size 10 is taken. (a) State the distribution of the sample mean with its mean and variance.

[2 marks]

(b) Find the probability that the sample mean is located between 18 cm and 24 cm.

[3 marks]

[Answer : (b) 0.676]

12. [STPM ] In a survey of 500 motorists on a certain highway, it is found that 120 of them have exceeded the speed limit. (a) Obtain a 95% confidence interval for the proportion of motorists who have exceeded the speed limit on the highway. [5 marks] (b) Determine the smallest sample size which should be surveyed so that the error of estimation is not more than 0.04 at the 90% confidence level. [5 marks] [Answer : (a) (0.203,0.277) ; (b) 309]

13. [STPM ] A survey carried out in an area to estimate the proportion of people who have more than one house. This proportion is estimated using 95% confidence interval. If the estimated proportion is 0.35, determine the smallest sample size required so that estimation error did not exceed 0.03 and deduce the smallest sample size required so that the estimation error did not exceed 0.01. [7 marks] [Answer : 972 , 8740]

14. [STPM ] The mean and standard deviation of the sleeping period of a sample of 100 students chosen at random in a school are 7.15 hours and 1.10 hours respectively. (a) Estimate the mean and standard deviation of the sleeping period of all the students in the school. (b) Estimate the standard error of the mean.

[3 marks]

[1 marks]

[Answer : (a) 7.15 , 1.106 ; (b) 0.1106]

15. [STPM ] A survey carried out by a manufacturer of decorative lamps finds that 136 out of 400 shops sell the decorative lamps at prices less than the recommended prices. (a) Find the 90% symmetric confidence interval for the proportion of shops selling the decorative lamps at prices less than the recommended prices. [5 marks]

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(b) Determine the smallest sample size required so that the estimated proportion of shops selling the decorative lamps at prices less than the recommended prices is within 2% of the actual proportion at the 90% confidence level. [5 marks] (c) Calculate the probability that more than 60% of a random sample of 500 shops sell the decorative lamps at prices less than the recommended prices. [3 marks] [Answer : (a) (0.301,0.379) ; (b) 1519 ; (c) 0]

16. [STPM ] A random sample X1 , X2 , . . . , Xn is taken from a normal population with mean µ and variance 1. ¯ lies within 0.2 of Determine the smallest sample size which is required so that the probability that X µ is at least 0.90. [5 marks] [Answer : 68]

17. [STPM ] A factory receives its supply of raw materials in packages. The mass of each package is normally distributed with mean 300 kg and standard deviation 5 kg. A random sample of four packages is selected. Find the probability that the mean mass of the sample lies between 292 kg and 296 kg. [4 marks]

[Answer : 0.0541 or 0.0548]

18. [STPM ] A telecommunications company wants to estimate the proportion of customers who require an additional line. A random sample of 500 customers is taken and it is found that 135 customers require an additional line. (a) Obtain the 99% symmetric confidence interval for the proportion of customers who require an additional line. Interpret the confidence interval obtained. [6 marks] (b) If the company wants to estimate the proportion of customers who require an additional line at a different location, determine the smallest sample size required so that the error of estimation does not exceed 0.03 at the 95% confidence level. [5 marks]

19. [STPM ] A normal population has mean µ and variance σ 2 .

(a) Explain briefly what a 95% confidence interval for µ means.

[Answer : (a) (0.219,0.321) ; (b) 842]

[2 marks]

(b) From a random sample, it is found that the 95% confidence interval for µ is (−1.5, 3.8). State whether it is true that the probability that µ lies in the interval is 0.95. Give a reason. [2 marks] (c) A total of 120 random samples of size 50 are taken from the population and for each sample a 95% confidence interval for µ is calculated. Find the number of 95% confidence intervals which are expected to contain µ. [1 marks] [Answer : (c) 114]

20. [STPM ] A marketing research firm believes that 40% of the subscribers of a magazine will participate in a competition held by the magazine. A preliminary survey of 100 subscribers is conducted to find out their participation in the competition. 182

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(a) Determine the sampling distribution of the proportion of the subscribers who will participate in the competition, stating its mean and variance. [3 marks] (b) Find the probability that at least 30% of the subscribers will participate in the competition. [3 marks]

[Answer : (b) 0.9839]

21. [STPM ] In a study on the petrol consumption of cars, it is found that the mean mileage per litre of petrol for 24 cars of the same engine capacity is 15.2 km with a standard deviation of 4.2 km. Calculate the standard error of the mean mileage and interpret this standard error. [3 marks] [Answer : 0.876]

22. [STPM ] The lifespan of a type of tyre is normally distributed with mean 70000 km and standard deviation 10 000 km. (a) Determine the probability that a randomly chosen tyre has a lifespan of less than 80 000 km. [2 marks]

(b) Find the probability that the mean lifespan of 10 randomly chosen tyres is more than 68 000 km but less than 75 000 km. [4 marks] (c) Determine the minimum number of tyres to be chosen so that the standard error does not exceed 3500 km at the symmetric 99% confidence interval. [4 marks] [Answer : (a) 0.8413 ; (b) 0.6795 ; (c) 55]

23. [STPM ] A survey is to be carried out to estimate the proportion p of households having personal computers. This estimate must be within 0.02 of the population proportion at a confidence level of 95%. (a) If p is estimated to be 0.12, find the smallest sample size required.

[4 marks]

(b) If the value of p is unknown, determine whether a sample size of 2500 is sufficient.

[4 marks]

[Answer : (a) 1015 ; (b) No]

24. [STPM ] A market survey is conducted at a number of shopping complexes. A random sample of 1250 shoppers are asked whether they consume vitamins and 83% of them say “Yes”. (a) Obtain a symmetric 95% confidence interval for the proportion of shoppers who say “Yes” and interpret this confidence interval. [6 marks] (b) Explain why an interval estimate is more informative than a point estimate.

[2 marks]

[Answer : (a) (0.809,0.851)]

25. [STPM ] The lengths of petals taken from a particular species of flowers have mean 80 cm and variance 30 cm2 . Determine the sampling distribution of the sample mean if 100 petals are chosen at random. [3 marks] Hence, find the probability that the sample mean is at least two standard deviations from the mean. [3 marks]

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[Answer : X¯ ∼ (80, 0.3) ; 0.0456] 26. [STPM ] It is found that 5% of doctors in a particular country play golf. Find, to three decimal places, the probability that, in a random sample of 50 doctors, two play golf. [2 marks] Hence, state the sampling distribution of the proportion of the doctors who play golf, and construct a 98% confidence interval for the proportion. [5 marks] [Answer : 0.2611 , pˆ ∼ (0.05, 0.00095 , (-0.022,0.122)]

27. [STPM ] A researcher wishes to estimate the number of vehicles that pass by a location.

(a) According to a previous study, the standard deviation of the number of vehicles passing by the location per day is 245. Calculate the number of days required so that he is 99% confident that the estimate is within 100 vehicles of the true mean. [3 marks] (b) The standard deviation of the number of vehicles is actually 356. Based on the sample size obtained in (a), determine the confidence level for the estimate to be within 100 vehicles of the true mean. [3 marks] [Answer : (a) 40 days ; (b) 92.4%]

28. [STPM ] In a country, 78% of consumers are in favour of government control over prices. A random sample of 400 consumers is selected. (a) Find the mean and standard deviation of the distribution of the sample proportion.

[3 marks]

(b) Find the probability that the sample proportion is at least 5% lower than the population proportion. [4 marks] [Answer : (a) 0.78 , 0.0207 ; (b) 0.0093 or 0.00787 or 0.00790]

29. [STPM ] A machine is regulated to dispense a chocolate drink into cups. From a random sample of 100 cups of the chocolate drink dispensed, it is found that the cocoa content in one cup of the chocolate drink has mean 5 g and standard deviation 0.5 g. The owner of the machine uses the confidence interval (4.900 g, 5.100 g) to estimate the mean cocoa content in one cup of the chocolate drink. (a) Identify the population parameter under study.

(b) Determine the confidence level for the confidence interval used.

[1 marks]

[5 marks]

[Answer : (b) 95.34%]

30. [STPM ] On the average, a button making machine is known to produce 6% defective buttons. A random sample of 100 buttons is inspected and if eight or more buttons are found to be defective, the operation of the machine will be stopped. (a) State the sampling distribution for the sample proportion of defective buttons.

[1 marks]

(b) Find the probability that the operation of the machine will be stopped.

[3 marks]

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[Answer : (a) pˆ ∼ (0.06, 0.000564) ; (b) 0.264 or 0.2638] 31. [STPM ] A random sample of 200 students of a university is selected and it is found that 120 of them stay in university hostels. (a) Estimate the proportion of the students who stay in the hostels and determine the standard error. [3 marks]

(b) Construct a 95% confidence interval for the proportion of the students who stay in hostels, and interpret your answer. [4 marks] (c) What is the effect on the confidence interval if the confidence level is increased from 95% to 99%? [3 marks]

[Answer : (a) 0.60 , 0.0346 ; (b) (0.532,0.668) ; (c) (0.511,0.689), wider]

32. [STPM ] A census conducted in a school shows that the total hours per week pupils spent watching television has a mean of 16.87 hours and a standard deviation of 5 hours. If a random sample of 100 pupils is taken, find 3 hour of the population mean, 4 (b) the probability that the sample mean is more than 17 hours. (a) the probability that the sample mean is within

[4 marks]

[2 marks]

[Answer : (a) 0.866 ; (b) 0.397]

33. [STPM ] According to a report, 80% of the adult population is in favour of banning cigarettes. A proportion of a random sample of 100 adults is found to be in favour of banning cigarettes. (a) State the sampling distribution.

[2 marks]

(b) Find the probability that the sample proportion in favour of banning cigarettes is i. at least 6% lower than the population proportion, ii. within one standard deviation of the population proportion.

[3 marks] [3 marks]

[Answer : (a) pˆ ∼ (0.8, 0.0016) ; (b) (i) 0.0845 ; (ii) 0.6186 if without continuity correction, (i) 0.0668 ; (ii) 0.683]

34. [STPM ] A food company carries out a market survey in a state on its new flavoured yoghurt. Three hundred randomly chosen consumers taste the yoghurt. Their responses are shown in the table below. Response Number of consumers

Like 195

Dislike 70

Neutral 35

(a) Estimate the proportion of consumers in the state who like the yoghurt. Hence, calculate the probability that the proportion of consumers who like the yoghurt is at least 0.70. [5 marks] (b) Construct a 95% confidence interval for the proportion of consumers in the state who like the yoghurt. [4 marks] 185

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[Answer : (a) 0.65 , 0.0397 ; (b) (0.596,0.704)] 35. [STPM ] In a preliminary sample of 40 postgraduate students in a university, 32 students are satisfied with the services at the main library of the university. (a) Determine the smallest sample size needed to estimate the population proportion with an error not exceeding 0.05 at the 90% confidence level. State any assumption made. [5 marks] (b) State the effect on the sample size

i. if the error is larger than 0.05 with the confidence level unchanged. ii. if the confidence level is higher than 90% with the error unchanged.

[1 marks] [1 marks]

[Answer : (a) 174]

36. [STPM ] The age of women in country A suffering from kidney problems is found to be normally distributed with mean 40 years and standard deviation 5 years. (a) Find the probability that 10 randomly selected women who suffer from kidney problems have the mean age less than 42 years. [3 marks] (b) Find the probability that four randomly selected women who suffer from kidney problems have a total age of more than 145 years. [3 marks] (c) The ages of eight randomly selected women from country B who suffer from kidney problems are as follows: 52, 68, 22, 35, 30, 56, 39, 48. Assuming that the ages of the women who suffer from kidney problems are normally distributed, determine the 95% confidence interval for the mean age of the women. Hence, conclude whether the mean age differs from that of country A, and explain your answer. [6 marks] [Answer : (a) 0.8971 ; (b) 0.9332 ; (c) (33.3,54.2)]

37. [STPM ] A random sample of size n is taken to estimate the mean length of a particular aluminum rod produced by a factory. Assuming that the length of the rod is normally distributed with a standard deviation of 2 mm, determine the smallest value of n so that the width of the confidence interval for the mean length of the rod is 1 mm with a confidence level of at least 90%. [5 marks] [Answer : 44]

38. [STPM ] A random variable X is normally distributed with mean 20 and variance 6.25. The mean of a random ¯ sample of size n is X. ¯ (a) State the sampling distribution of X. ¯ < 18) = 0.0057, find the value of n. (b) If P(X

[1 marks] [4 marks]

6.25 [Answer : (a) X¯ ∼ N (20, ) ; (b) 10] n

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39. [STPM ] According to a recent census, children under 18 years of age spend an average of 16.87 hours per week surfing the Internet with a standard deviation of 5 hours per week. Find the probability that in a random sample of 100 children under 18 years of age, the mean time spent surfing the internet per week is (a) between 16.5 and 17.5 hours, inclusive.

[4 marks]

(b) within 0.75 hour of the population mean,

[3 marks]

(c) at least 0.75 hour lower than the population mean.

[3 marks]

[Answer : (a) 0.6666 ; (b) 0.8664 ; (c) 0.0668]

40. [STPM ] The masses of bags of flour produced in a factory have mean 1.004 kg and standard deviation 0.006 kg. (a) Find the probability that a randomly selected bag has a mass of at least 1 kg. State any assumptions made. [4 marks] (b) Find the probability that the mean mass of 50 randomly selected bags is at least 1 kg. [4 marks] [Answer : (a) 0.7477 ; (b) 1]

41. [STPM ] A population distribution has a mean of 205 and variance of 520. If 25 samples, each of size 40, are taken from this population, (a) calculate the probability that the sample mean is less than 200, (b) determine the number of samples with mean less than 200.

[4 marks]

[2 marks]

[Answer : (a) 0.0827 ; (b) 2]

42. [STPM ] The manufacturer of a closed-circuit television (CCTV) claims that the proportion of households installed with CCTV for security- purposes in a city is 0.15. A random sample of 250 households is taken from the city. (a) Assuming that the claim of the manufacturer is true, calculate the probability that the sample proportion is within 0.05 of the population proportion. [5 marks] (b) If 30 households from the random sample install CCTV. construct a 95% confidence interval for the proportion of households with CCTV. [5 marks] (c) If the number of persons, y, per household for the 30 households from the random sample is 30 30 X X summarised by yi = 130 and yi 2 = 967, construct a 95% confidence interval for the i=1

i=1

average size of households with CCTV.

[5 marks]

[Answer : (a) 0.9664 ; (b) (0.0799, 0.160) ; (c) (2.998,5.669)]

43. [STPM ] A random sample of 100 measurements taken from a population gives the following results: X X x = 2980 and (x − x ¯)2 = 3168. 187

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(a) Determine a 95% confidence interval for the population mean.

[7 marks]

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(b) Suggest two ways to reduce the width of the confidence interval that you obtain.

[2 marks]

44. [STPM ] A random of 15 independent measurements, in Pascal second, of the viscosity of a light machine oil is taken. The values obtained are shown below. 25.2 24.3

24.8 25.1

Find the unbiased point estimates for

25.0 25.3

24.0 25.2

24.5 24.5

24.6 24.4

25.0 24.5

25.6

(a) the mean and variance of the population from which the sample is drawn,

(b) the proportion of population having a viscosity of more than 25.0 Pascal seconds.

[6 marks]

[1 marks]

1 3

[Answer : (a) 24.8 , 0.196 ; (b) ]

45. [STPM ] A health survey is made on daily calcium intake and osteoporosis for senior citizens in a particular area. (a) It is known that the daily calcium intake per person has a mean of 1100 mg and a standard deviation of 450 mg. For a random sample of 50 senior citizens, i. determine the distribution of the sample mean, [3 marks] ii. find the probability that the mean daily calcium intake lies between 970 mg and 1230 mg. [4 marks]

State the effect on the probability in (a)(ii) if the sample size is increased and justify your answer. [2 marks]

(b) It is estimated that 64% of the senior citizens have osteoporosis. If a sample of 125 senior citizens are selected at random, i. determine the distribution of the sample proportion, ii. find the probability that at least 70% of them have osteoporosis.

[3 marks] [3 marks]

[Answer : (a)(i) X¯ N (100, 4050) , (ii) 0.9588 ; (b)(i) pˆ N (0.64, 0.0018432) ; (ii) 0.0962]

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17: Hypothesis Testing

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Hypothesis Testing

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17


1. [STPM ] A random sample of 40 batteries produced by a company is found to have a mean lifespan of 17 months and standard deviation of lifespan of 4 months. Determine, at 5% significance level, whether the mean lifespan of the batteries produced by the company is less than 18 months. [6 marks] [Answer : Test statistic=−1.581 > −1.645]

2. [STPM ] A random sample of 80 voters in an area showed that 57 voters supported party Y . Test, at 2% significance level, the claim by party Y that more than 65% of the voters in that area supported them. [7 marks]

[Answer : Test statistic=1.172 < 2.054]

3. [STPM ] A company sells two brands of batteries A and B. For a random sample of 50 brand A batteries, its 50 50 X X x2i = 51641. For a random sample of xi = 1600 and lifespan (in months), are summarised by i=1

i=1

40 brand B batteries, its lifespan (in months), are summarised by

40 X i=1

yi = 1240 and

40 X

yi2 = 39064.

i=1

(a) Calculate the unbiased estimate of the mean and variance of the lifespan for brand A battery and brand B battery. Explain what is meant by unbiased estimate. [7 marks] [Answer : (a) 32 , 9 ; 31 , 16]

4. [STPM ] A box contains a large number of identical beads of various colours. The proportion of white beads is p. A random sample of size 100 is taken to test the null hypothesis H0 : p = 0.5 against the alternative hypothesis H1 : p < 0.5. If the significance level is fixed as 1%, determine the critical region. [5 marks] [Answer : pˆ < 0.3837]

5. [STPM ] Random variable X is normally distributed with mean µ and variance 36. The significance tests performed on the null hypothesis H0 : µ = 70 versus the alternative hypothesis H1 : µ 6= 70 with a probability of type I error equal to 0.01. A random sample of 30 observations of X are taken and ¯ taken as the test statistic. Find the range of the test statistic lies in the critical region. sample mean X [8 marks]

[Answer : x¯ < 67.18, x¯ > 72.82]

6. [STPM ] The mean and standard deviation of the yield of a type of rice in Malaysia are 960 kg per hectare and 192 kg per hectare respectively. From a random sample of 30 farmers in Kedah who plant this rice, the mean yield of rice is 996 kg per hectare. Test, at the 5% significance level, the hypothesis that the mean yield of rice in Kedah is more than the mean yield of rice in Malaysia. Give any assumptions that need to be made in the test of this hypothesis. [8 marks] [Answer : Test statistic=1.027<1.645] 189

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7. [STPM ] In a survey on the quality of service provided by a bank, 12 out of 150 customers think that the service is unsatisfactory while the other 138 customers think otherwise. Test the hypothesis that the proportion of customers who think that the service is unsatisfactory is 0.10 (a) by using a 99% confidence interval,

[7 marks]

(b) by carrying out a significance test at the 1% significance level.

[6 marks]

Comment on methods (a) and (b) used in the test of the hypothesis.

[2 marks]

[Answer : (a) (0.0229, 0.1371)]

8. [STPM ] The lifespan of a type of bulb is known to be normally distributed with standard deviation 150 hours. The supplier of the bulbs claims that the mean lifespan is more than 5500 hours. The lifespans, in hours, of a random sample of 15 bulbs are as follows. 5260 5600

5400 5780

5820 5520

5530 5500

5380 5360

5460 5620

5510 5430

5520

(a) State the appropriate hypotheses to test the supplier’s claim and carry out the hypothesis test at the 5% significance level. [8 marks] (b) If the true mean lifespan is 5550 hours, find the probability that the test gives a correct decision. [4 marks]

[Answer : (a) Test statistic=0.328<1.645 ; (b) 0.3617]

9. [STPM ] The mean mark of an English test for a random sample of 50 form five students in a particular state is 47.7. A hypothesis test is to be carried out to determine whether the mean mark for all the form five students in the state is greater than 45.0. Using a population standard deviation of 13 marks, carry out the hypothesis test at the 10% significance level. [6 marks] [Answer : Test statistic=1.469]

10. [STPM ] The proportion of fans of a certain football club who are able to explain the offside rule correctly is p. A random sample of 9 fans of the football club is selected and 6 fans are able to explain the offside rule correctly. Test the null hypothesis H0 : p = 0.8 against the alternative hypothesis Hl : p < 0.8 at the 10% significance level. [6 marks] 11. [STPM ] A random sample of 48 mushrooms is taken from a farm. The diameter in centimetres, of each 48 48 X X mushroom is measured. The results are summarised by xi = 300.4 and xi 2 = 2011.01. i=1

i=1

(a) Calculate unbiased estimates of the population mean and variance of the diameters of the mushrooms. [3 marks] (b) Determine a 90% confidence interval for the mean diameter of the mushrooms.

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(c) Test, at the 10% significance level, the null hypothesis that the mean diameter of the mushrooms is 6.5cm. [6 marks] (d) Relate the confidence interval obtained in (b) with the result of the test in (c).

[2 marks]

[Answer : (a) 6.2583 , 2.787 ; (b) (5.862,6.655)]

12. [STPM ] A farmer claims that 95% tomatoes produced in his farm meet the food safety specifications. A random sample of 100 tomatoes is taken from the farm and it is found that 92 tomatoes meet the food safety specifications. Carry out a test, at the 5% significance level, whether the tomatoes from the farm meets the food safety specifications. [7 marks] [Answer : Insufficient evidence to reject the farmer’s claim]

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18: Chi-squared Tests

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Chi-squared Tests

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18


1. [STPM ] A car manufacturing company claims that the colours preferred by its customers are white, grey, blue, and red in the ratio 7 : 3 : 3 : 1. It is found that, from 420 prospective buyers listed in the companys record of order, 225 persons choose white, 90 grey, 85 blue, and 20 red. Based on this record of orders, test whether the claim made by the company can be accepted at the 5% significance level. [6 marks]

2. [STPM ]

2

In a chi-square test, the test statistic is given by χ =

[Answer : Test statistic=4.683]

k X (Oi − Ei )2 i=1

Ei

, where Oi is the observed

frequency, Ei the expected frequency, and k > 2. Show that

where N =

k X i=1

Oi =

k X (Oi − Ei )2 i=1 k X

Ei .

i=1

Ei

=

k X O2 i=1

i

Ei

− N,

[2 marks]

A car model has four colours: white, red, blue, and green. From a random sample of 60 cars of that model in Kuala Lumpur, 13 are white, 14 are red, 16 are blue, and 17 are green. Test, at the 5% significance level, whether the number of cars of each colour of that model in Kuala Lumpur is the same. [5 marks] [Answer : Test statistic=0.668<7.815]

3. [STPM ] In order to investigate whether the level of education and the opinion on a social issue are independent, 1300 adults are interviewed. The following table shows the results of the interviews. Level of education University College High school Total

Opinion on the social issue Agree Disagree 450 18 547 30 230 25 1227 73

Total 468 577 255 1300

Determine, at the 1% significance level, whether the level of education and the opinion on the social issue are independent. [9 marks] [Answer : Test statistic=11.392>9.21]

4. [STPM ] Explain what is meant by significance level in the context of hypothesis testing.

[2 marks]

In a goodness-of-fit test for the null hypothesis that the binomial distribution is an adequate model for the data, the test statistic is found to have the value 19.38 with 7 degrees of freedom. Find the smallest significance level at which the null hypothesis is rejected. [2 marks] [Answer : 1%] 192



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5. [STPM ] What is meant by a contingency table?

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A survey is carried out on a random sample of 820 persons who are asked whether students who break school rules should be caned. The results of the survey are as follows. hhh hh

hhh

Education level High school College University

Opinion

hhhh h

hhh h

Yes

No

Not sure

125 98 100

65 68 80

100 64 120

Carry out chi-squared tests to determine whether educational level is related to the opinion on caning at the significance levels of 1% and 2%. [10 marks] Comment on the conclusions of these tests.

[2 marks]

[Answer : Test statistic=12.59<13.28]

6. [STPM ] In a psychological study, 50 persons are asked to answer four multiple-choice questions. The answers obtained are compared with the predetermined answers. The data is recorded as a frequency distribution as follows: Number of matched answers Frequency

0 10

1 15

2 8

3 5

4 12

A psychologist suggests that the data fits the following probability distribution. Number of matched answers Probability

0 0.08

1 0.10

2 0.15

3 0.25

4 0.42

(a) Calculate the expected frequencies based on the probability distribution.

[2 marks]

(b) Determine whether there is significant evidence, at the 5% significant level, to reject the suggestion. [8 marks] proposed model in (a).

[7 marks]

7. [STPM ] It is thought that there is an association between the colour of a person’s eyes and the reaction of the person’s skin to ultraviolet light. In order to investigate this, each of a random sample of 120 persons is subjected to a standard dose of ultraviolet light. The degree of the reaction for each person is noted, ”-” indicating no reaction, ”+” indicating a slight reaction and ”++” indicating a strong reaction. The results are shown in the table below. ```

```

Reaction

colour ``Eye Blue ``` ```

+ ++

7 29 21

Grey or green

Brown

8 10 9

18 16 2

Test whether the data provide evidence, at the 5% significance level, that the colour of a person’s eyes and the reaction of the person’s skin to ultraviolet light are independent. [10 marks] 193

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Presence Absence

Heavy smokers 22 16

Smoking habit Moderate smokers 25 18

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Symptom of hypertension

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8. [STPM ] The symptom of hypertension and smoking habit for 120 individuals is shown in the contingency table below.

Non-smokers 15 24

Using a chi-squared test, at the 5% significance level, determine whether the symptom of hypertension is independent of smoking habit. [11 marks] [Answer : Test statistic=4.035<5.991]

9. [STPM ] The power lines across a range of mountains are constantly struck by lightning. The number of occurrence per week is recorded for the past 72 weeks. The results obtained are shown below. Number of lightning strikes per week Number of weeks

0 14

1 21

2 17

3 12

≥4 8

Perform a χ2 goodness-of-fit test, at 1% significance level, to determine whether the data fits a Poisson distribution with a mean of 2. [8 marks] You may use the probability distribution of a Poisson random variable with mean 2 given below. Number of occurrence Probability

0 0.1353

1 0.2707

2 0.2707

3 0.1804

≥4 0.1429

[Answer : Test statistic=2.88<13.28]

10. [STPM ] An environmentalist investigates whether there is association between temperature and quality. Temperature and air quality of 200 randomly selected days are recorded as follows:

Temperature

Low Moderate High

Good 25 28 18

Air quality Moderate Unhealthy 20 11 27 16 21 34

Perform a chi-squared test at the 5% significance level to determine whether there is an association between temperature and air quality. [11 marks] [XXAnswer : There is sufficient evidence to conclude that there is an association between temperature and air quality at the 5% significance level.]

11. [STPM ] A survey of 200 participants to study the preferred communication methods of two generations is carried out. The results of the survey is shown in the table below.

194

Generation

X Y

Preferred communication method Internet Phone call SMS 28 43 39 47 17 26


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Determine, at the 5% significance level, whether there is evidence to reject the null hypothesis that preferred communication methods and the generations are independent. [10 marks]

195

New STPM Mathematics (T) Chapter Past Year Question

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