CHAPTER 1: FUNCTIONS SPM 1993
SPM 1995
1. Given the function f : x → 3 – 4x and function g : x → x2 – 1, find (a) f -1 (b) f -1g(3) [5 marks]
1. Given the function f(x) = 3x + c and inverse function f -1(x) = mx + (a) the value of m and c (b) (i) f(3) (ii) f -1f(3)
2. Given the functions f, g and h as a f : x → 2x
2. Given the function f : x → mx + n, g : x → (x + 1)2 – 4 and fg : x → 2(x + 1)2 – 5. Find (i) g2(1) (ii) the values of m and n (iii) gf -1 [5 marks]
h : x → 6x2 – 2
determine function f h(x) find the value of g -1(-2) [7 marks]
3. Function m given that m : x → 5 – 3x2 . If p is a another function and mp given that mp : x → -1 – 3x2, find function p. [3 marks]
SPM 1996 hx + k , x≠2 x −2 2x − 5 and inverse function f -1 : x → , x≠3 x −3
1. Given the function f : x →
SPM 1994 1. Given the functions f(x) = 2 – x and function g(x) = kx2 + n. If the composite function gf(x) = 3x2 – 12x + 8, find (a) the values of k and n [3 marks] (b) the value of g2(0) [2 marks]
Find (a) the values of h and k [3 marks] (b) the values of x where f(x) = 2x [3 marks] 2. Given the function f : x → 2x + 5 and fg : x →13 – 2x, Find (i) function gf (ii) the values of c if gf(c2 + 1) = 5c - 6 [5 marks]
2. The function f is defined as f:x→
[3 marks] [3 marks]
3 g:x→ ,x≠2 x −2
(i) (ii)
4 . Find 3
p +x , for all value of x except 3 + 2x
x = h and p is a constant. (i) determine the value of h (ii) the value of 2 maps by itself under function f. Find (a) the value of p (b) the value of another x which is mapped onto itself (c) f -1(-1) [7 marks]
SPM 1997 1. Given the functions g: x → px + q and g2 : x→ 25x + 48 (a) Find the value of p and q (b) Assume that p>0, find the value of x so that 2g(x) = g(3x + 1) b\
1
SPM 1998
SPM 2001
1. Given the functions h(t) = 2t + 5t2 and v(t) = 2 + 9t Find (a) the value of h(t) when v(t) = 110 (b) the values of t so that h(t) = v-1(2) (c) function hv
1. Given the function f : x → ax + b, a > 0 and f 2 : x → 9x – 8 Find (a) the values of a and b [3 marks] (b) (f -1)2(x) [3 marks] −1
2. Given the function f -1(x) = p − x , x ≠ p and g(x) = 3 + x. Find (a) f(x) [2 marks] -1 2 (b) the value of p if ff (p –1) = g[(2-p)2] ( c) range of value of p so that fg-1(x) = x no real roots [5 marks]
1. Given the functions f(x) = 6x + 5 and g(x) = 2x + 3 , find (a) f g-1(x) (b) the value of x so that gf(-x) = 25 SPM 1999 1. Given the function f : x → k – mx. Find (a) f -1(x) in terms of k and m [2 marks] (b) the values of k and m, if f -1(14) = - 4 and f(5) = -13 [4 marks]
SPM 2002 1. Given the function f(x) = 4x -2 and g(x) = 5x +3. Find (i) fg -1(x)
2. (a) The function g is defined as g : x → x + 3. Given the function fg : x → x2 +6x + 7. Find (i) function f(x) (ii) the value of k if f(2k) = 5k [7 marks]
(ii)
f(x) = 3x2 – 5. Find (a) g(x)
x 2 )= 2 5
[5 marks] 2. (a) Given the function f : x →3x + 1, find f -1(5) [2 marks] (b) Given the function f(x) = 5-3x and g(x) = 2ax + b, where a and b is a constants. If fg(x) = 8 – 3x, find the values of a and b [3 marks]
SPM 2000 1. Given the function g -1(x) =
the value of x so that fg-1(
5 − kx and 3
[2 marks]
(b) the value of k when g(x2) = 2f(-x) [3 marks] 2. Given the function f : x → 4 – 3x. (a) Find (i) f2(x) (ii) (f2)-1(x) (iii) (f -1)2 [6 marks]
2
SPM 2003
3. Given the function h(x) = P = {1, 2, 3} Q = {2, 4, 6, 8, 10}
6 , x ≠ 0 and x
the composite function hg(x) = 3x, find (a) g(x) (b) the value of x so that gh(x) = 5 [4 marks]
1. Based on the above information, the relation between P and Q is defined by set of ordered pairs {(1,2), (1,4), (2,6), (2,8)}. State (a) the image of 1 (b) the object of 2 [2 marks]
SPM 2005 1. In Diagram 1, the function h maps x to y and the function g maps y to z
2. Given that g : x → 5x + 1 and h : x → x2 – 2x +3, find (a) g-1(3) (b) hg(x) [4 marks] SPM 2004 1. Diagram 1 shows the relation between set P and set Q
d∙
∙ ∙ ∙ ∙
e∙ f∙ Set P
Determine (a) h-1(5) (b) gh(2)
w x y z
[2 marks]
2. The function w is defined as w(x) =
5 , x ≠ 2. Find 2 −x
(a) w-1(x) (b) w-1(4)
Set Q Diagram 1
[3 marks]
3. The following information refers to the functions h and g.
State (a) the range of the relation (b) the type of the relation [2 marks]
h : x → 2x – 3 g : x → 4x - 1
2. Given the function h : x → 4x + m and h-1 : x → 2hk +
5 , where m and k are 8
Find gh-1 [3 marks]
constants, find the value of m and of k. [3 marks]
3
SPM 2006 Paper 1 1. In diagram 1, set B shows the image of certain elements of set A
Paper 2 1. Given that f : x → 3x − 2 and x + 1 , find 5 (a) f −1 ( x ) [1 m] −1 (b) f g ( x) [2 m] h ( x ) hg ( x ) = 2 x +6 ( c) such that
g:x→
[3 m] SPM 2007 Paper 1 1. Diagram 1 shows the linear function h. DIAGRAM 1 (a) State the type of relation between set A and set B (b) Using the function notation, write a relation between set A and set B [2 marks] 2. Diagram shows the function h:x →
m−x , x ≠ 0 , where m is a constant x
(a) State the value of m (b) Using the function notation, express h in terms of x [2 m] −
2.
1 2
Given the function , find the value of x such that f ( x) = 5 [2m] f : x →x −3
DIAGRAM 2 Find the value of m [2 marks]
4
3. The following information is about the function h and the composite function
(b) the range of f(x) corresponding to the given domain [3 m]
h2
2. Given the function g : x → 5 x + 2 and h : x → x 2 − 4 x + 3 , find
, where a and b are constants and
a) g −1 (6) b) hg (x ) [4m]
3. Given the functions f ( x) = x −1 and g ( x ) = kx + 2 , find a) f(5) b) the value of k such that gf(5)=14 [3m]
Find the value of a and b [3m] SPM 2008 Paper 1 1. Diagram 1 shows the graph of the function f ( x) = 2 x −1 , for the domain 0 ≤ x ≤ 5 .
State (a) the value of t
5
(2x – 3)(x + 4) + k = 0 and m = 4n, find the value of k [5 marks] 2. Find the values of λ so that (3 – λ)x2 – 2(λ + 1)x + λ + 1 = 0 has two equal real roots. [2 marks] CHAPTER 2: QUADRATIC EQUATIONS SPM 1994
SPM 1997 1. Given that m + 2 and n - 1 are the roots of the equation x2 + 5x = -4. Find the possible value of m and n.
1. If α and β are the roots of the quadratic equation 2x2 – 3x – 6 = 0, form another β α quadratic equation with roots and 3
SPM 1998
3
[4 marks]
1. The equation of px2 + px + 3q = 1 + 2x 1
SPM 1995
have the roots p and q (a) Find the value of p and q (b) Next, by using the value of p and q in (a) form the quadratic equation with roots p and -2q
1. One of the roots of the equation x2 + px + 12 = 0 is one third of the other root. Find the possible values of p. [5 marks] 2. Given that
SPM 1999
1 and -5 are the roots of the 2
1. One of the roots of the equation 2x2 + 6x = 2k - 1 is double of the other root, where k is a constant. Find the roots and the possible values of k. [4 marks]
quadratic equation. Write a quadratic equation in a form ax2 + bx + c = 0 [2 marks] 3. Find the range of value of k if the equation x 2 + kx + 2k − 3 = 0 has no real roots [3 marks]
2. Given the equation x2 – 6x + 7 = h(2x – 3) have two equal real roots. Find the values of h. [4 marks]
4. Prove that the roots of the equation (1 – p)x2 + x + p = 0 has a real and negative roots if 0 < p < 1 [5 marks]
3. Given that α and β are the roots of the equation x2 – 2x + k = 0, while 2α and 2β are the roots of the equation x2 +mx +9=0. Find the possible values of k and m. [6 marks]
SPM 1996 1. Given that a and b are the roots of the equation x2 – (a + b)x + ab = 0. If m and n are the roots of the equation
SPM 2000
6
1. The equation 2x2 + px + q = 0 has the roots -6 and 3. Find (a) the values of p and q [3 marks] (b) the range of values of k if the Equation 2x2 + px + q = k has no real roots [2 marks]
has two distinct roots. Find the range of values of p [3 marks] SPM 2004 1. Form the quadratic equation which has the roots -3 and
the form ax2 + bx + c =0, where a, b and c are constants [2 marks] SPM 2005
SPM 2001 1. Given that 2 and m are the roots of the equation (2x -1)(x + 3) = k(x – 1), where k is a constant. Find the values of m and k [4 marks]
1. Solve the quadratic equation x(2x – 5) = 2x – 1. Give your answer correct to three decimal places. [3 marks]
2. If α and β are the roots of the quadratic equation 2 x 2 + 3x − 1 = 0 , form another quadratic equation with roots 3α + 2 and 3β + 2. [5 marks]
SPM 2006 1. A quadratic equation x 2 + px + 9 = 2 x has two equal roots. Find the possibles values of p. [3 marks]
SPM 2002 1. Given the equation x2 + 3 = k(x + 1) has the roots p and q, where k is a constant, find the range of value of k if the equation has two different real roots. [5 marks] 2. Given that
1 . Give your answer in 2
SPM 2007 1. (a) Solve the following quadratic equation: 3x 2 + 5 x − 2 = 0
α β and are the roots of the 2 2
(c) The quadratic equation hx 2 + kx + 3 = 0, where h and k are constants, has two equal roots Express h in terms of k [4 marks]
equation kx(x – 1) = 2m – x. If α + β = 6 and αβ = 3, find the values of k and m. [5 marks]
SPM 2008 1. It is given that -1 is one of the roots of the quadratic equation
SPM 2003 1. Solve the quadratic equation 2x(x – 4) = (1 – x)(x + 2). Give your answer correct to four significant figures. [3 marks]
x 2 − 4x − p = 0
Find the value of p [2 marks]
2. The quadratic equation x(x + 1) = px - 4
7
(b) Find the range of value of p if x2 – (p + 1)x + 1 – p2 = 0 has no real roots. [3 marks]
CHAPTER 3: QUADRATIC FUNCTIONS SPM 1993
SPM 1995
1. Given the quadratic equation f(x) = 6x – 1 – 3x2. (a) Express quadratic equation f(x) in the form k + m(x + n)2, where k, m and n are constants. Determine whether the function f(x) has the minimum or maximum value and state the value of the minimum or maximum value.
1. Without using differentiation method or drawing graph, find the minimum or maximum value of the function y = 2(3x – 1)(x + 1) – 12x – 1. Then sketch the graph for the function y. [5 marks] 2. Given that 3x + 2y – 1 = 0, find the range of values of x if y < 5. [5 marks]
(b) Sketch the graph of function f(x) (c ) Find the range of value of p so that the equation 6x – 4 - 3x2 = p has two different real roots. [10 marks] SPM 1994
3. Find the range of values of n if 2n2 + n ≥ 1 [2 marks] SPM 1996 1. f(x) = 0 is a quadratic equation which has the roots -3 and p. (a) write f(x) in the form ax2 + bx + c [2 marks] (b) Curve y = kf(x) cut y-axis at the point (0,60). Given that p = 5, Find (i) the value of k (ii) the minimum point [4 marks]
1. In the diagram 1, the minimum point is (2, 3) of the function y = p(x + h)2 + k. Find (a) the values of p, h and k (b) the equation of the curve when the graph is reflected on the x-axis [2 marks]
2. Find the range of values of x if (a) x(x + 1) < 2 [2 marks] (b)
2. (a) Find the range of value of x if 5x ≥ x2 [2 marks]
−3 ≥x 1 −2x
SPM 1997
8
[3 marks]
(b) Find the range of value of x that satisfy inequality (x – 2)2 < (x – 2)
2
1. Quadratic function f(x) = 2[(x – m) + n], with m and n are constants, have a minimum point p(6t,3t2). (a) state the value of m and n in terms of t (b) if t = 1, find the range of value of k so that the equation f(x) = k has a distinct roots
SPM 1999 1. (a) Find the range of value of x so that 9 + 2x > 3 and 19 > 3x + 4 (b) Given that 2x + 3y = 6, find the range of value of x when y < 5 2. Find the range of value of x if (x – 2)(2x + 3) > (x – 2)(x + 2)
2. Find the range of values of x if (a) 2(3x2 – x) ≤ 1 – x (b) 4y – 1 = 5x and 2y > 3 + x
SPM 2000 1. Without using differentiation method or drawing graph, determine the minimum or maximum point of the function y = 1 + 2x – 3x2. Hence, state the equation of the axis of symmetry for the graph. [4 marks]
3. Given that y = x2 + 2kx + 3k has a minimum value 2. (a) Without using differentiation method, find two possible value of k. (b) By using the value of k, sketch the graph y = x2 + 2kx + 3k in the same axis (c) State the coordinate of minimum point for the graph y = x2 + 2kx + 3k
2. The straight line y = 2x + k does not intersect the curve x2 + y2 – 6 =0 . Find the range of values of k [5 marks]
SPM 1998 1.
SPM 2001 1.(a) State the range of value of x for 5x > 2x2 – 3 (b) Given that the straight line 3y = 4 – 2x and curve 4x2 + 3y2 – k = 0. Show that the straight line and the curve does not intersect if k < 4 The graph show two curve y = 3(x-2)2 + 2p and y = x2 + 2x – qx + 3 that intersect in the two point at x-axis. Find (a) the value of p and q (b) the minimum value for the both curve
1
2. Given that f-1 (x) = p − x , x ≠ p and g(x) = 3 + x. Find the range of value of p so that f-1g(x) = x has no real roots SPM 2002 1. Given the quadratic equation x2 + 3 = k(x + 1), where k is a constant, which has the roots p and q. find the range of values of k if p and q
2
2. (a) Given that f(x) = 4x – 1 Find the range of value of x so that f(x) is a positive
9
has two distinct roots. SPM 2005 (paper 1) 2. Given that y = p + qx – x2 = k – (x + h)2 for all values of x (a) Find (i) h (ii) k in terms of p and/or q (b) the straight line y = 3 touches the curve y = p + qx – x2 (i) state p in terms of q (ii) if q = 2, state the equation of the axis of symmetry for the curve. Next, sketch the graph for the curve
1. The straight line y = 5x – 1 does not intersect the curve y = 2x2 + x + p. Find the range of values of p [3 marks] 2. Diagram 2 shows the graph of a quadratic functions f(x) = 3(x + p)2 + 2, where p is a constant.
SPM 2003 (paper 2) 1. The function f(x) = x2 – 4kx + 5k2 + 1 has a minimum value of r2 + 2k, where r and k are constants. (a) By using the method of completing square, show that r = k -1 [4 marks] (b) Hence, or otherwise, find the values of k and r if the graph of the function is symmetrical about x = r2 - 1 [4 marks]
The y = f(x) has the minimum point (1, q), where q is a constant. State
curve
(a) the value of p (b) the value of q (c ) the equation of the axis of symmetry SPM 2005 (paper 1) 1. Diagram 2 shows the graph of a quadratic function f(x)=3(x + p)2 + 2, where p is a constant
SPM 2004 (paper 1) 1. Find the range of values of x for which x(x – 4) ≤ 12 [3 marks] 2. Diagram 2 shows the graph of the function y = -(x – k)2 – 2, where k is a constant.
Diagram 2 The curve y = f(x) has the minimum point (1,q), where q is a constant. State a) the value of p b) the value of q
Find Diagram 2 (a) the value of k (b) the equation of the axis of symmetry (c) the coordinates of the maximum point [3 marks]
10
c)
the equation of the axis of symmetry [3 m]
SPM 2008 (paper 1)
SPM 2006 1. The quadratic function f ( x) = p ( x + q ) 2 + r , where p, q and r are constants, has a minimum value of -4. The equation of the axis of symmetry is x = 3 State a) the range of values of p b) the value of q c) the value of r [3 m]
1. Diagram 3 shows the graph of quadratic function y = f (x) . The straight line y = −4 is a tangent to the curve y = f (x )
a)
write the equation of the axis of
2. Find the range of the value of x for ( x − 3) 2 < 5 − x . [3 m] SPM 2008 (paper 2) 1. Diagram 2 shows the curve of a quadratic function f ( x) = −x 2 + kx − 5 . The curve has a maximum point at B(2,p) and intersects the f(x)-axis at point A
symmetry of the curve b) express f ( x) in the form ( x + b) 2 + c , where b and c are constants. [3 marks] 3. Find the range of the values of x for (2 x −1)( x + 4) > 4 + x
[2 marks] SPM 2007(paper 1) 1. Find the range of values of x for which 2 x 2 ≤ 1 + x [3 marks]
Diagram 2 a) State the coordinates of A
2. The quadratic function f ( x ) = x 2 + 2 x − 4 can be expressed in the form f ( x) = ( x + m) 2 − n , where m and n are constants. Find the value of m and of n [3 marks] Answer m=………….. n=…………..
[1m] b) By using the method of completing square, find the value of k and of p. [4m] c) determine the range of values of x, if f ( x ) ≥ −5
[2m]
11
CHAPTER 4: SIMULTENOUS EQUATIONS SPM 1993 1. Solve the simultaneous equation x2 – y + y2 = 2x + 2y = 10
SPM 1997 1. Given that (3k, -2p) is a solution for the simultaneous equation x – 2y = 4 and 3 2 + 2 y =1. Find the values of k and p x
SPM 1994 1. Solve the following simultaneous equation and give your answer correct to two decimal places 2x + 3y + 1 = 0, x2 + 6xy + 6 = 0
2. Diagram 2 shows a rectangular pond JKMN and a quarter part of a circle KLM with centre M. If the area of the pond is 10 π m2 and the length JK is longer than the length of the curve KL by π m, Find the value of x.
2. Diagram 2 shows a rectangular room. shaded region is covered by perimeter of a rectangular carpet which is placed 1 m away from the walls of the room. If the area and the perimeter of the carpet are 8
3 2 m and 12 m, find the measurements 4
of the room. 1m 1m
1m SPM 1998 1. Solve the simultaneous equation:
1m
x 2 + = 4 , x + 6y = 3 3 y
Diagram 2 SPM 1995 1. Solve the simultaneous equation 4x + y + 8 = x2 + x – y = 2
2. Diagram 2 shows the net of an opened box with cuboids shape. If perimeter of the net box is 48 cm and the total surface area is 135 cm3, Calculate the possible values of v and w.
2. A cuboids aquarium measured u cm × w cm × u cm has a rectangular base. The top part of it is uncovered whilst other parts are made of glass. Given the total length of the aquarium is 440 cm and the total area of the glass used to make the aquarium is 6300 cm2. Find the value of u and w SPM 1996 1. Given that (-1, 2k) is a solution for the equation x2 + py – 29 = 4 = px – xy , where k and p are constants. Determine the value of k and p
12
SPM 1999 1. Given the curve y2 = 8(1 – x) and the
1. Given that x + y – 3 = 0 is a straight line cut the curve x2 + y2 – xy = 21 at two different point. Find the coordinates of the point
y straight line = 4. Without drawing the x
graph, calculate the coordinates of the intersection for the curve and the straight line. 2. Solve the simultaneous equation 2x + 3y = 9 and
2.
x 6y − y = −1 x
yam
SPM 2000 1. Solve the simultaneous equation 3x – 5 = 2y , y(x + y) = x(x + y) – 5 2. Solve the simultaneous equation y y x 3 1 − + 3 = 0 and + − =0 3 2 x 2 2
Pak Amin has a rectangular shapes of land. He planted padi and yam on the areas as shown in the above diagram. The yam is planted on a rectangular shape area. Given the area of the land planted with padi is 115 m2 and the perimeter of land planted with yam is 24 m. Find the area of land planted with yam.
SPM 2001 1. Given the following equation: M = 2x − y N = 3x + 1 R = xy − 8 Find the values of x and y so that 2M = N = R 4. Diagram 2 shows, ABCD is a piece of paper in a rectangular shape. Its area is 28 cm2. ABE is a semi-circle shape cut off from the paper. the perimeter left is 26 cm. Find the integer values of x and y
SPM 2003 1. Solve the simultaneous equation 4x + y = −8 and x2 + x − y = 2 SPM 2004 1. Solve the simultaneous equations p − m = 2 and p2 + 2m = 8. Give your answers correct to three decimal places. SPM 2005 1. Solve the simultaneous equation
[use
π
=
x+
22 ] 7
1 y = 1 and y2 − 10 = 2x 2
SPM 2006
SPM 2002
13
1. Solve the simultaneous equations 2 x 2 + y = 1 and 2 x 2 + y 2 + xy = 5 Give your answer correct to three decimal places [5 m] SPM 2007 1. Solve the following simultaneous equations: 2 x − y − 3 = 0 , 2 x 2 −10 x + y + 9 = 0 [5 m] SPM 2008 1. Solve the following simultaneous equations : x −3 y + 4 = 0 x 2 + xy − 40 = 0
[5m]
CHAPTER 5: INDICES AND LOGARITHMS SPM 1993 1. If 3 − log 10 x = 2log 10 y, state x in terms of y
14
equation T = 30(1.2) x when the metal is heated for x seconds. Calculate (i) the temperature of the metal when heated for 10.4 seconds (ii) time, in second, to increase the temperature of the metal from 30 0 C to 1500 0 C
2. (a) If h = log m 2 and k = log m 3, state in terms of h and /or k (i) log m 9 (ii) log 6 24 (b)Solve the following equations: (i)
4 2x =
1 32
(ii)
log
16 − log
x
x
SPM 1996 1. (a) Express 2 n +2 − 2 n + 10(2 n −1 ) in a simplify terms (c) Solve the equation 3 x +2 − 5 = 0
2=3
SPM 1994 1. Solve the following equations: (a) log 3 x + log 9 3x = −1 1 (b) 8 x +4 = x x +3 4 2 2. (a) Given that log 8 n =
2. (a)Solve the following equations: (i) 4 log 2 x =5 (ii) 2 x . 3 x = 5 x +1 (b) Given that log 5 3 = 0.683 and log 5 7 = 1.209. without using a calculator scientific or four-figure table , calculate (i) log 5 1.4 (ii) log 7 75
1 , find the 3
value of n (b) Given that 2 r = 3 s = 6 t . Express t in terms of r and s ( c) Given that y = kx m where k and m are constants. y = 4 when x = 2 and y = 8 when x = 5. Find the values of k and m
SPM 1997 1. Show that log 3 xy = 2 log 9 x + 2 log 9 y. Hence or otherwise, find the value of x and y which satisfies the equation log 9 xy 3 log 3 xy = 10 and = log 9 y 2
SPM 1995 1. Solve the following equations: (a) 81(27 2 x ) = 1 (b) 5 t = 26.3
2.(a) Find the value of 3 log 3 7 without using a scientific calculator or four figure table.
2. (a) Given that m = 2 r and n = 2 t , state in terms of r and/or t mn 3 32
(b) Solve the equation 5 log x 3 + 2 log x 2 - log x 324 = 4 and give your answer correct to four significant figures.
,
(i)
log 2
(ii)
log 8 m − log 4 n
b) The temperature of a metal increased from 30 0 C to T 0 C according to the
3. (a) Given that 15
using scientific calculator or fourfigure mathematical tables (i) prove that log a 27a = 3.3772 (ii) solve the equation 3 × a n −1 = 3
2 log 3 (x + y) = 2 + log 3 x + log 3 y, show that x 2 + y 2 = 7xy (b) Without using scientific calculator or four-figure mathematical tables, solve the equation log 9 [log 3 (4x – 5)] = log 4 2
SPM 2000 1. (a) Solve 3 log 2 x = 81 (b) If 3 2 x = 8(2 3 x ), prove that
(c ) After n year a car was bought the 7 8
n
9 8
price of the car is RM 60 000 .
x log a = log a 8
Calculate after how many years will the car cost less than RM 20 000 for the first time
log12 49 × log 64 12 2. (a)Simplify log16 7 Without using scientific calculator or four-figure mathematical tables
SPM 1998 1. Given that log x 4 = u and log y 5 = y State log 4 x 3 y in terms of u and/or w
(b) Given that 3 lg xy 2 = 4 + 2lgy lgx
2. (a) Given that log a 3 = x and log a 5 = y. 45 Express log a 3 in terms of x and a y (b) Find the value of log 4 8 + log r r (c ) Two experiments have been conducted to get relationship between two variables x and y. The equation x 3(9 ) = 27 y and log 2 y = 2 + log 2 (x – 2) were obtain from the first and second experiment respectively
with the condition x and y is a positive integer. Show that xy = 10 (c) The total savings of a cooperation after n years is given as 2000(1 + 0.07) n . Calculate the minimum number of years required for the savings to exceed RM 4 000. SPM 2001 1. Given that log 2 k = p and log 3 k = r Find log k 18 in terms of p and r
SPM 1999 1. Given that log 2 3 = 1.585 and log 2 5 = 2.322. Without using scientific calculator or four-figure mathematical tables, Find (a) log 2 45
2. (a) Given that log 10 x = 2 and log 10 y = -1, show that xy – 100y 2 = 9
9 5
(b) log 4
(b) Solve the equation (i) 3 x +2 = 24 + 3 x (ii) log 3 x =log 9 ( 5 x + 6)
2. (a) Given that x = log 2 3, find the value of 4 x . Hence find the value of 4 y if y=1+x (b) Given that log a 3 = 0.7924. Without
SPM 2002 16
1. (a) Given that log 5 3 = k. If 5 2 λ−1 = 15, Find λ in terms of k
2 x −3 = 1. Solve the equation 8
2. Given that
log 2 xy = 2 + 3 log 2 x − log 2 y ,
2. (a) Given that 2 log 4 x − 4 log 16 y = 3 State x in terms of y
express y in terms of x [3 marks] 3. Solve the equation
(b) Solve the simultaneous equation 2 m −1 × 32 k +2 = 16 and 5
×125
3−k
4 x +2
[3 marks]
(b) Solve the equation log 2 ( 7t − 2 ) − log 2 2t = −1
−3 m
1
2 + log 3 ( x + 1) = log 3 x
=1
[3 marks]
where m and k are constants
SPM 2007
SPM 2003,P1 1. Given that log 2 T − log 4 v = 3 , express T in terms of V [4 marks] 2. Solve the equation 4 2 x −1 = 7 x [4 marks]
1. Given that log 2 = x and log 2 c = y , 8b in terms of x and y c
express log 4
[4 marks] 2. Given that 9(3 n −1) = 27 n [3 marks]
SPM 2004,P1 1. Solve the equation 32 4 x = 4 8 x +6 [3 marks]
SPM 2008(paper 1) 1. Solve the equation
2. Given that log 5 2 = m and log 5 7 = p , express log 5 4.9 in terms of m and p
16 2 x −3 = 8 4 x
[3 m] 2. Given that log 4 x = log 2 3 , find the value of x. [3 m]
SPM 2005,P1 1. Solve the equation 2 x + 4 − 2 x +3 = 1 [3 marks] 2. Solve the equation
log 3 4 x − log 3 ( 2 x − 1) = 1
[3 marks] 3.Given that log m 2 = p and log m 3 = r , 27m in terms of p and r 4
express log m
CHAPTER 6: COORDINATE GEOMETRY SPM 2006
SPM 1993 17
1. Solutions to this question by scale drawing will not be accepted Point P and point Q have a coordinate of (4,1) and (2, 4). The straight line QR is perpendicular to PQ cutting x-axis at point R. Find (a) the gradient of PQ (b) the equation of straight line QR ( c) the coordinates of R SPM 1993 2. The above diagram show, a parallelogram KLMN. (a) Find the value of T. Hence write down the equation of KL in the form of intercepts (b) ML is extended to point P so that L divides the line MP in the ratio 2 : 3. Find the coordinates of P SPM 1994 1. From the above diagram, point K(1, 0) and point L(-2, 0) are the two fixed points. Point P moves such that PK:PL = 1:2 (a) Show that the equation of locus P is x 2 + y 2 − 4x = 0
(b) Show that the point M(2, 2) is on the locus P. Find the equation of the straight line KM (c ) If the straight line KM intersects again locus P at N, Find the coordinates of N (d) Calculate the area of triangle OMN
2. (a)The above diagram, P, Q and R are three points are on a line 2 y − x = 4 where PQ : QR = 1:4 Find (i) the coordinates of point P (ii) the equation of straight line passing through the point Q and perpendicular with PR (iii) the coordinates of point R
SPM 1994 1. Solutions to this question by scale drawing will not be accepted. Points A, B, C and D have a coordinates (2, 2), (5, 3), (4, -1) and (p, q) respectively. Given that ABCD is a parallelogram, find (a) the value of p and q (b) area of ABCD
(b) A point S moves such that its distance from two fixed points E(-1, 0) and F(2, 6)
SPM 1993
18
in the ratio 2SE = SF Find (i) the equation of the locus of S (ii) the coordinates of point when locus S intersect y-axis SPM 1995 1. Solutions to this question by scale drawing will not be accepted. 1. In the diagram, the straight line y = 2 x + 3 is the perpendicular bisector of straight line which relates point P(5, 7) and point Q(n, t) (a) Find the midpoint of PQ in terms of n and t (b) Write two equations which relates t and n ( c) Hence, find the distance of PQ
Graph on the above show that the straight line LMN Find (a) the value of r (b) the equation of the straight line passing through point L and perpendicular with straight line LMN 2. The straight line y = 4 x − 6 cutting the curve y = x 2 − x − 2 at point P and point Q (a) calculate (i) the coordinates of point P and point Q (ii) the coordinates of midpoint of PQ (iii) area of triangle OPQ where Q is a origin
2. The diagram shows the vertices of a rectangle TUVW on the Cartesian plane (a) Find the equation that relates p and q by using the gradient of VW (b) show that the area of ∆TVW can
(b) Given that the point R(3, k) lies on straight line PQ (i) the ratio PR : RQ (ii) the value of k
5 2
be expressed as p − q + 10 ( c) Hence, calculate the coordinates of point V, given that the area of rectangular TUVW is 58 units2 (d) Fine the equation of the straight line TU in the intercept form
SPM 1996
SPM 1997
19
(c) A point move such that its distance from point S is point T. (i) (ii)
1 of its distance from 2
Find the equation of the locus of the point Hence, determine whether the locus intersects the x-axis or not
SPM 1998
1. In the diagram, AB and BC are two straight lines that perpendicular to each other at point B. Point A and point B lie on x-axis and y-axis respectively. Given the equation of the straight line AB is 3y + 2x −9 = 0
(a) Find the equation of BC [3m] (b) If CB is produced, it will intersect the xaxis at point R where RB = BC. Find the coordinates of point C [3m] 1. In the diagram, ACD and BCE are straight lines. Given C is the midpoint of AD, and BC : CE = 1:4 Find (a) the coordinates of point C (b) the coordinates of point E (c ) the coordinates of the point of intersection between lines AB and ED produced [3m] 2. Point P move such that distance from point Q(0, 1) is the same as its distance from point R(3, 0). Point S move so that its distance from point T(3, 2) is 3 units. Locus of the point P and S intersects at two points. (a) Find the equation of the locus of P (b) Show that the equation of the locus of point S is x 2 + y 2 − 6 x − 4 y + 4 = 0 ( c) Calculate the coordinates of the point of intersection of the two locus (d) Prove that the midpoint of the straight line QT is not lie at locus of point S
2. The diagram shows the straight line graphs of PQS and QRT on the Cartesian plane. Point P and point S lie on the x-axis and y-axis respectively. Q is the midpoint of PS (a) Find (i) the coordinates of point Q (ii) the area of quadrilateral OPQR [4m] (b)Given QR:RT = 1:3, calculate the coordinates of point T
20
2. The diagram shows the curve y 2 = 16 − 8 x that intersects the xaxis at point B and the y-axis at point A and D. Straight line BC, which is perpendicular to the straight line AB, intersects the curve at point C. Find (a) the equation of the straight line AB [3m] (b) the equation of the straight line BC [3m] (c) the coordinates of point C [4m]
3. In the 1 2
diagram, P(2, 9), Q(5, 7) and R 4 ,3 are midpoints of straight lines JK, KL and LJ respectively, where JPQR forms a parallelogram. (a) Find (i) the equation of the straight line JK (ii) the equation of the perpendicular bisector of straight line LJ [5m] (b) Straight line KJ is produced until it intersects with the perpendicular bisector of straight line LJ at point S. Find the coordinates of point S [2m] (c ) Calculate the area of ∆PQR and hence, find the area of ∆JKL [3m]
SPM 2000
SPM 1999 1. Given point A( −2,−4) and point B ( 4,8) . Point P divides the line segment AB in the ratio 2 : 3. Find (a) the coordinates of point P (b) the equation of straight line that is perpendicular to AB and passes through P.
1. The diagram shows a triangle ABC where A is on the y-axis. The equations of the straight line ADC and BD are y − 3 x +1 = 0 and 3 y + x − 7 = 0 respectively. Find (a) the coordinates of point D (b) the ratio AD : DC
produced=diperpanjangkan
21
gradient of line BC is 1 and straight line AC is
perpendicular to the straight line AB. Find (a) the coordinates of points A and B [1m] (b) the equation of the straight lines AC and BC [5m] (c) the coordinates of point C [2m] (d) the area of triangle ABC [2m]
2. The diagram shows a trapezium ABCD. Given the equation of AB is 3 y − 2 x −1 = 0 Find (a) the value of k [3m] (b) the equation of AD and hence, find the coordinates of point A [5m] (c) the locus of point P such that triangle BPD is always perpendicular at P [2m] SPM 2001 1. Given the points P(8, 0) and Q(0, -6). The perpendicular bisector of PQ intersects the axes at A and B. Find (a) the equation of AB [3m] (b) the area of ∆AOB , where O is the origin. [2m]
3. In the diagram, the equation of BDC is y = −6 . A point P moves such that its distance from A is always
1 the 2
distance of A from the straight line BC. Find (a) the equation of the locus of P (b) the x-coordinates of the point of intersection of the locus and the x-axis [5m]
2. Solutions to this question by scale drawing will not be accepted. Straight line x − 2 y = 6 intersects the x-axis and y-axis at point A and point B respectively. Fixed point C is such that the
22
SPM 2002
[4m] (c) the equation of the straight line that passes through point B and is perpendicular to the straight lineAC [3m] 3. Given A(-1, -2) and B(2, 1) are two fixed points. Point P moves such that the ratio of AP and PB is 1 : 2. (a) Show that the equation of the locus of point P is x 2 + y 2 + 4 x + 6 y + 5 = 0 [2m] (b) Show that point C(0, -5) lies on the locus of point P [2m] (c) Find the equation of the straight line AC [3m] (d) Given the straight line AC intersects the locus of point P at point D. Find the coordinates of point D [3m]
1. The diagram shows a triangle ABC with an area 18 units2 . the equation of the straight line CB is y − x +1 = 0. Point D lies on the x-axis and divides the straight line CB in the ratio m : n. Find (a) the coordinates of point B (b) m : n
SPM 2003(P1) 1. The points A(2h, h) , B ( p, t ) and C ( 2 p,3t ) are on a straight line. B divides AC internally in the ratio 2 : 3 Express p in terms of t [3m] 2. The equations of two straight lines are y x + = 1 and 5 y = 3 x + 24 . 5 3
Determine whether the lines are perpendicular to each other
2. A(1, 3), B and C are three points on the straight line y = 2 x +1 . This straight line is tangent to curve x 2 + 5 y + 2 p = 0 at point B. Given B divides the straight lines AC in the ratio 1 : 2. Find (a) the value of p [3m] (b) the coordinates of points B and C
[3m] 3. x and y are related by the equation y = px 2 + qx , where p and q are constants. A straight line is obtained by plotting Diagram 1. y x
23
y against x, as shown in x
Given that y = 6 x − x 2 , calculate the value of k and of h [3m]
Diagram 1 Calculate the values of p and q [4m] P2(section B) 1. solutions to this question by scale drawing will not accepted. A point P moves along the arc of a circle with centre A(2, 3). The arc passes through Q(-2, 0) and R(5, k). (a) Find (i) the equation of the locus of the point P (ii) the values of k [6m]
2. Diagram 4 shows a straight line PQ with the equation
x y + = 1 . The point P lies 2 3
on the x-axis and the point Q lies on the yaxis
(b) The tangent to the circle at point Q intersects the y-axis at point T. Find the area of triangle OQT [4m]
Find the equation of the straight line perpendicular to PQ and passing through the point Q [3m] 3. The point A is (-1, 3) and the point B is (4, 6). The point P moves such that PA : PB = 2 : 3. Find the equation of the locus of P [3m] P2(section A) 4. Digram 1 shows a straight line CD which meets a straight line AB at the point D . The point C lies on the y-axis
SPM 2004(P1) 1. Diagram 3 shows a straight line graph of y against x x y x
24
P2(section B) 2. Solutions to this question by scale drawing will not accepted.
(a) write down the equation of AB in the form of intercepts [1m] (b) Given that 2AD = DB, find the coordinates of D [2m] (c) Given that CD is perpendicular to AB, find the y-intercepts of CD [3m] SPM 2005(P1) 1. The following information refers to the equations of two straight lines, JK and RT, which are perpendicular to each other. JK RT
: :
(a) Find (i)
the equation of the straight line AB (ii) the coordinates of B [5m] (b) The straight line AB is extended to a point D such that AB : BD = 2 : 3 Find the coordinates of D [2m] (c) A point P moves such that its distance from point A is always 5 units. Find the equation of the locus of P [3m] SPM 2006(P1) 1. Diagram 5 shows the straight line AB which is perpendicular to the straight line CB at the point B
y = px + k
y = ( k − 2) x + p
where p and k are constant Express p in terms of k
[2m]
The equation of the straight line CB is y = 2 x −1
Find the coordinates of B [3 marks]
25
P2(section B) 1. Solutions to this question by scale drawing will not be accepted
SPM 2007 Section A (paper 2) 1. solutions by scale drawing will not be accepted
Diagram 3 shows the triangle AOB where O is the origin. Point C lies on the straight line AB
(a) (b) (c)
In diagram 1, the straight line AB has an equation y + 2 x + 8 = 0 . AB intersects the x-axis at point A and intersects the y-axis at point B
Calculate the area, in unit2, of triangle AOB Given that AC:CB = 3:2, find the coordinates of C A point P moves such that its distance from point A is always twice its distance from point B (i) Find the equation of the locus of P (ii) Hence, determine whether or not this locus intercepts the y-axis
y + 2x + 8 = 0
Diagram 1 Point P lies on AB such that AP:PB = 1:3 Find (a) the coordinates of P [3 m] (b) the equations of the straight line that passes through P and perpendicular to AB [3 m] SPM 2007 (paper 1) 1. The straight line
x y + = 1 has a 6 h
y- intercept of 2 and is parallel to the straight line y + kx = 0 .Determine the value of h and of k [3 marks] 2. The vertices of a triangle are A(5,2), B(4,6) and C(p,-2). Given that the area of the triangle is 30 unit 2 , find the values of p. [3 marks] SPM 2008(paper 1)
26
1. Diagram 13 shows a straight line passing through S(3,0) and T(0,4)
SPM 2008 Section B (paper 2) 1. Diagram shows a triangle OPQ. Point S lies on the line PQ.
(a)
Diagram 13 Write down the equation of the straight line ST in the form
(a) A point W moves such that its
x y + =1 a b
(b)
distance from point S is always 2
A point P(x,y) moves such that PS = PT. Find the equation of the locus of P [4 m]
1 2
units. Find the equation of the locus of W [3m] (b) It is given that point P and point Q lie on the locus of W. Calculate (i) the value of k, (ii) the coordinates of Q [5m] (c) Hence, find the area, in unit2, of triangle OPQ [2m]
2. The points (0,3), (2,t) and (-2,-1) are the vertices of a triangle. Given that the area of the triangle is 4 unit2, find the values of t. [3 m]
27
CHAPTER 7: STATISTICS SPM 1993 1. The mean for the numbers 6, 2, 6, 2, 2, 10, x, y is 5 (a) show that x + y = 12 (b) hence, find the mode for the numbers when (i) x = y (ii) x ≠ y (c) if standard deviation is
1 2
37 , find
the values of x 2. The below table shows the marks obtained by a group of students in a monthly test . Marks Number
1-20
21-40
41-60
61-80
5
8
12
11
81-100 4
students
(a) On a graph paper, draw a histogram and use it to estimate the modal mark (b) By calculating the cumulative frequency, find the median mark, without drawing an ogive (c) Calculate the mean mark SPM 1994 1. The below table shows the marks obtained by a group of students in a monthly test .
28
Marks
1
2
3
4
5
Number of students
4
6
2
x
1
Find (a) the maximum value of x if modal mark is 2 (b) the minimum value of x if mean mark more than 3 (c) the range of value of x if median mark is 2 2. Set A is a set that consist of 10 numbers. The sum of these numbers is 150 whereas the sum of the squares of these numbers is 2890.
The table shows the age distribution of 200 villagers. Without drawing a graph, calculate (i) the median (ii) the third quartile of their ages
Length (mm) Numbers of fish 20-29 2 30-39 3 40-49 7 50-59 12 60-69 14 70-79 9 80-89 3 (a) Find the mean and variance of the numbers in set A (b) If another number is added to the 10 numbers in set A, the mean does not change. Find the standard deviation of these numbers. [6m] SPM 1995 1. (a) Given a list of numbers 3, 6, 3, 8. Find the standard deviation of these number (b) Find a possible set of five integers where its mode is 3, median is 4 and mean is 5.
SPM 1996 1. The list of numbers x −2, x + 4, 2 x + 5, 2 x −1, x + 7 and x − 3 has a mean of 7.Find (a) the value of x (b) the variance [6m] 2. The table shows the length of numbers of 50 fish (in mm) (a) calculate the mean length (in mm) of the fish (b) draw an ogive to show the distribution of the length of the fish Numbers of classes Numbers of pupils 6 35 5 36 4 30 (c) from your graph, find the percentage of the numbers of fish which has a length more than 55 mm
2. (a) The table shows the results of a survey of the number of pupils in several classes in a school. Find (i) the mean (ii) the standard deviation, of the number of pupils in each class (b)
Age 1-20 21-40 41-60 61-80 81-100
Numbers of villagers 50 79 47 14 10
SPM 1997 1. The table shows a set of numbers which has been arranged in an ascending order where m is a positive integer 29
Set numbers Frequency
1
m-1
5
m+3
8
10
1
3
1
2
2
1
SPM 1998 1. The mean of the data 2, k, 3k, 8, 12 and 18 which has been arranged in an ascending order, is m. If each element of the data is reduced by 2, the new median
(a) express median for the set number in terms of m (b) Find the possible values f m (c) By using the values of m from (b), find the possible values of mode
is
Find (a) the values of m and k (b) the variance of the new data
(ii)
[4m] [2m]
2. Set X consist of 50 scores, x, for a certain game with a mean of 8 and standard deviation of 3 (a) calculate Σx and Σx 2 (b) A number of scores totaling 180 with
2. (a) The following data shows the number of pins knocked down by two players in a preliminary round of bowling competition. Player A: 8, 9, 8, 9, 8, 6 Player B: 7, 8, 8, 9, 7, 9 Using the mean and the standard deviation, determine the better player to represent the state based on their consistency [3m] (b) use a graph paper to answer this question The data in the table shows the monthly salary of 100 workers in a company.
(i)
5m . 8
Monthly Salary Numbers of (RM) workers 500-1 000 10 1 001-1 500 12 1 501-2 000 16 2 001-2 500 22 2 501-3 000 20 3 001-3 500 12 3 501-4 000 6 4 001-4 500 2 a mean of 6 and the sum of the squares of these scores of 1 200, is taken out from set X. Calculate the mean and variance of the remaining scores in set X. [7m] SPM 1999 1. The set of numbers integer positive 2, 3, 6, 7, 9, x, y has a mean of 5 and a standard deviation of 6. Find the possible values of x and y
Based on the data, draw an ogive to show distribution of the workers’ monthly salary From your graph, estimate the number of workers who earn more than RM 3 200
2. The frequency distribution of marks for 30 pupils who took a additional mathematics test is shown in the table Marks 20-39
30
Frequency 6
49-59 60-79 80-99
5 14 5
[3m] 2. The table shows the frequency distribution of the marks obtained by 100
(a) By using a graph paper, draw a histogram and estimate the modal mark [4m] (b) Without drawing an ogive, calculate the median mark [3m] (c) Find the mean mark [3m]
Marks Number of students
<20
<30
<40
<50
<60
<70
<80
2
8
21
42
68
87
98
100
pupils Marks Number of pupils 6-10 12 11-15 20 16-20 27 21-25 16 26-30 13 31-35 10 36-40 2 (i) Calculate the variance [3m]
SPM 2000 1. The table shows the results 100 students in a test (a) Based on the table above, copy complete the table below [2m]
Marks Frequency
(b) Without drawing an ogive, estimate the interquartile range of this distribution. [4m]
0-9
(ii) Construct a cumulative frequency table Marks Number of pupils
2. The table shows the distribution of marks in a physics test taken by 120 pupils. Calculate (a) (b) (c)
<10
20-29 2
30-39
40-49
50-59
60-69
70-79
14
35
50
17
2
and draw an ogive to show the distribution of their marks. From the ogive, find the percentage of pupils who scored between 6 to 24. [7m] SPM 2002 1. The table shows the distribution of scores obtained by 9 pupils in a competition. The scores are arranged in an ascending order. Given the mean score is 8 and the third quartile is 11.
the mean [4m] the median [3m] the standard deviation [3m] of the distribution
SPM 2001 1. (a) Given that four positive integers have a mean of 9.When a number y is added to these four integers, the mean becomes 10. Find the value of y [2m] (b) Find the standard deviation of the set of numbers below: 5, 6, 6, 4, 7
Scores
1
3
6
x
y
14
Number of pupils
1
1
2
3
1
1
Find the values of x and y
31
[5m]
2. The table shows the scores obtained by a number of pupils in a quiz. The number of pupils is 40. By drawing an ogive, find Scores
≤
5
≤1
0
≤1
5
≤2
0
≤2
5
≤3
0
≤3
5
Number of pupils
By drawing an ogive, find (a) The median (b) The percentage of excellent pupils if the score for the excellent category is 31.5 SPM 2003,p2 section A 1. A set of examination marks x1 , x 2 , x3 , x 4 , x5 , x 6 has a mean of 5 and a standard deviation of 1.5 (a) Find (i) the sum of the marks,
SPM 2005, paper 1 1. The mean of four numbers is m . The sum of the squares of the numbers is 100 and the standard deviation is 3k Express m in terms of k [3] paper 2,section A
∑x
(ii)
(b)
the sum of the squares of the marks, ∑x 2 [3m] Each mark is multiplied by 2 and then is added to it. Find, for the new set of marks, (i) the mean (ii) the variance [4m]
1. Diagram 2 is a histogram which represents the distribution of the marks obtained by 40 pupils in a test.
SPM 2004,p2 section A 1. A set of data consist of 10 numbers. the sum of the number is 150 and the sum of the squares of the data is 2 472. (a) Find the mean and variance of the 10 numbers [3] (b) Another number is added to the set of data and the mean is increased by 1 Find (i) the value of this number (ii) the standard deviation of the set 11 numbers [4 marks] 32
(a) Without using an ogive, calculate the median mark [3m] (b) Calculate the standard deviation of the distribution [4m]
Score Number of students
<10
<20
<30
<40
<50
2
8
21
42
68
Score 10-19 20-29 30-39 40-49 50-59 60-69 Marks 10-19 20-29 30-39 40-49 50-59
SPM 2006 paper 1 1. A set of positive integers consists of 2, 5 and m. The variance for this set of integers is 14. Find the value of m [3 marks]
Number of pupils 1 2 8 12 k 1 Number of candidates 4 x y 10 8
paper 2,section A 1. Table 1 shows the frequency distribution of the scores of a group of pupils in a game. (a) It is given that the median score of the distribution is 42. Calculate the value of k [3 marks] (b) Use the graph paper to answer this question Using a scale of 2 cm to 10 scores on the horizontal axis and 2 cm to 2 pupils on the vertical axis, draw a histogram to represent the frequency distribution of the scores. Find the mode score [4 marks] (c) What is the mode score if the score of each pupil is increased by 5? [1 mark]
SPM 2007 Paper 2 1. Table 1 shows the cumulative frequency distribution for the scores of 32 students in a competition
Table 1 (a) Based on table 1, copy and complete Table 2
33
Marks Freque ncy
0-9
10-19
20-29
30-39
40-49
Table 2 [1 m] (b) Without drawing an ogive, find the interquatile range of the distribution [5 m] SPM 2007 Paper 1 1. A set of data consists of five numbers. The sum of the numbers is 60 and the sum of the squares of the numbers is 800 Find for the five numbers (a) the mean (b) the standard deviation [3 m]
CHAPTER 8: CIRCULAR MEASURE SPM 1993
SPM 2008(Paper 1) 1. A set of seven numbers has a mean of 9 (a) Find ∑x (b) When a number k is added to this set, the new mean is 8.5 [3m]
1. The diagram shows two arcs, PS and QR, of two circles with centre O and with radii OS and OR respectively. Given the ratio OS:SR = 3:1, Find (a) the angle θ in radian (b) the area of the shaded region PQRS [6m]
SPM 2008(Paper 2) 1. Table 5 shows the marks obtained by 40 candidates in a test. Given that the median mark is 35.5, find the value of x and of y. Hence, state the modal class [6m]
SPM 1994 1.
34
1. The diagram shows a semicircle with centre O and diameter AOC. Find the value of the angle θ (in degrees and minutes) so that the length of arc of the circle AB same with the total of diameter AOC and length of arc of the circle BC
The diagram shows a semicircle ABCD with centre D. Point P move such that PB = BC = BA. Locus for the point P is a circle with centre B. (a) Find the distance of BC (b) Show that the equation of locus P is x 2 + y 2 = 4 x + 6 y + 37 (c) (i) Find the area of major sector BAPC in terms of π (ii) Hence, show that the area of the shaded region is 25(π +1) unit 2
2. The diagram shows a piece of wire in the shape of a sector OPQ of a circle with centre O . The length of the wire is 100 cm. Given the length of arc PQ is 20 cm, find (a) The angle θ in radians (b) the area of sector OPQ [5m] SPM 1996 1.
2. In the diagram, M and N are the centers of two congruent circles with radius r cm respectively. a. Show that ∠PMQ =120 0 b. Find, in terms of π and r, the area of the shaded region [6m] SPM 1995
35
OPQ and ORS of two concentric circle with centre O. Given ∠OPQ =θ rad, the length of arc PQ is twice the length of radius OQ, and the length of radius OS =6 Find (i) the value of θ (ii) the perimeter of the shaded region [4m]
The diagram shows a piece of cake with a uniform cross-section in the shape of a sector OPQ of a circle with centre O and radius 20 cm. The length of arc PQ is 15 cm and the thickness of the cake is 8 cm. Find (a) the angle of this sector in radians (b) the total surface area of the cake [5m]
2.
2. The diagram shows, AOB is a semicircle with centre D and AEB is a length of arc of the sector with centre C. The equation of AB is
The diagram show semicircle PQR with centre O and sector QST of a circle with centre S. Given ST = 5, OR = 4 cm, and the length of arc QT = 4.5 cm. Find ∠QST in rad (a) (b) the area of the shaded region [4m]
x y + =1 12 6
Calculate (a) the area of ∆ABC (b) ∠ACB in radians (c) the area of the shaded region
SPM 1997 1. (a) Convert (i) (ii)
SPM 1998 1.
64020' into radians 4.36 radians into degrees [2m]
(b)
The diagram shows two sectors OPQR and OST of two concentric circle with centre
The diagram shows two sectors
36
O having the same area. Given OPS and OQT are straight lines, ∠POQ = 0.6 rad, OR = 8 cm, and the length of arc PQ same as that arc QR Find (a) the length of PS (b) the length of arc ST
(b)
the area of region swept by the pendulum [2m]
2.
2.
The diagram shows a traditional Malay kite, wau bulan, that has an axis of symmetry OR. Given that APB is an arc of a circle with centre O and radius 25 cm. ANBQ is a semicircle with centre N and diameter 30 cm. TQS is an arc of circle with centre R and radius 10 cm. Given that the length of arc DCF is 18.75 cm. Calculate (a) ∠AOB (b) the area of segment AGBH ( c) the area of the shaded region (Use π = 3.142 )
The diagram shows a sector MJKL with centre M and two sectors PJM and QML, of two circles with centre P and Q respectively. Given the angle of major JML is 3.6 radians. Find (a) the radius of sector MJKL [2m] (b) the perimeter of the shaded region [2m] (c) the area of sector PJM [2m] (d) the area of the shaded region [4m] SPM 1999 1.
SPM 2000 1.
The diagram shows the position of a simple pendulum that swings from P to Q. If the angle POQ is 80 and the length of arc PQ is 14.4 cm, find (a) the length of OQ [3m]
The diagram shows two sectors OAP and OBQ, of two concentric circle with centre O. Given ∠AOP = 0.5 rad, OB = 3 AO , and the ratio of the length of arc AP to the length of arc BQ is 2:3 Calculate ∠BOQ in degrees
37
[5m]
(b) the area of the shaded region [4m]
2. 2.
The diagram shows semicircle DAECF with centre Q and rhombus QAPC. Calculate (a) the radius of semicircle DAECF [1m] (b) the angle θ in radians [3m] (c) the area of sector QAEC [2m] (d) the area of the shaded region [4m]
The diagram shows a circle, KATBL, with centre O and radius 6 cm. KOL is an arc of a circle with centre T. Given AB is parallel to KL, AB = 6 cm and ∠KOL = 1200 (a) Find ∠AOB [1m] (b) Calculate the area of segment ABT [4m] (c) Show that the perimeter of the shaded region is 6 3 + 6 + 2π [5m]
SPM 2001 1.
SPM 2002 1.
The diagram shows a sector, OPQR of a circle with centre O and radius 5 cm. Given the length of arc PQR is 7.68 cm, find (a) ∠POR in radians [2m]
The diagram shows two sectors OAB and OCD of two concentric circles with centre O
38
, where AOD and BOC are straight lines. Given OB = (k + 2) cm, OD = k cm and perimeter of the figure is 35 cm. Find (a) the value of k [3m] (b) the difference between the areas of sector OAB and OCD [2m] 2.
The length of the arc RS is 7.24 cm and the perimeter of the sector ROS is 25 cm. Find the value of θ in rad [3m] paper 2(section A) 1. Diagram 1 shows the sector POQ, centre O with radius 10 cm The point R on OP is such that OR : OP = 3 : 5
In the diagram, ABCD is a rectangle and OAED is a sector of a circle with centre O and radius 6 cm. Given O is the midpoint of AC.Calculate (a) ∠AOD in radians [2m] (b) the perimeter of the shaded region [4m] ( c) the area of the shaded region [4m]
Diagram 1 Calculate (a) the value of θ , in rad, [3m] (b) the area of the shaded region , in cm2. [4m]
SPM 2004 paper 1 1. Diagram 1 shows a circle with centre O SPM 2003 paper 1 1. Diagram 1 shows a sector ROS with centre O
Diagram 1
39
Given that the length of the major arc AB is The length of the minor arc AB is 16 cm and the angle of the major sector AOB is 2900 . Using π = 3.142, find (a) the value of θ , in radians, (Give your answer correct to four significant figures) (b) the length, in cm, of the radius of the circle [3m] paper2 (section B) 1. Diagram 1 shows a sector POQ of a circle, centre O. The point A lies on OP, the point B lies on OQ and AB is perpendicular to OQ. The length of OA = 8 cm and π ∠POQ = radian
45.51 cm, find the length, in cm, of the radius. (use π = 3.142) [3m] paper 2(section B)
6
1. Diagram 4 shows a circle PQRT, centre O and radius 5 cm. JQK is a tangent to the circle at Q. The straight lines, JO and KO, intersect the circle at P and R respectively. OPQR is a rhombus. JLK is an arc of a circle, centre O Calculate (a) the angle α , in terms of π [2m] (b) the length, in cm, of the arc JKL [4m] 2 (c) the area, in cm , of the shaded region
It is given that OA : OP = 4 : 7 (Use π = 3.142 ) Calculate (a) the length, in cm, of AP (b) the perimeter, in cm, of the shaded region, (c) the area, in cm2, of the shaded region SPM 2006 paper 1 1. Diagram 7 shows sector OAB with centre O and sector AXY with centre a
[4m] SPM 2005 paper 1 1. Diagram 1 shows a circle with centre O
40
RAQ is sector of a circle with centre A and has a radius of 14 m.
Diagram 7 Given that OB = 10 cm, AY = 4 cm, ∠XAY =1.1 radians and the lengths of arc AB = 7 cm, calculate (c) the value of θ in radian (d) the area in cm2, of the shaded region
Sector COQ is a lawn. The shaded region is a flower bed and has to be fenced. It is given that AC = 8 cm and ∠COQ =1.956 radians [use π = 3.142] Calculate (a) the area, in m2 of the lawn [2m] (b) the length, in m, of the fence required for fencing the flower bed [4m] (c) the area, in m2, of the flower bed [4m]
paper 2 1. Diagram 4 shows the plan of a garden. PCQ is a semicircle with centre O and has a radius of 8 m.
SPM 2007
Paper 1
1. Diagram 4 shows a sector BOC of a 41
circle with centre O
[5 m] (b) the area in cm , of shaded region [5 m] 2
SPM 2008 paper 1 1. Diagram 18 shows a circle with centre O and radius 10 cm.
It is given that AD = 8 cm and BA =AO = OD = DC = 5 cm Find (a) the length, in cm, of the arc BC (b) the area, in cm 2 , of the shaded region [4 m]
Given that P, Q and R are points such that OP = PQ and ∠OPR = 900, [Use π = 3.142] Find (a) ∠QOR, in radians (b) the area, in cm2 of the coloured Region [4m]
SPM 2007 paper 2 1. Diagram 4 shows a circle, centre O and radius 10 cm inscribed in a sector APB of a circle, centre P. The straight lines, AP and PB, are tangents to the circle at point Q and point R, respectively.
[use π = 3.142] SPM 2008 paper 2 1. Diagram shows two circles. The
Calculate (a) the length, in cm, of the arc AB
42
larger circle has centre X and radius 12 cm. The smaller circle has centre Y and radius 8 cm. The circle touch at point R. The straight line PQ is a common tangent to the circle at point P and point Q.
[use π = 3.142] Given that ∠PXR = θ radian, (a) show that θ = 1.37 (to two decimal places) [2m] (b) calculate the length, in cm of the minor arc QR [3m] (c) calculate the area, in cm2, of the colored region. [5m]
CHAPTER 9: DIFFERENTATION SPM 1993
43
1. Given that f ( x ) =
3
1 − 2x 2 , find f '(x) 4x − 3
3. Given p = 2 x − 3 and y = − p 2 . Find (a) the approximate change in x if thE rate of change in p is 3 units per second
SPM 1994 1. (a) Given that y = 3 x 2 + 5 , find
dy using dx
(b)
the first principle (c) Find
(c) the small change in y, when x decreases from 2 to 1.98
d 1 dx 2 x +1
SPM 1997
16 dy 2. Given y = 4 , find if x = 2 . dx x Hence,
estimate the value of
dy in terms of x dx
1. (a) Find the value of
(b) Given f ( x ) = ( 2 x − 3) 5 find f ′′ (x)
16
(1.98)
4
2. Differentiate y =
SPM 1995 1. Given f ( x) =
lim n 2 − 4 n → 2 n − 2
4 − 3 using the first x
principle
1 − 2x3 find f ' (x) x −1
3. (a)
2. Given y = x(3 − x) , express d2y dy y 2 +x + 12 in terms of x. dx dx Hence, find the value of x that satisfy the d2y dy equation y 2 + x + 12 dx dx 3. Find the coordinates at the curve y = (2 x − 5) 2 where the gradient of the normal for the curve is
The
1 4
SPM 1996 diagram shows a container in the shape of a pyramid. The square base of the pyramid has an area of 36 cm2 and the height of the pyramid is 4 cm. Water is poured into the container so that its surface area is 4p2 cm2 and its height from the vertex of the pyramid is h cm.
1. Differentiate x 4 (1 + 3 x) 7 with respect to x. 2. The gradient of the curve y = hx +
k at x2
7 2
the point −1,− is 2. Find the values of h and k
44
(i) Show that the volume of the container that is filled with water is V =
2.
3 (64 − h 3 ) 4
(ii) If the rate of change in the height of water is 0.2 cm s-1, calculate the rate of change in the volume of the space that is not filled with water if h = 2 cm. [5m] (b)
The diagram shows a wooden block consisting of a cone on top of a cylinder with radius of x cm. Given the slant height of the cone is 2x cm. and the volume of the cylinder is 24 π cm 3 a) Prove that the total surface area of the block, A cm 2 , is given by
The diagram shows a rectangle JKLM inscribed in a circle. Given JK = x cm and KL = 6 cm (a) show that the area of the shaded region, A cm2, is given by
2 A = 3π x +
16 x
[3m]
b)Calculate the minimum surface area of the block [3m]
πx 2 A= − 6 x + 9π 4
(b) Calculate the value of x so that the area of the shaded region is a minimum
c) Given the surface area of the block changes at a rate of 42 π cm 2 s −1 . Find the of change of its radius when its radius is 4 cm. [2m]
[5m]
d) Given the radius of the cylinder increases from 4 cm to 4.003 cm. find the approximate increase in the surface area of the block [2m]
SPM 1998 1. Given that f ( x ) = 4 x (2 x −1) 5 , find f ' ( x)
SPM 1999 1. Given f ( x) =
(x
f ' (0)
)
5
−2 , find 1 − 3x 2
[4m]
2. Given y = t − 2t 2 and x = 4t + 1 (a)Find
dy , in terms of x dx
(b) If x increases from 3 to 3.01,
45
2. Given y = 3 x 2 − 4 x + 6 . When x = 5 , x increases by 2%. Find the corresponding rate of change of y. 3. Find the equation of the tangent to the curve y = 2 x 2 + r at the point x = k . If the tangent passes through the point (1,0), find r in terms of k
find the corresponding small increase in t. [2m] 3 (a)
4.(a) The straight line 4 y + x = k is the normal to the curve y = ( 2 x −1) 2 − 3 at point A. Find (i) the coordinates of point A and the value of k (ii) the equation of the tangent at
The diagram shows a box with a uniform cross section ABCDE . Given AB = ED = (30-6x) cm, BC = 3x cm, CD = 4x and AF = 2 cm (i) Show that the volume of the box V cm 3 , is given by equation V = 300 x − 48 x 2 (ii) Calculate (b) the value of x that makes V a maximum (c) the maximum value of V
point A
3 (b) A piece of wire 60 cm long is bent to form a circle. when the wire is heated, its length increases at a rate of 0.1 cm s −1 (use π = 3.142 ) (i) Calculate the rate of change in the radius of the circle (ii) Hence, calculate the radius of the circle after 4 second
4.(b) The diagram shows a toy in the shape of a semicircle with centre O. Diameter AB can be adjusted so that point C which lies on the circumference can move such that AC + CB = 40 cm. Given that AC = x cm and the area of triangle ABC is A
SPM 2000 1. Differentiate the following expressions with respect to x (a) 1 + 3 x 4 [2m] 2x + 5 (b) 4 x +3
cm, find an expressions for
dL in dx
terms of x and hence, find the maximum area of triangle ABC
[2m]
SPM 2001
46
1. Given f (r ) = of f (r ) when
4 + 3r find limited value 5 − 2r
(b)
Given y = 2 x 3 − 5 x 2 + 7 , find the value of
r→ ∞
dy at the point dx
(2,
3). Hence, find (i) the small change in x, when y decreases from 3 to 2.98 (ii) the rate of change in y, at the instant when x = 2 and the rate of change in x is 0.6 unit per second [5m] SPM 2002
2. Given that graph of function k f ( x) = hx 3 + 2 has gradient function x 96 f ' ( x) = 3 x 2 − 3 where h and k are x constants, Find a. the values of h and k b. x-coordinate of the turning point of the graph of the function 3. (a)
5 2
3 2 1. Given p = (1 + t ) + t
dp and hence find the values of t dt dp =7 where dt
Find
2.
y = 2x – x2
The diagram shows a circle inside rectangle ABCD such that the circle is constantly touching the two sides of the rectangle. Given the perimeter of ABCD is 40 cm a. Show that the area of the shaded region A = 4 +π 2
(a) The diagram shows the curve y = 3 x − x 2 that passes through the origin. Given straight lines AB and PQ touch the curve at point O and point R respectively, where AB and PQ are perpendicular to each other. Find the coordinates of point R [4m]
20 y −
y 4 b. Using π = 3.142 , find
(b) A drop of ink falls on a piece of paper and forms an expanding ink blot in the shape of a circle. (i) If the radius of the ink blot increases at a constant rate of 18 mm for every 6 second, find the rate of change in the area of ink blot at the instant when its
the length and width of the rectangle that make the area of the shaded region a maximum
47
radius is 5 mm (ii) Using differentiation, find the approximate value of the area ink blot at the instant when its radius is 5.02 mm
Calculate the rate of change of the height of the water level at the instant when the height of the water level is 0.4 m (use π = 3.142; Volume of a cone =
SPM 2003 paper2(section A)
[4m]
SPM 2004 1. Differentiate 3 x 2 (2 x − 5) 4 with respect to x [3m]
1. Given that y = 14 x(5 − x) , calculate (a) the value of x when y is a maximum (c) the maximum value of y [3m]
2. Two variables x and y are related by the equation y = 3x +
2 . x
Given that y increases at a constant rate of 4 units per second, find the rate of change of x when x = 2 [3m]
2. Given that y = x 2 + 5 x , use differentiation to find the small change in y when x increases from 3 to 3.01 [3m]
paper 2(section B) 3. The gradient function of a curve which passes through A(1, -12) is 3x 2 − 6 x . Find (a) the equation of the curve [3m] (b) the coordinates of the turning points of the curve and determine whether each of the turning points is a maximum or a minimum [5m]
dy = 2 x + 2 and 3. (a) Given that dx y = 6 when x = −1 , find y in terms of
x [3m] (b) Hence, find the value of x if x2
1 2 πr h ) 3
d2y dy + ( x − 1) + y =8 2 dx dx
SPM 2005
[4m] paper2(sectionB) 4. (a) Diagram 2 shows a conical container of diameter 0.6 m and height 0.5m.Water is poured into the container at a constant rate of 0.2 m3 s-1
1. Given that h( x) = h" (1)
1
( 3x − 5) 2 , evaluate [4m]
2. The volume of water, V cm3, in a 1 3
container is given by V = h 3 + 8h , where h cm is the height of the water in the container. Water is poured into the container at the rate of 10 cm3 s-1. Find the rate of change of the height of water, in cm s-1, at the instant when its height is 2 cm [3m] 48
1. The curve y = f (x) is such that paper2(sectionA) 3. A curve has a gradient function px 2 − 4 x , where p is a constant. The tangent to the curve at the point (1,3) is parallel to the straight line y + x − 5 = 0 . Find (a) the value of p (b) the equation of the curve SPM 2006 Paper 1 1. The point P lies on the curve y = ( x − 5) 2 . It is given that the gradient of the normal at P is −
dy = 3kx + 5 , where k is a constant. dx The gradient of the curve at x = 2 is 9
Find the value of k [2 m] 2. The curve y = x 2 − 32 x + 64 has a
minimum point at x = p , where p is a constant. Find the value of p [3 m]
SPM 2008 Paper 1 1. Two variables x and y are related by the 16 equation y = 2 . x Express, in terms of h, the approximate change in y when x changes from 4 to 4 + h, where h is a small value [3m]
1 4
Find the coordinates of P [3m] 2 3 dy u = 3 x − 5 . Find in terms of x dx
2. It is given that y = u 7 , where
[4m] 2. The normal to the curve y = x 2 − 5 x at point P is parallel to the straight line y = −x +12 . Find the equation of the normal to the curve at point P. [4m]
3. Given that y = 3x 2 + x − 4 (a) find the value of
dy when x =1 dx
(b) express the approximate change in y, in terms of p, when x changes from 1 to 1 + p, where p is small value SPM 2007 Paper 2 1. A curve with gradient function 2 x −
2 x2
has a turning point at (k, 8) (a) Find the value of k [3 m] (b) determine whether the turning point is a maximum or minimum point [2 m] ( c) find the equation of the curve [3 m] SPM 2007 Paper 1
49
SPM 1993 1.
(a) if the fence want to build along the boundary BC, calculate the total length is needed CHAPTER 10: SOLUTION OF TRIANGLE (b) Calculate ∠BCA (c ) Given that the area of ∆FCG same with the area ∆ADE . Calculate the length of GC SPM 1994 1.
The diagram shows a ∆PQR (a) Calculate obtuse angle PQR
[2m]
(b) Sketch and label another triangle which is different from triangle PQR in the diagram, where the lengths of PQ and QR as well as angle PQR are maintained. [1m]
In the diagram, BCD is a straight line, calculate the length of CD
(c ) If the length of PR is reduced while the length of PQ and angle PQR are maintained, calculate the length of PR so that only one [2m] ∆PQR can be form
2.
2.
The diagram shows a pyramid with ∆ABC as the horizontal base. Given that AB = 3 cm, BC = 4 cm and ∠ABC = 90 0 and vertex D is 4 cm vertically above B, calculate the area of the slanting face. [5m]
The diagram shows a land form triangle, ABC, divide by three parts. ADB, BFC, and AEGC is a straight line Given that sin ∠BAC =
12 13
50
In the diagram, points A, B, C, D and E lie on a flat horizontal surface. Given BCD is a straight line, ∠ACB is an obtuse angle and the area of ∆ADE = 20 cm2, calculate (a) the length of AD (b) ∠DAE
SPM 1995 1.
SPM 1997
In the diagram, sin ∠ADC =
4 where 5
∠ADC is an obtuse angle. Calculate
1. The diagram shows a triangle ABC Calculate (a) the length of AB (b) the new area of triangle ABC if AC is lengthened while the lengths of AB, BC and ∠BAC are maintained [3m]
(a) the length of AC correct to two decimal places [3m] (b) ∠ABC [2m] SPM 1996 1.
SPM 1998
The diagram shows a cuboid. Calculate (a) ∠JQL [4m] (b) the area of ∆JQL [2m] 1. In the diagram, BD = 5 cm, BC = 7cm, CD = 8 cm and AE = 12 cm, BDE and ADC are a straight lines. Find (a) ∠BDC (b) the length of AD
2.
51
(a) ∠JLK (b) the area of ∆JKL 2.
SPM 2000
1. The diagram shows a cyclic quadrilateral ABCD. The lengths of straight lines DC and CB are 3 cm and 6 cm respectively. Express the length of BD in terms of (a) α (b) β Hence, show that cos α =
The diagram shows a pyramid VABCD with a square base ABCD. VD is vertical and base ABCD is horizontal. Calculate (a) ∠VTU (b) the area of plane VTU
11 29
2.
SPM 1999 1.
In the diagram, PQR is a straight line. Calculate the length of PS The diagram shows a trapezium ABCD Calculate (a) ∠CBD (b) the length of straight line AC 2. JKL is a triangle with side JK = 10 cm. Given that sin ∠KJL = 0.456 and sin ∠JKL = 0.36 , Calculate
52
SPM 2001 1.
The diagram shows a prism with a uniform triangular cross-section PTS. Given the volume of the prism is 315 cm3. Find the total surface area of the rectangular faces
The diagram shows a pyramid with a triangular base PQR whish is on a horizontal plane. Vertex V is vertically above P. Given PQ = 4 cm, PV = 10 cm, VR = 15 cm and
[5m]
∠VQR = 80 0
SPM 2003
Calculate (a) the length of QR (b) the area of the slanting face
1. The diagram shows a tent VABC in the shape of a pyramid with triangle ABC as the horizontal base. V is the vertex of the tent and the angle between the inclined plane VBC and the base is 500
SPM 2002 1.
The diagram shows a quadrilateral ABCD. Given AD is the longest side of triangle ABD and the area of triangle ABD is 10 cm2 Calculate (a) ∠BAD (b) the length of BD ( c) the length of BC
Given that VB = VC = 2.2 m and AB = AC = 2.6 m, calculate (a) the length of BC if the area of the
2. 53
base is 3 m2 (b) the length of AV and the base is 250 (c ) the area of triangle VAB
SPM 2005 SPM 2004 1. The diagram shows triangle ABC 1. The diagram shows a quadrilateral ABCD such that ∠ABC is acute
(a) Calculate the length, in cm, of AC [2m]
(a) Calculate (i) ∠ABC (ii) ∠ADC (iii) the area, in cm2, of quadrilateral ABCD [8m]
(b) A quadrilateral ABCD is now formed so that AC is a diagonal, ∠ACD = 40 0 and AD = 16 cm Calculate the two possible values of ∠ADC [2m]
(b) A triangle A'B'C' has the same measurements as those given for triangle ABC, that is, A'C' = 12.3 cm, C'B' = 9.5 cm and ∠B 'A'C' = 40.50, but which is different in shape to triangle ABC
(c ) By using the acute ∠ADC from (b), calculate (i) the length, in cm, of CD (ii) the area, in cm2, of the quadrilateral ABCD
(i) Sketch the triangle A'B'C' (ii) State the size of ∠ A'B'C' [2m]
54
cyclic quadrilateral=sisi empat kitaran vertical = mencancang horizontal = mengufuk obtuse angle = sudut cakah slanting face = permukaan condong acute = tirus formed = dibentuk diagonal = pepenjuru SPM 2006
SPM 2006
1. Diagram 5 shows a quadrilateral ABCD
1. Diagram 7 shows quadrilateral ABCD
Diagram 5 The area of triangle BCD is 13 cm2 and ∠BCD is acute Calculate (a) ∠BCD [2 m] (b) the length, in cm, of BD [2 m] (c) ∠ABD [3 m] (d) the area, in cm2, quadrilateral ABCD [3 m]
i.
Calculate (a) the length, in cm, of AC (b) ∠ACB [4 M] ii. Point A’ lies on AC such that A’ B = AB (i) sketch ∆A ’BC (ii) calculate the area, in cm 2 , of ∆A ’BC [6 M]
55
Monthly expenses Food House rental Entertainment CHAPTER 11: INDEX NUMBER Clothing Others SPM 1993 1. The table below shows the monthly expenses of Ali’s family Year Expenses Food Transportation Rental Electricity & water
1998
1992
RM 320 RM 80 RM 280 RM 40
RM 384 RM 38 RM 322 RM 40
Price Index 130 115 110 115 130
Calculate (a) the composite price index, correct to the nearest integer, of the monthly expenses in the Yusnis’ household (b) the total monthly expenses in the year 1993, correct to the nearest ringgit, if the total monthly Item Shirt Trousers Bag Shoes
Find the composite index in the year 1992 by using the year 1998 as the base year. Hence, if Ali’s monthly income in the year 1998 is RM 800, find the monthly income required in the year 1992 so that the increases in his income is in line with the increases in his expenses [5m]
Price Index Weightage 100 N 110 6 140 2 100 4 expenses of the Yusris’ household in the year 1990 is RM 850
SPM 1995 1. The table below shows the price indices and weightages of four items in the year 1994 based on the year 1990. Given the composite price index in the year 1994 is RM 114
SPM 1994 1. The pie chart below shows the distribution of the monthly expenses in the Yusnis’ household in the year 1990. The table that follows shows the price indices in the year 1993 based on the year 1990
Calculate (a) the value of n (b) the price of a shirt in 1994 if its price in 1990 is RM 40 SPM 1996 1. (a) In the year 1995, the price and price index of a kilogram of a certain grade of rice are RM 2.40 and 160. Using the year 1990 as the base year, calculate the price of a kilogram of rice in the year 1990. 56
[2m]
(b) the value n
SPM 1998 1. The price index of a certain item in the year 1997 is 120 when 1995 is used as the Index number, Ii 105 94 120 Weightages, Wi 5-x x 4 base year and 150 when 1993 is used as the base year. Given the price of the item in the year 1995 is RM 360, calculate its price in the year 1993 SPM 1999
(b) The above table shows the price indices in the year 1994 using 1992 as the base year, changes to price indices from the year 1994 to 1996 and their weightages respectively. Item
Price Index 1994
Changes to Price index from 1994 to 1996
Weightages
180 116 140 124
Increases 10% Decreases 5% No change No change
5 4 2 1
Wood Cement Iron Steel
1. The composite index number of the data in the below table is 108 Find the value of x
1. The table below shows the price indices and weightages of 5 types of food items in the year 1998 using the year 1996 as the base year. Given the composite price index in the 1998, using the year 1996 as the base year, is 127. Food item Fish Prawn Chicken Beef Cuttlefish
SPM 1997 1. The below table shows the price indices and weightages of three items in the year 1995 based on the year 1990.Given the price of item R in the year 1990 and 1995 are RM 30 and RM 33 respectively, and the composite price index in the year 1995 is 130. Price index 120 150 m
Calculate (a) the value of m
[4m]
SPM 2000
Calculate the composite price index in the year 1996 [3m]
Item P Q R
[2m]
Price index
Weightages
140 120 125 115 130
4 2 4 3 X
Calculate (a) the value of x [4m] (b) the price of a kilogram of chicken in the year 1998 if the price of a kilogram of chicken in the year 1996 is RM 4.20 [6m]
Weightage 2 n 3 [2m] 57
Type of item A B C
SPM 2001 1. The table below shows the prices indices of items A, B, and C with their respective weightages. Given the price of P in the year 1996 is RM 12.00 and increases to RM 13.80 in the year 1999. By using 1998 as the base year, calculate the value of x. Hence, find the value of y if the composite price index is 113 Item A B C
Price index x 98 123
Weightage 5 y 14 - y
(RM)
A B C
Year 1999 55 40 80
Year 2000 66 x 100
Price Index (Base year 1999) 120 150 125
Weightage % Y X 2x
(a) Using the year 1996 as the base year, calculate the price indices of items A, B and C (b) Given the composite price index of these items in the year 1998 based on the year 1996 is 140, find the values of x and y [5m]
1. The table below shows the prices, price indices and the number of three items Pric e
Price (RM) in 1998 105 100 67.50
2. The table below shows the prices of three items A, B and C in the year 1996 and 1998, as well as their weightages
SPM 2002
Item
Price (RM) in 1996 70 80 60
Number of items 200 500 y
(a) Find the value x (b) If the composite price index of the three items in the year 2000 using year 2000 as the base year is 136.5, find the value of y
58
( c) The total monthly cost of the items in the year 1990 is RM 456 (d) The cost of the items increases by 20% from the year 1995 to the year 2000. Find the composite index for the year 2000 based on the year 1990
SPM 2004
1. The diagram below show is a bar chart indicating the weekly cost of the items P, Q, R, S and T for the year 1990. Table 1 shows the prices and the price indices for the items.
1. The table below shows the price indices and percentage of usage of four items, P, Q, R and S which are the main ingredients in the production of a type of biscuits
WEEKLY COST (RM)
SPM 2003
(a) Calculate (i) the price of S in the year 1993 if its price in the year 1995 is RM 37.70 (ii) the price index of P in the year 1995 based on the year 1991 if its price index in the year 1993 based on the year 1991 is 120 [5m]
35 30 25 20 15 10 5 0 P
Q
R
S
T
Item
ITEMS
Items
P Q R S T
Price in 1990
Price in 1995
x RM 2.00 RM 4.00 RM 6.00 RM 2.50
RM 0.70 RM 2.50 RM 5.50 RM 9.00 z
P Q R S
Price Index in 1995 based on 1990 175 125 y 150 120
Price index for the year 1995 based on the year 1993 135 x 105 130
Percentage of usage (%) 40 30 10 20
(b) The composite index number of the lost of biscuits production for the year 1995 based on the year 1993 is 128. Calculate (i) the value of x (ii) the price of a box of biscuits in the year 1993 if the corresponding price in the year 1995 is RM 32 [5m]
(a) Find the value of (i) x (ii) y (iii) z (b) Calculate the composite index for the items in the year 1995 based on the year 1990
59
(c) The cost of making these biscuits is expected to increase by 50% from the year 2004 to the year 2007 Find the expected composite index for the year 2007 based on the year 2001 [2m]
SPM 2005 1. The table below shows the prices and the price indices for the four ingredients P, Q, R and S used in making biscuits of a particular kind. Diagram below shows a pie chart which represents the relative amount of the ingredients P, Q, R, and S used in making these biscuits Ingredients
P Q R S
Price per kg Year Year 200 2004 1 0.80 1.00 2.00 y 0.40 0.60 z 0.40
SPM 2006 1. A particular kind of cake is made by using four ingredients P, Q, R and S. Table shows the prices of the ingredients
Ingredient P Q R S
Price index for the year 2004 based on the year 2001 x 140 150 80
Price per kilogram (RM) Year 2004 Year 2005 5.00 W 2.50 4.00 x Y 4.00 4.40
(a) The index number of ingredient P in the year 2005 based on the year 2004 is 120. Calculate the value of w [2m] (b) The index number of ingredient R in the year 2005 based on the year 2004 is 125. The price per kilogram of ingredient R in the year 2005 is RM 2.00 more than its corresponding price in the year 2004. Calculate the value of x and of y [3m]
(a) Find the value of x, y and z [3m] (b) (i) calculate the composite index for the cost of making these biscuits in the year 2004 based on the year 2001 (ii) Hence, calculate the corresponding cost of making these biscuits in the year 2001 if the cost in the year 2004 was RM 2985 [5m]
(c) The composite index for the cost of making the cake in the year 2005 based on the year 2004 is 127.5 Calculate (i) The price of a cake in the year 60
2004 if its corresponding price in the year 2005 is RM30.60 (ii) the value of m if the quantities of ingredients P, Q, R and S used are in the ratio of 7 : 3 : m : 2 [3m]
(c) The price of each component increases by 20% from the year 2006 to the year 2008 Given that the production cost of one toy in the year 2004 is RM 55, calculate the corresponding cost in the year 2008 [4m]
Component P Q R S T
Price (RM) for the year 1.20 x 4.00 3.00 2.00
1.50 2.20 6.00 2.70 2.80
Price index for the year 2006 based on the year 2004 125 110 150 y 140
SPM 2007 1. Table 4 shows the prices and the price indices of five components, P, Q, R, S and T, used to produce a kind of toy Diagram 6 shows a pie chart which represents the relative quantity of components used Diagram 6 (a) Find the value of x and of y [3m] (b) Calculate the composite index for the production cost of the toys in the year 2006 based on the year 2004 [3m]
61
