SPM Add Maths pass year question

CHAPTER 1: FUNCTIONS

 SPM 1993

 SPM 1995

1. Given the function  f :  f : x → 3 – 4 x and 2 function g  function g :: x → x  – 1, find (a) f  (a) f -1 (b) f  (b) f -1 g (3) [5 marks] marks]

1. Given the function f  function f ( x)  x) = 3 x + c and 4 inverse function f  function f -1( x)  x) = mx + . Find 3 (a) the value of m of m and c [3 marks] (b) (i) f  (i) f (3) (3) (ii) f  (ii) f -1 f (3) (3) [3 marks]

2. Given the functions f  functions f , g and  g and h as a  f :  f : x → 2x 3  g :  g : x → , x ≠ 2 x−2 h : x → 6 x2 – 2 (i) (ii)

2. Given the function f  function f :: x → mx + n,  g :  g : x → ( x  x + 1)2 – 4 and  fg :  fg : x → 2( x  x + 1)2 – 5. Find (i)  g 2(1) (ii) the values of  m  m and n -1 (iii)  gf  [5 marks]

determine function f h( h( x)  x) -1 find the value of g    (-2) [7 marks] marks]

3. Function m given that m : x → 5 – 3 x2 . If  p  p is a another function and mp given that mp : x → -1 – 3 x2, find function p function p.. [3 marks]

 SPM 1996 

1. Given the function f  function f :: x →

hx + k 

, x≠2  x≠2 x−2 2x − 5 and inverse function f  function f -1 : x → , x≠3  x≠3 x −3 Find (a) the values of h of h and k  [3 marks] (b) the values of  x  x where f  where f ( x)  x) = 2 x [3 marks]

 SPM 1994

1. Given the functions f  functions f ( x)  x) = 2 –  x  x and 2 function g  function g ( x)  x) = kx + n. If the composite function gf  function gf ( x)  x) = 3 x2 – 12 x + 8, find (a) (a) the the value valuess of k  of k and and n [3 marks] 2 (b) (b) the the value value of  g  (0) [2 marks]

2. Given the function f  function f :: x → 2 x + 5 and  fg :  fg : x →13 – 2 x,  x, Find (i) function gf  (ii) the values of  c  c if  gf   gf (c2 + 1) = 5c 5c - 6 [5 marks]

2. The function f is defined as  p + x  f :  f : x → , for all value of  x  x except 3 + 2x  x = h and p and p is a constant. (i) determine the value of h of h (ii) the value value of 2 maps by itself itself under  function f  function f . Find (a) (a) the the valu valuee of  p (b) the value value of anot another  her  x which is mapped onto itself  (c)  f -1(-1) [7 marks]

 SPM 1997 

1. Given the functions g: functions g: x → px + q and 2  g  : x→  x→ 25 x + 48 (a) Find the value of  p of  p and q (b) Assume that p that p>0, >0, find the value of  x  x so that 2 g ( x)  x) = g  = g (3 (3 x + 1)  b\

1

 SPM 1998  SPM 2001 2

1. Given the functions h(t ) = 2t  2t + + 5t  5t  and v(t ) = 2 + 9t  9t  Find (a) (a) the the valu valuee of h of h(t ) when v(t ) = 110 (b) the values values of t  of t so so that h(t ) = v-1(2) (c) (c) funct functio ion n hv

1. Given the function f  function f :: x → ax + b, a > 0 2 and  f  : x → 9 x – 8 Find (a) (a) the the value valuess of a of a and b [3 marks] -1 2 (b) ( f   f  ) ( x)  x) [3 marks]

1. Given the functions f  functions f ( x)  x) = 6 x + 5 and  g ( x)  x) = 2 x + 3 , find (a)  f g -1( x)  x) (b) (b) the the value value of  x so that gf  that gf ((- x)  x) = 25

2. Given the function f  function f -1( x)  x) =

−1 , x ≠ p  p − x

and g  and g ( x)  x) = 3 + x + x.. Find (a) f  (a) f ( x)  x) [2 marks] -1 2 (b) the value of  p  p if  ff   ff  ( p  p  –1) = g  = g [(2[(2- p)  p)2]  SPM 1999 ( c) range of value of  p  p so that fg  that fg -1( x)  x) = x = x no real roots 1. Given the function f  function f :: x → k –  k – mx mx.. Find [5 (a) f  (a) f -1(x) in terms of k  of k and and m [2 marks] marks] (b) the values of k  of k and and m, if  f   f -1(14) = - 4 and f  and f (5) (5) = -13 [4 marks] SPM 2002 2. (a) The function g  function g is is defined as  g :  g : x → x + 3. Given the function  fg :  fg : x → x2 +6 x + 7. Find (i) function f ( x)  x) (ii) the value of  k   k if  if  f   f (2k  (2k ) = 5k  5k  [7 marks]

1. Given the function f  function f ( x)  x) = 4 x -2 and  g ( x)  x) = 5 x +3. Find (i)  fg -1( x)  x) x 2 (ii) the value of x   so that fg  that fg -1( ) = 2 5 [5 marks] 2. (a) Given the function f  function f :: x →3 x + 1, find -1  f  (5) [2 marks] (b) Given the function f  function f ( x)  x) = 5-3 x and  g ( x)  x) = 2ax 2ax + b, where a and b is a constants. If  fg  If  fg ( x)  x) = 8 – 3 x,  x, find the values of a of a and b [3 marks]

 SPM 2000

1. Given the function g  function g -1( x)  x) = f ( x)  x) = 3 x2 – 5. Find (a)  g ( x)  x)

5 − kx 3

and

[2 marks]

(b) (b) the the value value of k  of k when when g   g ( x  x2) = 2 f ((- x)  x) [3 marks] 2. Given the function f  function f :: x → 4 – 3 x.  x. (a) Find 2 (i)  f  ( x)  x) (ii) ( f 2)-1( x)  x) -1 2 (iii) ( f  ) [6 marks]

2

3. Given the function h( x)  x) =

6

, x ≠ 0 and x the composite function hg ( x)  x) = 3 x,  x, find (a)  g ( x)  x) (b) (b) the the value value of  x so that gh that gh(( x)  x) = 5 [4 marks]

 SPM 2003

P = {1, 2, 3} Q = {2, 4, 6, 8, 10} 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) (a) the the image image of of 1 (b) the object object of 2 [2 marks]

 SPM 2005

1. In Diagram 1, the function h maps x maps x to y to y and the function g  function g maps maps y  y to z  to z 

2. Given that g  that g :: x → 5 x + 1 and h : x → x2 – 2 x +3, find (a) g  (a) g -1(3) (b) hg ( x)  x) [4 marks]  SPM 2004

1. Diagram 1 shows the relation between set P and set Q

d ∙ d ∙

∙ ∙ ∙ ∙

e∙

Determine (a) h-1(5) (b) gh (b)  gh((2)

w  x  y  z 

2. The function w is defined as 5 w( x)  x) = , x ≠ 2. Find 2−x (a) w-1( x)  x) -1 (b) w (4) [3 marks]

 f ∙  f ∙ Set P

[2 marks]

Set Q

3. The following information refers to the functions h and g. and g.

Diagram 1

State (a) the range range of the the relation relation (b) the type type of the the relation relation [2 marks]

h : x → 2 x – 3  g :  g : x → 4 x - 1

2. Given the function h : x → 4 x + m and 5 h-1 : x → 2hk  2hk + + , where m and k are k are 8 constants, find the value of m of m and of k  of k . [3 marks]

Find gh Find gh-1 [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 → 3 x − 2 and  x  g : x → + 1 , find 5 −1 (a)  f  ( x) [1 m] − (b)  f   g ( x) [2 m] ( c) h( x) such that hg ( x) = 2 x + 6 [3 m] 1

 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 m − x h : 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]

−

1

2.

2

Given the function  f  : x →  x − 3 , find the value of   x such that  f ( x) = 5

DIAGRAM 2

[ [2m]

Find the value of m [2 marks]

4

(a) the value of t  (b) the range of  f ( x) corresponding to the given domain [3 m]

3. The following information is about the function h and the composite function h 2

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

−1

a)  g  (6)  b) hg ( x) [4m] x − 1 and

Find the value of a and b

3. Given the functions  f  ( x) =  g ( x) = kx + 2 , find a)  f (5)  b) the value of k such that  gf (5)=14 [ [3m]

[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

5

equation x2 – (a + b) x + ab = 0. If  m and n are the roots of the equation (2 x – 3)( x + 4) + k = 0 and m = 4n, find the value of k  [5 marks]

CHAPTER 2:

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] UADRATIC E UATIONS

 SPM 1994

 SPM 1997  1. Given that m + 2 and n - 1 are the roots of the equation x2 + 5 x = -4. Find the  possible value of m and n.

1. If α and β  are the roots of the quadratic equation 2 x2 – 3 x – 6 = 0, form another  quadratic equation with roots

β 

and

α   SPM 1998

3 3 [4 marks]

1. The equation of  px2 + px + 3q = 1 + 2 x 1 have the roots and q  p (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

 SPM 1995

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

1

 SPM 1999

and -5 are the roots of the

2 quadratic equation. Write a quadratic equation in a form ax2 + bx + c = 0 [2 marks]

1. One of the roots of the equation 2 x2 + 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]

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 – 6 x + 7 = h(2 x – 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 – 2 x + 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

6

 SPM 2000

2. The quadratic equation x( x + 1) = px - 4 has two distinct roots. Find the range of  values of  p [3 marks]  SPM 2004 1. Form the quadratic equation which has 1 the roots -3 and . Give your answer in 2 2 the form ax + bx + c =0, where a, b and c are constants [2 marks] SPM 2005

2

1. The equation 2 x + 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 2 x2 + px + q = k has no real roots [2 marks]

 SPM 2001

1. Given that 2 and m are the roots of the equation (2 x -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(2 x – 5) = 2 x – 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 + 3 x − 1 = 0 , form another  quadratic equation with roots 3α + 2 and 3β + 2. [5 marks]

 SPM 2006 

1. A quadratic equation 2  x +  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

α

and

 SPM 2007 

1. (a) Solve the following quadratic equation: 3 x 2 + 5 x − 2 = 0

β

(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]

are the roots of the 2 2 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  x 2 − 4 x −  p = 0 Find the value of  p [2 marks]

 SPM 2003 1. Solve the quadratic equation 2 x( x – 4) = (1 –  x)( x + 2). Give your answer correct to four  significant figures. [3 marks]

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2. (a) Find the range of value of  x if  5 x ≥  x2 [2 marks] (b) Find the range of value of  p if   x2 – ( p + 1) x + 1 –  p2 = 0 has no real roots. [3 marks]

CHAPTER 3:

UADRATIC FUNCTIONS

 SPM 1993

 SPM 1995

1. Given the quadratic equation  f ( x) = 6 x – 1 – 3 x2. (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(3 x – 1)( x + 1) – 12 x – 1. Then sketch the graph for the function y. [5 marks] 2. Given that 3 x + 2 y – 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 6 x – 4 - 3 x2 = 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] −3 ≥ x (b) 1 − 2 x

8

2. (a) Given that f ( x) = 4 x2 – 1 Find the range of value of  x so that  f ( x) is a positive (b) Find the range of value of  x that satisfy inequality ( x – 2)2 < ( x – 2)

[3 marks]  SPM 1997 

1. Quadratic function f ( x) = 2[( x – m)2 + n], with m and n are constants, have a minimum point p(6t ,3t 2). (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 + 2 x > 3 and 19 > 3 x + 4 (b) Given that 2 x + 3 y = 6, find the range of value of  x when y < 5 2. Find the range of value of  x if  ( x – 2)(2 x + 3) > ( x – 2)( x + 2)

2. Find the range of values of  x if  (a) 2(3 x2 –  x) ≤ 1 –  x (b) 4 y – 1 = 5 x and 2 y > 3 + x

SPM 2000 1. Without using differentiation method or  drawing graph, determine the minimum or  maximum point of the function  y = 1 + 2 x – 3 x2. Hence, state the equation of the axis of symmetry for the graph. [4 marks]

2

3. Given that y = x + 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 = 2 x + 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  5 x > 2 x2 – 3 (b) Given that the straight line 3 y = 4 – 2 x and curve 4 x2 + 3 y2 – 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 + 2 p and y = x2 + 2 x – 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

2. Given that f -1 ( x) =

1

, x ≠ p and  p − x  g ( x) = 3 + x. Find the range of value of  p so that f -1 g ( x) = x has no real roots

 SPM 2002

1. Given the quadratic equation

9

 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 has two distinct roots.

(b) the equation of the axis of symmetry (c) the coordinates of the maximum point [3 marks]

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

 SPM 2005 (paper 1)

1. The straight line y = 5 x – 1 does not intersect the curve y = 2 x2 + 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 + 5k 2 + 1 has a minimum value of r 2 + 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 = r 2 - 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.

Find Diagram 2 (a) the value of k 

Diagram 2 The curve y = f(x) has the minimum point

10

a)  b)

(1,q), where q is a constant. State the value of p the value of q c) the equation of the axis of  symmetry [3 m]  SPM 2006 

 SPM 2008 (paper 1)

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 2 quadratic function  f ( x) = − x + 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]

Dia gram 2

 SPM 2007(paper 1)

a) State

1. Find the range of values of  x for  which 2 x 2 ≤ 1 + x [3 marks]

coordinates of A [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]

2. The quadratic function  f ( x) =  x 2 + 2 x − 4 can be expressed in the form  f ( x) = ( x + m) − n , where m and n are constants. Find the value of m and of n [3 marks] Answer m=………….. n=………….. 2

11

the

the value of k and p CHAPTER 4: SIMULTENOUS EQUATIONS  SPM 1993 1. Solve the simultaneous equation  x2 –  y + y2 = 2 x + 2 y = 10

 SPM 1997  1. Given that (3k , -2 p) is a solution for the simultaneous equation x – 2 y = 4 and 2 3 + =1. Find the values of k and p  x 2 y

 SPM 1994 1. Solve the following simultaneous equation and give your answer correct to two decimal places 2 x + 3 y + 1 = 0,  x2 + 6 xy + 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 3 8 m2 and 12 m, find the measurements 4 of the room. 1m 1m

1m  SPM 1998 1. Solve the simultaneous equation:  x 2 + = 4 , x + 6 y = 3 3  y

1m Diagram 2  SPM 1995 1. Solve the simultaneous equation 4 x + y + 8 =  x2 + x –  y = 2

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

12

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.

[use π  =

22 7

]

 SPM 2002

 SPM 1999 1. Given the curve y2 = 8(1 –  x) and the  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  x 6 y 2 x + 3 y = 9 and − = −1  y  x  SPM 2000 1. Solve the simultaneous equation 3 x – 5 = 2 y ,  y( x + y) = x( x + y) – 5

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

2. yam

2. Solve the simultaneous equation  y  y 1  x 3 − + 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 = 2 x − y  N = 3 x + 1  R = xy − 8 Find the values of  x and y so that 2 M = 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 4 x + y = −8 and x2 + x − y = 2  SPM 2004

13

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 1  x +  y = 1 and y2 − 10 = 2 x 2

 SPM 2006  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]

14

(a) 81(27 2 x ) = 1 (b) 5 t  = 26.3 2. (a) Given that m = 2 r  and n = 2 t  , state in terms of r and/or t  (i)

 mn 3     log 2   , 32    

(ii) log 8 m − log 4 n CHAPTER 5: INDICES AND LOGARITHMS  b) The temperature of a metal increased from 30 0 C to T 0 C according to the 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

 SPM 1993 1. If 3 − log 10  x = 2log 10  y, state x in terms of  y

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: 1 (i) 4 2 x = 32 (ii)

log

 x

16 − log

 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

2. (a)Solve the following equations: (i) 4 log 2  x =5 (ii) 2  x . 3  x = 5 x +1

 SPM 1994 1. Solve the following equations: (a) log 3 x + log 9 3 x = −1 1 (b) 8 x + 4 =  x  x +3 4 2

2. (a) Given that log 8 n =

1 3

(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

, find the 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

(ii) log 7 75 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

 SPM 1995 1. Solve the following equations:

15

log 3  xy = 10 and

log 9  xy log 9  y

=

3  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

2.(a) Find the value of 3 log3 7 without using a scientific calculator or four figure table. (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.

(b) log 4

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 using scientific calculator or fourfigure mathematical tables (i) prove that log a 27a = 3.3772 (ii) solve the equation 3 × a n−1 = 3

3. (a) Given that 2 log 3 ( x + y) = 2 + log 3 x + log 3  y, show that x 2 + y 2 = 7 xy (b) Without using scientific calculator or  four-figure mathematical tables, solve the equation log 9 [log 3 (4 x – 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  9    x log a    = log a 8  8  

(c ) After n year a car was bought the n

 7    price of the car is RM 60 000    .  8   Calculate after how many years will the car cost less than RM 20 000 for  the first time

2. (a)Simplify

 log12 49 × log 64 12        log 7 16    

Without using scientific calculator  or four-figure mathematical tables

 SPM 1998

1. Given that log

 9       5  

 x

4 = u and log y 5 = y (b) Given that 3 lg xy 2 = 4 + 2lg y - lg x with the condition x and y is a  positive integer. Show that xy = 10

State log 4  x 3 y in terms of u and/or w 2. (a) Given that log a 3 = x and log a 5 = y. Express log a

 45    3   in terms of  x and y  a  

(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.

(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

 SPM 2001 1. Given that log 2 k = p and log 3 k  = r 

16

2. Solve the equation log 3 4 x − log 3 ( 2 x − 1)

Find log k  18 in terms of  p and r 

=1 [3 marks]

2. (a) Given that log 10 x = 2 and log 10 y = -1, show that  xy – 100 y 2 = 9

3.Given that log m 2 =  p and log m 3 = r ,

 27m     in terms of  p and r    4  

express log m 

(b) Solve the equation (i) 3  x + 2 = 24 + 3  x (ii) log 3 x =log 9 ( 5 x + 6 )

 SPM 2006 

 SPM 2002 1. (a) Given that log 5 3 = k . If 5 2λ −1 = 15, Find λ  in terms of k 

1. Solve the equation 8

(b) Solve the equation log 2 ( 7t − 2 ) − log 2 2t  = −1 2. (a) Given that 2 log 4  x − 4 log16  y State x in terms of  y

 SPM 2003,P1 1. Given that log 2 T  − log 4 v in terms of V 

2. Solve the equation 4 2 x −1

 SPM 2004,P1 1. Solve the equation 32 4 x

=

1

4 x + 2 [3 marks]

2. Given that log 2  xy = 2 + 3 log 2  x − log 2  y , express y in terms of  x [3 marks]

=3

(b) Solve the simultaneous equation 2 m −1 × 32 k + 2 = 16 and 5 −3m × 125 3− k  where m and k are constants

2 x − 3

3. Solve the equation 2 + log 3 ( x + 1) = log 3  x

=1

[3 marks]  SPM 2007 

= 3 , express T 

1. Given that log 2 = x and log 2 c

=  y ,

 8b     in terms of  x and y   c  

[4 marks] = 7 x [4 marks]

express log 4 

[4 marks] 2. Given that 9(3

= 4 8 x +6

n −1)

= 27 n [3 marks]

[3 marks] SPM 2008(paper 1) 1. Solve the equation 16 2 x −3 = 8 4 x

2. Given that log 5 2 = m and log 5 7 =  p , express log 5 4.9 in terms of m and p  SPM 2005,P1 1. Solve the equation 2 x + 4

[3 m] 2. Given that log 4  x = log 2 3 , find the value of  x. [3 m]

− 2 x+3 = 1 [3 marks]

17

locus P at N , Find the coordinates of  N  (d) Calculate the area of triangle OMN   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 CHAPTER 6: COORDINATE GEOMETRY (b) area of ABCD  SPM 1993 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

 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 − 4 x = 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

18

(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 − 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

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 

(b) Given that the point R(3, k ) lies on straight line PQ (i) the ratio PR : RQ (ii) the value of  k 

 SPM 1996 

(b) A point S moves such that its distance from two fixed points E(-1, 0) and F(2, 6) 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 

19

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 5  be expressed as  p − q + 10 2 ( 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

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 (c) A point move such that its distance 1 from point S is of its distance from 2  point T. (i) Find the equation of the locus of the point (ii) Hence, determine whether  the locus intersects the  x-axis or not

 SPM 1997 

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 3 y + 2 x − 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]

 SPM 1998

20

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  2 2  point S is  x +  y − 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

(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 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 .

produced=diperpanjangkan

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]

3. In the diagram, P (2, 9), Q(5, 7) and R  4 1 ,3      are midpoints of straight lines JK ,   2    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] 21

(c) the coordinates coordinates of point point C  [4m] SPM 2000

1. The diagram shows a triangle ABC  triangle ABC  where A where A is on the y the y-axis. -axis. The equations of the straight line ADC  line ADC  and BD and BD are  y − 3 x + 1 = 0 and 3 y + x − 7 = 0 respectively. Find (a) the coordi coordinates nates of point D point D (b) (b) the the rati ratio o AD : DC 

2. The diagram shows a trapezium ABCD trapezium ABCD.. Given the equation of  AB  AB is 3 y − 2 x − 1 = 0 Find (a) (a) the the valu valuee of k  of k  [3m] (b) the equa equatio tion n of  AD  AD and hence, find the coordinates of point A point A [5m] (c) the locu locuss of poin pointt P such  P such that triangle  BPD is always perpendicular at P  at P  [2m]  SPM 2001 1. Given the points P  points P (8, (8, 0) and Q(0, -6). The  perpendicular bisector of  PQ  PQ intersects the axes at A at A and B and B.. Find (a) the equati equation on of  of  AB  AB [3m] (b) (b) the the area area of  of  ∆ AOB , where O is the origin. [2m] 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 point A and point B point B

respectively. Fixed point C is C is such that the gradient of line BC  line BC is is 1 and straight line AC  line AC  is perpendicular to the straight line AB line AB.. Find (a) the coordin coordinates ates of points points A  A and B and B [1m] (b) the equation equation of the the straight straight lines lines AC  and BC  and BC  [5m] (c) the coordi coordinates nates of point point C  [2m] (d) the area area of triangle triangle  ABC  [2m]

22

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 Point D lies on the x the x-axis -axis and divides the straight line CB in the ratio m : n. Find (a) the coordinates coordinates of point point B  B (b) m : n

3. In the diagra diagram, m, the the equat equation ion of  BDC  is  y = −6 . A point P moves such that its 1 distance from A from A is always the distance 2 of  A  A from the straight line BC  line  BC . Find (a) the equation equation of the locus of  P   P  (b) the x-coordinates  x-coordinates of the point of  intersection of the locus and the  x-axis  x-axis [5m]  SPM 2002

2. A 2. A(1, (1, 3), B 3), B and C are C are three points on the straight line  y = 2 x + 1 . This straight line 2 is tangent to curve  x + 5 y + 2 p = 0 at  point B  point B.. Given B Given B divides the straight lines  AC in  AC in the ratio 1 : 2. Find (a) (a) the the valu valuee of  p [3m] (b) the coordina coordinates tes of points points B and C  [4m]

(c) the equation equation of the straigh straightt line that that  passes through point B point B and is  perpendicular to the straight line AC  [3m] 3. Given A Given A(-1, (-1, -2) and B and B(2, (2, 1) are two fixed  points. Point P  Point P moves moves such that the ratio of  AP  of  AP and and PB  PB is 1 : 2. (a) Show that that the equatio equation n of the locus locus of  2 2  point P   point P is is  x +  y + 4 x + 6 y + 5 = 0 [2m] (b) Show that that point point C (0, (0, -5) lies on the locus of point P  point P  [2m] (c) Find the equatio equation n of the straight straight line line  AC  [3m] (d) Given the the straight straight line AC  line AC intersects intersects the locus of point P  point P at at point D point D.. Find the coordinates of point D point D [3m]

23

 SPM 2003(P1) 2003(P1) 1. The points  A(2h, h) ,  B ( p, t ) and C ( 2 p,3t ) are on a straight line. B line. B divides  AC internally  AC internally in the ratio 2 : 3 Express p Express p in terms of t  of t  [3m]

(b) The tangent tangent to the circle circle at point point Q intersects the y-axis at point T . Find the area of triangle OQT  [4m]

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  [3m] 3.  x and y and y are related by the equation  y =  px 2 + qx , where p where p and q are constants. A straight line is obtained  y  by plotting against x against x,, as shown in  x Diagram 1.  y

 SPM 2004(P1) 2004(P1)

1. Diagram 3 shows a straight line graph of   y against x against x  x  y  x

 x

Given that  y of k  of k and and of h of h

Diagram 1

= 6 x − x 2 , calculate the value [3m]

Calculate the values of  p of  p and q [4m] P2(section B) 1. solutions 1. solutions to this question by scale drawing will not accepted. A point P  point P moves moves along the arc of a circle with centre A centre A(2, (2, 3). The arc  passes through Q(-2, 0) and R and R(5, (5, k ). ). (a) Fin Find (i) the equation of the locus of the  point P   point P  (ii) the values of k  of k  [6m]

2. Diagram 4 shows a straight line PQ with  x  y the equation + = 1 . The point P lies 2 3 on the x the x-axis -axis and the point Q lies on the y the y-axis

24

Find the equation of the straight line  perpendicular to PQ and passing through the  point Q [3m]

Express p in terms of k 

[2m]

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 B) 2. Solutions to this question by scale drawing will not accepted.

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

(a) Find (i)

(a) write down the equation of  AB in the form of intercepts [1m] (b) Given that 2 AD = 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 

: :

 y  y

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

=  px + k  = (k − 2) x +  p

where p and k are constant

25

The equation of the straight line CB is  y = 2 x − 1 Find the coordinates of  B [3 marks] 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

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

Diagram 3 shows the triangle AOB where O is the origin. Point C lies on the straight line AB

(a) (b) (c)

(i)

 y + 2 x + 8 = 0

2

Calculate the area, in unit , 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 Find the equation of the locus

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)

of P (ii)

Hence, determine whether or   not this locus intercepts the y-axis

26

 x

of the triangle is 4 unit2, find the values of t . [3 m]

 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) 1. Diagram 13 shows a straight line passing through S (3,0) and T (0,4) 1. The straight line

 SPM 2008 Section B (paper 2) 1. Diagram shows a triangle OPQ. Point S  lies on the line PQ.

Diagram 13 (a) Write down the equation of the straight line ST in the form  x  y + =1 a b (b) A point P( x, y) moves such that  PS = PT . Find the equation of the locus of  P  [4 m]

(a)

A point W moves such that its

distance from point S is always 2

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 ,

2. The points (0,3), (2,t ) and (-2,-1) are the vertices of a triangle. Given that the area

27

(ii) the coordinates of Q [5m] (c) Hence, find the area, in unit2, of triangle OPQ [2m]

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 1 37 , find (c) if standard deviation is 2 the values of  x

2. The below table shows the marks obtained by a group of students in a monthly test . Marks

1-20

21-40

41-60

61-80

81-100

5

8

12

11

4

 Number  students

(a) On a graph paper, draw a histogram

28

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 

(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

 SPM 1994 1. The below table shows the marks obtained by a group of students in a monthly test . 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. (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.

Numbers of villagers 50 79 47 14 10

The table shows the age distribution of  200 villagers. Without drawing a graph, calculate (i) the median (ii) the third quartile of their ages  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.

2. (a) The table shows the results of a survey of the number of pupils in several classes in a school. Find

 Numbers of classes 6 5 4 29

Numbers of pupils 35 36 30

Length (mm) 20-29 30-39 40-49 50-59 60-69 70-79 80-89

(b) use a graph paper to answer this question The data in the table shows the monthly salary of 100 workers in a company.

Numbers of fish 2 3 7 12 14 9 3

(i)

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 (c) from your graph, find the percentage of the numbers of fish which has a length more than 55 mm

(ii)

Monthly Salary (RM) 500-1 000 1 001-1 500 1 501-2 000 2 001-2 500 2 501-3 000 3 001-3 500 3 501-4 000 4 001-4 500

 SPM 1997  1. The table shows a set of numbers which has been arranged in an ascending order  where m is a positive integer  Set numbers Frequency

1

m-1

5

m+3

8

10

1

3

1

2

2

1

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  Numbers of  workers 10 12 16 22 20 12 6 2

(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

2.

 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 5m is . 8 Find (a) the values of m and k  [4m] (b) the variance of the new data [2m]

(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] 30

2. Set X consist of 50 scores, x, for a certain game with a mean of 8 and standard deviation of 3 (a) calculate Σ x and Σ x 2 (b) A number of scores totaling 180 with 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

(b) Without drawing an ogive, estimate the interquartile range of  this distribution. [4m] Marks <10  Number  of  2 students

<30

<40

<50

<60

<70

<80

8

21

42

68

87

98

100

2. The table shows the distribution of marks in a physics test taken by 120 pupils. Calculate the mean [4m] the median [3m] the standard deviation [3m] of the distribution

2. The frequency distribution of marks for  30 pupils who took a additional mathematics test is shown in the table Marks 20-39 49-59 60-79 80-99

<20

 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

Frequency 6 5 14 5

(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

20-29

30-39

40-49

50-59

60-69

70-79

 Number  of pupils

2

14

35

50

17

2

[2m] (b) Find the standard deviation of the set of numbers below: 5, 6, 6, 4, 7

[3m] 2. The table shows the frequency distribution of the marks obtained by 100  pupils

 SPM 2000 1. The table shows the results 100 students in a test

Marks 6-10 11-15 16-20 21-25 Marks26-30 0-9 31-35 Frequency 36-40

(a) Based on the table above, copy complete the table below [2m]

31

Number of pupils 12 20 27 16 13 10 2

(i) Calculate the variance

[3m] (b)

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]

(ii) Construct a cumulative frequency table 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. Scores

1

3

6

 x

y

14

 Number  of   pupils

1

1

2

3

1

1

Find the values of  x and y

 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 the value of this number  (ii) the standard deviation of the set 11 numbers [4 marks]

[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

≤

≤1

≤1

≤2

≤2

≤3

5

0

5

0

5

0

≤ 35

 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

 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

1. A set of examination marks  x1 , x 2 , x3 , x4 , x5 , x6 has a mean of 5 and a (a)

standard deviation of 1.5 Find (i) the sum of the marks, ∑ x (ii) the sum of the squares of the marks, ∑ x 2 [3m]

1. Diagram 2 is a histogram which represents the distribution of the marks

32

obtained by 40 pupils in a test.

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 Score Number of pupils 10-19 1 20-29 2 30-39 8 40-49 12 50-59 k  60-69 1 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]

(a) Without using an ogive, calculate the median mark [3m] (b) Calculate the standard deviation of the distribution [4m]

 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]

 paper 2,section A 1. Table 1 shows the frequency distribution of the scores of a group of   pupils in a game.

 SPM 2007   Paper 2

1. Table 1 shows the cumulative frequency distribution for the scores of 32 students in a competition

(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

33

set, the new mean is 8.5 [3m]

SPM 2008(Paper 2) 1. Table 5 shows the marks obtained by 40 candidates in a test.

Table 1

Score <10  Number  of  2 students

<20

<30

<40

<50

8

21

42

68

Given that the median mark is 35.5, find the value of  x and of  y. Hence, state the modal class

(a) Based on table 1, copy and complete Table 2 Marks Freque ncy

0-9

10-19

20-29

30-39

Marks 10-19 20-29 30-39 40-49 50-59

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]

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

 SPM 1993

34

Number of candidates 4  x  y 10 8 [6m]

CHAPTER 8: CIRCULAR MEASURE

2. In the diagram, M and N are the centers of two congruent circles with radius r cm respectively. 0 Show that ∠ PMQ = 120  b. Find, in terms of  π  and r , the area of the shaded region [6m]  SPM 1995

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 1994 1.

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 2 2 is  x +  y = 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

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 

35

(c) the area of the shaded region

 SPM 1997  (a) Convert

(a)

(i) (ii)

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 The angle θ  in radians (b) the area of sector  OPQ [5m]  SPM 1996  1.

64020' into radians 4.36 radians into degrees [2m]

(b)

The diagram shows two sectors 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  x  y + =1 is 12 6 Calculate (a) the area of ∆ ABC  (b) ∠ ACB in radians

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)

36

(b)

the area of the shaded region [4m]

(b) the perimeter of the shaded region [2m] (c) the area of sector  PJM  [2m] (d) the area of the shaded region [4m]

 SPM 1998

1.

 SPM 1999 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] (b) the area of region swept by the  pendulum [2m]

The diagram shows two sectors OPQR and OST of two concentric circle with centre 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 

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

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 J ML is 3.6 radians. Find (a) the radius of sector  MJKL [2m]

37

(b) the area of segment AGBH  ( c) the area of the shaded region (Use π  = 3.142 )

(a) (b) (c) (d)

 SPM 2000 1.

 SPM 2001 1.

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 [5m]

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] (b) the area of the shaded region [4m]

2.

2.

The diagram shows semicircle DAECF with centre Q and rhombus QAPC . Calculate the radius of semicircle DAECF  [1m] the angle θ  in radians [3m] the area of sector  QAEC  [2m] 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]

38

 SPM 2002 1.

 SPM 2003 paper 1 1. Diagram 1 shows a sector  ROS  with centre O

The diagram shows two sectors OAB and OCD of two concentric circles with centre O , 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.

Diagram 1 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]

39

the length, in cm, of the arc JKL [4m] 2

the area, in cm , of the shaded region [4m]  SPM 2004 paper 1 1. Diagram 1 shows a circle with centre O

 SPM 2005 paper 1 1. Diagram 1 shows a circle with centre O

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

45.51 cm, find the length, in cm, of the radius. (use π  = 3.142) [3m] paper 2(section B)

(a)

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 the angle α  , in terms of  π  [2m]

∠ POQ = 40

π  6

radian

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

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.  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] 2 (c) the area, in m , of the flower bed 41

[4m]

AP and PB, are tangents to the circle at  point Q and point R, respectively.

[use π  = 3.142]  SPM 2007

Paper 1

Calculate (a) the length, in cm, of the arc AB [5 m] 2 (b) the area in cm , of shaded region [5 m]

1. Diagram 4 shows a sector BOC of a circle with centre O

SPM 2008 paper 1 1. Diagram 18 shows a circle with centre O and radius 10 cm.

(a) (b)

It is given that AD = 8 cm and  BA = AO = OD = DC = 5 cm Find the length, in cm, of the arc BC  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

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,

42

[4m]

(b) calculate the length, in cm of the minor arc QR [3m] 2 (c) calculate the area, in cm , of the colored region. [5m]

SPM 2008 paper 2 1. Diagram shows two circles. The 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]

43

equation d 2 y dy  y 2 + x dx dx

+ 12

3. Find the coordinates at the curve  y = ( 2 x − 5) 2 where the gradient of the normal for the curve 1 is 4  SPM 1996  4 7 CHAPTER 9: DIFFERENTATION 1. Differentiate  x (1 + 3 x) with respect to  x.

 SPM 1993 1. Given that  f ( x)

=

1 − 2 x 2 4 x − 3

, find f '( x)

2. The gradient of the curve  y the point

 SPM 1994

1. (a) Given that  y

= 3 x 2 + 5 , find

dy dx

d    1

     dx  2 x + 1  

=

16  x

4

, find

dy dx

estimate the value of 

= 2 x − 3 and  y = −

 p 2

. Find

dy

if  x = 2 . Hence,

in terms of  x dx the small change in y, when x decreases from 2 to 1.98

16

(1.98) 4

=

1 − 2 x 3  x − 1

=  x(3 −

+ x

 n 2 − 4      (a) Find the value of  n → 2   n − 2     lim

find f ' ( x)

x) , express

5 (b) Given  f ( x) = ( 2 x − 3) find  f ′′ ( x)

dy

+ 12 in terms of  x. 2 dx dx Hence, find the value of  x that satisfy the

 y

3

 SPM 1997 

1. Given  f ( x)

d 2 y

at

(a) the approximate change in x if  thE rate of change in p is 3 units per  second

 SPM 1995

2. Given  y

 x 2

and k 

using

3. Given  p

2. Given  y

k 

 − 1,− 7      is 2. Find the values of h 2    

the first principle (c) Find

= hx +

2.

Differentiate  y

first principle

44

4

= −3  x

using the

minimum 3.

(a) [5m]  SPM 1998 5 ' 1. Given that  f ( x) = 4 x( 2 x − 1) , find  f  ( x)

The 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 4 p2 cm2 and its height from the vertex of the pyramid is h cm. (i) Show that the volume of the container  that is filled with water is 3 V  = (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]

2.

(b)

(a)

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 2 x 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   2 16   A = 3π  x +   [3m]  x    

The diagram shows a rectangle JKLM inscribed in a circle. Given JK = x cm and KL = 6 cm show that the area of the shaded region, A cm2, is given by π  x 2 − 6 x + 9π   A = 4 (b) Calculate the value of  x so that the area of the shaded region is a

 b)Calculate the minimum surface area of the  block [3m] c) Given the surface area of the block  changes at a rate of 42 π  cm 2 s −1 . Find the

45

of change of its radius when its radius is 4 cm. [2m]

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

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 1999

( x − 2 ) 1. Given  f ( x) = 2

1 − 3 x

5

 SPM 2000 1. Differentiate the following expressions with respect to x

, find

 f ' (0)

[4m]

(a)

2. Given  y = t − 2t  and  x = 4t  + 1 dy (a)Find , in terms of  x dx (b) If  x increases from 3 to 3.01, find the corresponding small increase in t . [2m]

1 + 3 x 4

[2m]

2

(b)

2 x + 5  x

4

+3

[2m]

2 Given  y = 3 x − 4 x + 6 . When  x = 5 , x increases by 2%. Find the corresponding rate of change of  y. Find the equation of the tangent to 2 the curve  y = 2 x + r  at the point  x = k  . If  the tangent passes through the point (1,0), find r in terms of k 

3 (a)

4.(a) The straight line 4 y + x

= k  is the 2 normal to the curve  y = ( 2 x − 1) − 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 = (30-6 x) cm,  BC = 3 x cm, CD = 4 x 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

46

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    y 2 20 y −      4    b. Using π  = 3.142 , find the length and width of the rectangle that make the area of the shaded region a maximum

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 dL cm, find an expressions for  in dx terms of  x and hence, find the maximum area of triangle ABC   SPM 2001

1. Given  f ( r ) =

Given  y = 2 x − 5 x + 7 , find dy the value of  at the point (2, 3). Hence, dx 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 5 2 3 1. Given  p = (1 + t ) + t  2 dp Find and hence find the values of t dt  dp =7 where dt  3

4 + 3r 

5 − 2r  of   f (r ) when r  → ∞

find limited value

2.

Given that graph of function k  3  f ( x) = hx + 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)

2.

 y = 2 x –  x2

47

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 

 x

2

d 2 y dx 2

+ ( x − 1)

dy dx

+  y = 8 [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

[4m]

(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 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 1 2 cone = π r  h ) [4m] 3

 SPM 2003 paper2(section A)

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]

 SPM 2004 2 4 1. Differentiate 3 x ( 2 x − 5) with respect to x

[3m]

2. Two variables x and y are related by the 2 equation  y = 3 x + .  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 + 5 x , use differentiation to find the small change in y when x increases from 3 to 3.01 [3m] 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  2

paper 2(section B) 3. The gradient function of a curve which  passes through A(1, -12) is 3 x 2 − 6 x . Find (a) the equation of the curve [3m] (b) the coordinates of the turning points of 

48

the curve and determine whether each of the turning points is a maximum or  a minimum [5m]

(a) find the value of 

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 2005

1. Given that h( x) =

1

( 3 x − 5) 2

h" (1)

, evaluate

 SPM 2007   Paper 2

[4m]

2. The volume of water, V cm3, in a 1 3 container is given by V  = h + 8h , 3 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]

1. A curve with gradient function 2 x −

u

= 3x − 5 .

Find

= dy dx

2 3

 x 2

(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 1. The curve  y =  f ( x) is such that 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 2. The curve  y =  x − 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]

u 7 , where

in terms of  x [4m]

3. Given that  y

2

has a turning point at (k , 8)

paper2(sectionA) 2 3. A curve has a gradient function  px − 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 1 gradient of the normal at P is − 4 Find the coordinates of  P  [3m]

2. It is given that  y

dy

= 3 x 2 + x − 4

49

2 2. The normal to the curve  y =  x − 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]

[1m] (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 ∆ PQR can be form [2m] 2.

 SPM 1993 1.

The diagram shows a land form triangle,  ABC , divide by three parts. ADB, BFC , and  AEGC is a straight line 12 CHAPTER 10: SOLUTION OF TRIANGLE Given that sin ∠ BAC  = 13 (a) if the fence want to build along the  boundary BC , calculate the total length is needed (b) Calculate ∠ BCA (c ) Given that the area of  ∆ FCG same with the area ∆ ADE . Calculate the length of GC 

The diagram shows a ∆ PQR (a) Calculate obtuse angle PQR

 SPM 1994 1.

[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.

50

(a) the length of AC correct to two decimal  places [3m] (b) ∠ ABC  [2m] In the diagram, BCD is a straight line, calculate the length of CD

 SPM 1996  1.

2.

The diagram shows a cuboid. Calculate (a) ∠ JQL [4m] (b) the area of  ∆ JQL [2m] 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]

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

51

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]

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   SPM 1999 1.

 SPM 1998

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 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 (a) ∠ JLK  (b) the area of  ∆ JKL

2.

 SPM 2000

52

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  11 Hence, show that cos α  = 29

 SPM 2002 1.

2. 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 

In the diagram, PQR is a straight line. Calculate the length of  PS 

2.

 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 ∠VQR = 80 0 Calculate (a) the length of QR (b) the area of the slanting face

[5m]  SPM 2003

53

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

Given that VB = VC = 2.2 m and AB = AC = 2.6 m, calculate (a) the length of  BC if the area of the  base is 3 m2 (b) the length of  AV and the base is 250 (c ) the area of triangle VAB

(a) Calculate (i) ∠ ABC  (ii) ∠ ADC  (iii) the area, in cm2, of quadrilateral  ABCD [8m] (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  (i) Sketch the triangle A'B'C'  (ii) State the size of  ∠  A'B'C'  [2m]

 SPM 2004

1. The diagram shows a quadrilateral ABCD such that ∠ ABC  is acute

54

kitaran vertical = mencancang horizontal = mengufuk  obtuse angle = sudut cakah slanting face = permukaan condong acute = tirus formed = dibentuk  diagonal = pepenjuru  SPM 2006   SPM 2005

1. Diagram 5 shows a quadrilateral ABCD 1. The diagram shows triangle ABC 

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] 2 (d) the area, in cm , quadrilateral ABCD [3 m]

(a) Calculate the length, in cm, of   AC  [2m] (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] (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

cyclic quadrilateral=sisi empat 55

CHAPTER 11: INDEX NUMBER   SPM 2006   SPM 1993 1. The table below shows the monthly expenses of Ali’s family

1. Diagram 7 shows quadrilateral ABCD

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

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]

Calculate the length, in cm, of  AC  (b) ∠ ACB [4 M]

 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

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]

56

Monthly expenses Food House rental Entertainment Clothing Others

Price Index

(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.

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 expenses of the Yusris’ household in the year 1990 is RM 850

Item

Price Index 1994

Changes to Price index from 1994 to 1996

Weightages

Wood

180

Increases 10%

5

Cement

116

Decreases 5%

4

Iron

140

No change

2

Steel

124

No change

1

Calculate the composite price index in the Item Price Index Weightage Shirt 100  N  Trousers 110 6 Bag 140 2 Shoes 100 4 year 1996 [3m]

 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

Calculate (a) the value of n (b) the price of a shirt in 1994 if its  price in 1990 is RM 40

Food item Fish Prawn Chicken Beef Cuttlefish

 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. [2m]

Price index

Weightages

140 120 125 115 130

4 2 4 3  X 

 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.

57

Item  P  Q  R

Price index 120 150 m

Calculate (a) the value of m (b) the value n

Weightage 2 n 3 Index number, I i  Weightages, W i 

[2m] [2m]

 SPM 1998 1. The price index of a certain item in the year 1997 is 120 when 1995 is used as the  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

94 x

120 4

 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

1. The composite index number of the data in the below table is 108

Find the value of  x

105 5-x

Item A B C

[4m]

 SPM 2000

Price index  x 98 123

Weightage 5  y 14 - y

 SPM 2002

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.

1. The table below shows the prices, price indices and the number of three items

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]

Item

Pric e

(RM)

 A B C

Year  1999 55 40 80

Year  2000 66 x 100

Price Index (Base year  1999) 120 150 125

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

 SPM 2003 Type of  item

Price (RM) in 1996 70

A

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

Price (RM) in 1998 105

Weightage %

Y 

B

80

100

 X 

C

60

67.50

2 x

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.

(a) Using the year 1996 as the base year, calculate the price indices of items A, B and C

30 25

(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]

20 15 10 5 0 P

Q

R

S

ITEMS

Items

Price in 1990

Price in 1995

P

 x

RM 0.70

Price Index in 1995 based on 1990 175

Q R S T

RM 2.00 RM 4.00 RM 6.00 RM 2.50

RM 2.50 RM 5.50 RM 9.00  z 

125  y 150 120

(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 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 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

 SPM 2004

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

Ingredients

 P  Q  R S

(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]

(i)

(b) The composite index number of the lost of biscuits production for the year 1995  based on the year 1993 is 128. Calculate 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]

Item

 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

Price index for the year 2004 based on the year 2001

Price index for the year 1995 based on the year 1993 135 x 105 130

 x 140 150 80

Percentage of  usage (%) 40 30 10 20

(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]

60

127.5 (c) The cost of making these biscuits is Calculate expected to increase by 50% from the (i) The price of a cake in the year  year 2004 to the year 2007 2004 if its corresponding price Find the expected composite index for  in the year 2005 is RM30.60 the year 2007 based on the year 2001 (ii) the value of m if the [2m] quantities of ingredients P , Q,  R and S used are in the ratio of 7 : 3 : m : Component Price (RM) for the Price index for the 2 year year 2006 based on the year 2004 [3m] P 1.20 1.50 125 Q  x 2.20 110 R 4.00 6.00 150 S 3.00 2.70  y T 2.00 2.80 140  SPM 2007  Ingredient  P  Q  R S 

 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

Price per kilogram (RM) Year 2004 Year 2005 5.00 W  2.50 4.00 x Y  4.00 4.40

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

(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.

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

Calculate the value of  x and of  y [3m] (c) The composite index for the cost of making the cake in the year  2005 based on the year 2004 is

61