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SPM 1993
1. Given the function f : x → 3 – 4x and function g : x → x2 – 1, find (a) f -1
(b) f -1g(3) [5 marks]
2. Given the functions f, g and h as a f : x → 2x
g : x → 2x
3
−, x ≠ 2
h : x → 6x2 – 2
(i) determine function f h(x)(ii) 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 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]
2. The function f is defined as
f : x → x23
xp
++
, for all value of x except
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 1995
1. Given the function f(x) = 3x + c and
inverse function f -1(x) = mx + 3
4. Find
(a) the value of m and c [3 marks] (b) (i) f(3) (ii) f -1f(3)
[3 marks]
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]
SPM 1996
1. Given the function f : x → 2x
khx
−+
, x≠2
and inverse function f -1 : x →3x
5x2
−−
, x≠3
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]
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
CHAPTER 1: FUNCTIONS
SPM 1998
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 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]
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]
SPM 2000
1. Given the function g -1(x) = 3
kx5 − and
f(x) = 3x2 – 5. Find (a) g(x) [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]
SPM 2001
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]
2. Given the function f -1(x) = xp
1
−−
, x ≠ p
and g(x) = 3 + x. Find (a) f(x) [2 marks] (b) the value of p if ff -1(p2–1) = g[(2-p)2]
( c) range of value of p so that fg-1(x) = x no real roots
[5 marks]
SPM 2002
1. Given the function f(x) = 4x -2 and g(x) = 5x +3. Find
(i) fg -1(x)
(ii) the value of x so that fg-1(2
x) =
5
2
[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]
2
SPM 2003
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]
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
Set P Set Q Diagram 1
State(a) the range of the relation(b) the type of the relation
[2 marks]
2. Given the function h : x → 4x + m and
h-1 : x → 2hk + 8
5, where m and k are
constants, find the value of m and of k.[3 marks]
3. Given the function h(x) = x
6, x ≠ 0 and
the composite function hg(x) = 3x, find (a) g(x)(b) the value of x so that gh(x) = 5
[4 marks]
SPM 2005
1. In Diagram 1, the function h maps x to y and the function g maps y to z
Determine(a) h-1(5)(b) gh(2) [2 marks]
2. The function w is defined as
w(x) = x2
5
−, x ≠ 2. Find
(a) w-1(x)(b) w-1(4) [3 marks]
3. The following information refers to the functions h and g.
Find gh-1 [3 marks][
, where a and b are constants and ,
3
P = {1, 2, 3}Q = {2, 4, 6, 8, 10}
d ∙
e ∙
f ∙
∙ w∙ x∙ y∙ z
h : x → 2x – 3g : x → 4x - 1
SPM 2006Paper 11. In diagram 1, set B shows the image of certain elements of set A
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
x
xmxh
−→: , 0≠x , where m is a constant
DIAGRAM 2
Find the value of m[2 marks]
Paper 2 1. Given that 23: −→ xxf and
15
: +→ xxg , find
(a) )(1 xf − [1 m]
(b) )(1 xgf − [2 m]
( c) )(xh such that 62)( += xxhg[3 m]
SPM 2007 Paper 1
1. Diagram 1 shows the linear function h.
(a) State the value of m(b) Using the function
notation, express h in terms of x [2 m]
2. Given the function 3: −→ xxf , find the value of
x such that 5)( =xf[2m][
, where a and b are constants and ,
4
2
1−
3. The following information is about the function h and the composite function 2h
Find the value of a and b[3m]
SPM 2008 Paper 1
1. Diagram 1 shows the graph of the
function 12)( −= xxf , for the
domain 50 ≤≤ x .
State
(a) the value of t(b) the range of f(x) corresponding to the
given domain[3 m]
2. Given the function 25: +→ xxg and 34: 2 +−→ xxxh , find
a) )6(1−g
b) )(xhg[4m]
3. Given the functions 1)( −= xxf and 2)( += kxxg , find
a) f(5)b) the value of k such that
gf(5)=14[3m][
, where a and b are constants and ,
5
SPM 19 94
1. If α and β are the roots of the quadratic equation 2x2 – 3x – 6 = 0, form another
quadratic equation with roots 3
β and
3
α
[4 marks]
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 2
1and -5 are the roots of the
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 0322 =−++ kkxx has no real roots
[3 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]
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 (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]
SPM 19971. Given that m + 2 and n - 1 are the roots of the equation x2 + 5x = -4. Find the possible value of m and n.
SPM 1998
1. The equation of px2 + px + 3q = 1 + 2x
have the roots p
1and 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
SPM 1999
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]
2. Given the equation x2 – 6x + 7 = h(2x – 3) have two equal real roots. Find the values of h.
[4 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]
6
CHAPTER 2: QUADRATIC EQUATIONS
SPM 2000
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]
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]
2. If α and β are the roots of the quadratic equation 0132 2 =−+ xx , form another quadratic equation with roots 3α + 2 and 3β + 2.
[5 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 2
α and
2
β are the roots of the
equation kx(x – 1) = 2m – x. If α + β = 6 and αβ = 3, find the values of k and m.
[5 marks]
SPM 20031. Solve the quadratic equation 2x(x – 4) = (1 – x)(x + 2). Give your answer correct to four significant figures.
[3 marks]
2. The quadratic equation x(x + 1) = px - 4 has two distinct roots. Find the range of values of p
[3 marks]SPM 20041. Form the quadratic equation which has
the roots -3 and 2
1. Give your answer in
the form ax2 + bx + c =0, where a, b and c are constants
[2 marks] SPM 2005
1. Solve the quadratic equation x(2x – 5) = 2x – 1. Give your answer correct to three decimal places.
[3 marks]
SPM 2006
1. A quadratic equation xpxx 292 =++ has two equal
roots. Find the possibles values of p.[3 marks]
SPM 2007
1. (a) Solve the following quadratic equation:
0253 2 =−+ xx
(c) The quadratic equation ,032 =++ kxhx where h and k are
constants, has two equal rootsExpress h in terms of k
[4 marks]
SPM 2008 1. It is given that -1 is one of the roots of the quadratic equation 042 =−− pxx Find the value of p
[2 marks]
7
SPM 19 93
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.
(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
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. (a) Find the range of value of x if 5x ≥ 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]
SPM 1995
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]
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]
2. Find the range of values of x if (a) x(x + 1) < 2
[2 marks]
(b) x21
3
−−
≥ x
8
CHAPTER 3: QUADRATIC FUNCTIONS
[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,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
2. Find the range of values of x if (a) 2(3x2 – x) ≤ 1 – x (b) 4y – 1 = 5x and 2y > 3 + x
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
SPM 19981.
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
2. (a) Given that f(x) = 4x2 – 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)
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 < 52. Find the range of value of x if (x – 2)(2x + 3) > (x – 2)(x + 2)
SPM 20001. 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]
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 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
2. Given that f-1 (x) = xp −
1, 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
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.
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 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]
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 (a) the value of k
(b) the equation of the axis of symmetry(c) the coordinates of the maximum point
[3 marks]
SPM 2005 (paper 1)
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.
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 (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
Diagram 2 The curve y = f(x) has the minimum point
10
Diagram 2
(1,q), where q is a constant. State a) the value of pb) the value of q
c) the equation of the axis of symmetry [3 m]
SPM 2006
1. Diagram 3 shows the graph of quadratic function )(xfy = . The straight line 4−=y is a tangent to the curve )(xfy =
a) write the equation of the axis of
symmetry of the curveb) express )(xf in the form
cbx ++ 2)( , where b and c are constants.
[3 marks]
3. Find the range of the values of x for xxx +>+− 4)4)(12(
[2 marks]
SPM 2007(paper 1)
1. Find the range of values of x for which xx +≤ 12 2
[3 marks]
2. The quadratic function 42)( 2 −+= xxxf can be expressed
in the form nmxxf −+= 2)()( , where m and n are constants. Find the value of m and of n
[3 marks]Answer m=…………..
n=…………..
SPM 2008 (paper 1)
1. The quadratic function rqxpxf ++= 2)()( , where p, q and r
are constants, has a minimum value of -4. The equation of the axis of symmetry is x = 3State a) the range of values of pb) the value of qc) the value of r
[3 m]
2. Find the range of the value of x for xx −<− 5)3( 2 .
[3 m] SPM 2008 (paper 2)
1. Diagram 2 shows the curve of a quadratic function 5)( 2 −+−= kxxxf . The curve has a maximum point at B(2,p) and intersects the f(x)-axis at point A
Diagram 2
a) State the
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 5)( −≥xf
[2m]
11
SPM 19931. Solve the simultaneous equation x2 – y + y2 = 2x + 2y = 10
SPM 19941. 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 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
84
3m2 and 12 m, find the measurements
of the room.
Diagram 2
SPM 19951. Solve the simultaneous equation 4x + 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 19961. 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
SPM 19971. Given that (3k, -2p) is a solution for the simultaneous equation x – 2y = 4 and
x
2+
y2
3=1. Find the values of k and p
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 10π m2 and the length JK is longer than the length of the curve KL by π m, Find the value of x.
SPM 19981. Solve the simultaneous equation:
3
x+
y
2= 4 , x + 6y = 3
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.
12
CHAPTER 4: SIMULTENOUS EQUATIONS
1 m
1 m 1m
1 m
SPM 19991. Given the curve y2 = 8(1 – x) and the
straight line x
y= 4. Without drawing the
graph, calculate the coordinates of the intersection for the curve and the straight line.2. Solve the simultaneous equation
2x + 3y = 9 and x
y6 −
y
x = −1
SPM 20001. Solve the simultaneous equation 3x – 5 = 2y , y(x + y) = x(x + y) – 5
2. Solve the simultaneous equation
3
x −
2
y + 3 = 0 and
x
3 +
2
y −
2
1 = 0
SPM 20011. Given the following equation: M = 2x − y
N = 3x + 1R = 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
[use π = 7
22]
SPM 2002
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.
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 20031. Solve the simultaneous equation 4x + y = −8 and x2 + x − y = 2
SPM 2004
13
yam
1. Solve the simultaneous equations p − m = 2 and p2 + 2m = 8. Give your answers correct to three decimal places.
SPM 20051. Solve the simultaneous equation
x + 2
1y = 1 and y2 − 10 = 2x
SPM 20061. Solve the simultaneous equations
12 2 =+ yx and 52 22 =++ xyyxGive your answer correct to three decimal places [5 m]
SPM 20071. Solve the following simultaneous equations:
032 =−− yx , 09102 2 =++− yxx [5 m]
SPM 20081. Solve the following simultaneous equations :
040
0432 =−+
=+−xyx
yx
[5m]
14
SPM 19931. If 3 − log10 x = 2log10 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:
(i) 4 x2 = 32
1
(ii) log x 16 − log x 2 = 3
SPM 19941. Solve the following equations: (a) log 3 x + log 9 3x = −1
(b) 48 +x = 324
1+xx
2. (a) Given that log 8 n = 3
1, 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
SPM 19951. Solve the following equations:
(a) 81(27 x2 ) = 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) log 2
32
3mn,
(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 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 19961. (a) Express 2 2+n − 2 n + 10(2 1−n ) in a simplify terms
(c) Solve the equation 3 2+x − 5 = 0
2. (a)Solve the following equations: (i) 4 x2log =5 (ii) 2 x . 3 x = 5 1+x
(b) Given that log53 = 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
SPM 19971. 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
15
CHAPTER 5: INDICES AND LOGARITHMS
log 3 xy = 10 and y
xy
9
9
log
log =
2
3
2.(a) Find the value of 3 7log3 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.
3. (a) Given that 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 (c ) After n year a car was bought the
price of the car is RM 60 000n
8
7.
Calculate after how many years will the car cost less than RM 20 000 for the first time
SPM 1998
1. Given that log x 4 = u and log y 5 = y
State log 4yx3 in terms of u and/or w
2. (a) Given that log a 3 = x and log a 5 = y.
Express log a
3
45
a in terms of x and 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 3(9 x ) = 27 y and log 2 y = 2 + log 2 (x – 2) were obtain from the first and second experiment respectively
SPM 19991. 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
(b) log 4
5
9
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 four- figure mathematical tables
(i) prove that log a 27a = 3.3772(ii) solve the equation
3 × a 1−n = 3
SPM 20001. (a) Solve 3 x2log = 81
(b) If 3 x2 = 8(2 x3 ), prove that
x log a
8
9 = log a 8
2. (a)Simplify
×7log
12log49log
16
6412
Without using scientific calculator or four-figure mathematical tables (b) Given that 3 lg xy 2 = 4 + 2lgy - lgx 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 k3 = r
16
Find log k 18 in terms of p and r
2. (a) Given that log x10 = 2 and
log y10 = -1, show that xy – 100y 2 = 9
(b) Solve the equation (i) 3 2+x = 24 + 3 x
(ii) log x3 =log 9 ( )65 +x
SPM 2002 1. (a) Given that log 35 = k. If 5 12 −λ = 15, Find λ in terms of k
(b) Solve the equation log 2 ( )27 −t − log 122 −=t
2. (a) Given that 3log4log2 164 =− yx
State x in terms of y
(b) Solve the simultaneous equation 16322 21 =× +− km and 11255 33 =× −− km where m and k are constants
SPM 2003,P11. Given that 3loglog 42 =− vT , express T in terms of V
[4 marks]2. Solve the equation xx 74 12 =−
[4 marks]
SPM 2004,P11. Solve the equation 684 432 += xx
[3 marks]
2. Given that m=2log5 and p=7log5 ,
express 9.4log5 in terms of m and p
SPM 2005,P11. Solve the equation 122 34 =− ++ xx
[3 marks]
2. Solve the equation ( ) 112log4log 33 =−− xx
[3 marks]
3.Given that pm =2log and rm =3log ,
express
4
27log
mm in terms of p and r
SPM 2006
1. Solve the equation 2
32
4
18
+
− =x
x
[3 marks]
2. Given that yxxy 222 loglog32log −+= ,
express y in terms of x[3 marks]
3. Solve the equation xx 33 log)1(log2 =++
[3 marks]
SPM 2007 1. Given that log x=2 and log yc =2 ,
express log
c
b84 in terms of x and y
[4 marks]
2. Given that nn 273(9 )1 =−
[3 marks]
SPM 2008(paper 1)1. Solve the equation
xx 432 816 =−
[3 m]2. Given that 3loglog 24 =x , find the
value of x.[3 m]
17
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
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 0422 =−+ xyx(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
SPM 19941. 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
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
18
CHAPTER 6: COORDINATE GEOMETRY
2. (a)The above diagram, P, Q and R are three points are on a line 42 =− xy 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) 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 19951. Solutions to this question by scale drawing will not be accepted.
Graph on the above show that the straight line LMNFind
(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 64 −= xy cutting the
curve 22 −−= xxy 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
(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
1. In the diagram, the straight line 32 += xy 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
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
be expressed as 102
5 +− qp
( 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 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
0923 =−+ xy (a) Find the equation of BC [3m](b) If CB is produced, it will intersect the x- axis at point R where RB = BC. Find the coordinates of point C [3m]
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
from point S is 2
1 of its distance from
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 1998
20
1. In the diagram, ACD and BCE are straight lines. Given C is the midpoint of AD, and BC : CE = 1:4Find(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 044622 =+−−+ yxyx ( 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
3. In the diagram, P(2, 9), Q(5, 7) and R
3,2
14 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 19991. Given point )4,2( −−A and point )8,4(B . 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 xy 8162 −= that intersects the x- axis 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]
21
(c) the coordinates of point C [4m]
SPM 2000
1. The diagram shows a triangle ABC where A is on the y-axis. The equations of the straight line ADC and BD are 013 =+− xy and 073 =−+ xy respectively.
Find(a) the coordinates of point D(b) the ratio AD : DC
2. The diagram shows a trapezium ABCD. Given the equation of AB is 0123 =−− xy 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 20011. 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]
2. Solutions to this question by scale drawing will not be accepted.Straight line 62 =− yx intersects the x-axis and y-axis at point A and point B
respectively. Fixed point C is such that the 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]
22
3. In the diagram, the equation of BDC is 6−=y . A point P moves such that its
distance from A is always 2
1the 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]SPM 2002
1. The diagram shows a triangle ABC with an area 18 units2 . the equation of the straight line CB is .01 =+− xy 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
2. A(1, 3), B and C are three points on the straight line 12 += xy . This straight line
is tangent to curve 0252 =++ pyx 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
[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 056422 =++++ yxyx
[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]
23
SPM 2003(P1)1. The points ),2( hhA , ),( tpB and )3,2( tpC 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
135
=+ xy and 2435 += xy .
Determine whether the lines are perpendicular to each other
[3m]
3. x and y are related by the equation qxpxy += 2 , where p and q are constants. A straight line is obtained
by plotting x
y against x, as shown in
Diagram 1.
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]
(b) The tangent to the circle at point Q intersects the y-axis at point T. Find the area of triangle OQT
[4m]
SPM 2004(P1)
1. Diagram 3 shows a straight line graph of
x
y against x
Given that 26 xxy −= , calculate the value of k and of h
[3m]
2. Diagram 4 shows a straight line PQ with
the equation 132
=+ yx. The point P lies
on the x-axis and the point Q lies on the y- axis
24
x
yx
y
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
(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.
Express p in terms of k [2m]
P2(section B)2. Solutions to this question by scale drawing will not accepted.
(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 : 3Find 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
25
JK : kpxy +=RT : pxky +−= )2(
where p and k are constant
The equation of the straight line CB is 12 −= xy Find the coordinates of B
[3 marks]P2(section B)1. Solutions to this question 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) Calculate the area, in unit2, of triangle AOB
(b) Given that AC:CB = 3:2, find the coordinates of C
(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
SPM 2007Section A (paper 2) 1. solutions by scale drawing will not be accepted In diagram 1, the straight line AB has an equation 082 =++ xy .AB intersects the x-axis at point A and intersects the y-axis at point B
Diagram 1
Point P lies on AB such that AP:PB = 1:3Find
(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)
26
082 =++ xy
1. The straight line 16
=+h
yx has a
y- intercept of 2 and is parallel to the straight line 0=+ kxy .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)
Diagram 13
(a) Write down the equation of the straight line ST in the form
1=+b
y
a
x
(b) A point P(x,y) moves such that PS = PT. Find the equation of the locus of P [4 m]
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]
SPM 2008 Section B (paper 2)1. Diagram shows a triangle OPQ. Point S lies on the line PQ.
(a) A point W moves such that its
distance from point S is always 2
12
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,
27
(ii) the coordinates of Q [5m]
(c) Hence, find the area, in unit2, of triangle OPQ
[2m]
SPM 19931. The mean for the numbers 6, 2, 6, 2, 2, 10, x, y is 5 (a) show that 12=+ yx (b) hence, find the mode for the numbers when (i) yx = (ii) yx ≠
(c) if standard deviation is 372
1, find
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
Number
students5 8 12 11 4
(a) On a graph paper, draw a histogram
28
CHAPTER 7: STATISTICS
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 19941. 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 22. 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 19951. (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.
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)
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 19961. The list of numbers ,2−x ,4+x
,52 +x ,12 −x 7+x and 3−x has a mean of 7.Find
(a) the value of x(b) the variance
[6m]2.
Numbers of classes Numbers of pupils6 355 364 30
Age Numbers of villagers1-20 5021-40 7941-60 4761-80 1481-100 10
29
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
SPM 19971. The table shows a set of numbers which has been arranged in an ascending order where m is a positive integer
Set numbers
1 m-1 5 m+3 8 10
Frequency 1 3 1 2 2 1
(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. (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, 6Player B: 7, 8, 8, 9, 7, 9Using 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 questionThe data in the table shows the monthly salary of 100 workers in a company.
(i) Based on the data, draw an ogive to show distribution of the workers’ monthly salary
(ii) From your graph, estimate the number of workers who earn more than RM 3 200
SPM 19981. 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
is 8
5m.
Find (a) the values of m and k [4m](b) the variance of the new data [2m]
Length (mm) Numbers of fish20-29 230-39 340-49 750-59 1260-69 1470-79 980-89 3
Monthly Salary(RM)
Numbers of workers
500-1 000 101 001-1 500 121 501-2 000 162 001-2 500 222 501-3 000 203 001-3 500 123 501-4 000 64 001-4 500 2
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 xΣ and 2xΣ(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 19991. 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
2. The frequency distribution of marks for 30 pupils who took a additional mathematics test is shown in the table
Marks Frequency20-39 649-59 560-79 1480-99 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]
SPM 20001. The table shows the results 100 students
in a test
(a) Based on the table above, copy complete the table below
[2m]
(b) Without drawing an ogive, estimate the interquartile range of this distribution.
[4m]
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
SPM 20011. (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
[3m]2. The table shows the frequency distribution of the marks obtained by 100 pupils
Marks Number of pupils6-10 1211-15 2016-20 2721-25 1626-30 1331-35 1036-40 2
Marks <10 <20 <30 <40 <50 <60 <70 <80Number of students
2 8 21 42 68 87 98 100
Marks <10 <20 <30 <40 <50 <60 <70 <80Number of students
2 8 21 42 68 87 98 100
Marks 0-9
Frequency
Marks <10 <20 <30 <40 <50 <60 <70 <80Number of students
2 8 21 42 68 87 98 100
Marks <10 <20 <30 <40 <50 <60 <70 <80Number of students
2 8 21 42 68 87 98 100
Marks 20-29 30-39 40-49 50-59 60-69 70-79
Number of pupils
2 14 35 50 17 2
Marks <10 <20 <30 <40 <50 <60 <70 <80Number of students
2 8 21 42 68 87 98 100
31
(i) Calculate the variance [3m]
(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 20021. 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 14Number
of pupils
1 1 2 3 1 1
Find the values of x and y [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
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
654321 ,,,,, xxxxxx has a mean of 5 and a standard deviation of 1.5
(a) Find (i) the sum of the marks,
x∑(ii) the sum of the squares
of the marks, 2x∑[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]
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
Findthe value of this number
(ii) the standard deviation of the set 11 numbers
[4 marks]
SPM 2005,paper 11. 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. Diagram 2 is a histogram which represents the distribution of the marks
Scores ≤5
≤ 10
≤ 15
≤ 20
≤ 25
≤ 30
≤ 35
Number of pupils
32
obtained by 40 pupils in a test.
(a) Without using an ogive, calculate the median mark [3m] (b) Calculate the standard deviation of the distribution [4m]
SPM 2006paper 11. 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.
(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 2007Paper 2
1. Table 1 shows the cumulative frequency distribution for the scores of 32 students in a competition
Score Number of pupils10-19 120-29 230-39 840-49 1250-59 k60-69 1
33
Table 1
(a) Based on table 1, copy and complete Table 2
Table 2[1 m]
(b) Without drawing an ogive, find the interquatile range of the distribution [5 m]
SPM 2007Paper 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
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.
Given that the median mark is 35.5, find the value of x and of y. Hence, state the modal class
[6m]
SPM 1993
Score <10 <20 <30 <40 <50Number of students
2 8 21 42 68
Marks Number of candidates10-19 420-29 x30-39 y40-49 1050-59 8
Marks 0-9 10-19 20-29 30-39 40-49
Frequency
34
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 19941.
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
2. In the diagram, M and N are the centers of two congruent circles with radius r cm respectively.Show that 0120=∠PMQ
b. Find, in terms of π and r, the area of the shaded region
[6m]SPM 1995
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 376422 ++=+ yxyx(c) (i) Find the area of major sector BAPC in terms of π
(ii) Hence, show that the area of the shaded region is )1(25 +π unit 2
35
CHAPTER 8: CIRCULAR MEASURE
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.
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.
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 1612
=+ yx
Calculate(a) the area of ABC∆(b) ACB∠ in radians
(c) the area of the shaded region
SPM 1997(a) Convert
(i) 64020' into radians(ii) 4.36 radians into
degrees[2m]
(b)
The diagram shows two sectors OPQ and ORS of two concentric circle with centre O. Given
θ=∠OPQ 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]
2.
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(a) QST∠ in rad
36
(b) the area of the shaded region
[4m]
SPM 1998
1.
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, 6.0=∠POQ rad, 8=OR cm, and the length of arc PQ same as that arc QRFind
(a) the length of PS(b) the length of arc ST
2.
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 19991.
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]
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∠
37
(b) the area of segment AGBH ( c) the area of the shaded region (Use 142.3=π )
SPM 20001.
The diagram shows two sectors OAP and OBQ, of two concentric circle with centre O. Given 5.0=∠AOP rad, AOOB 3= , and the ratio of the length of arc AP to the length of arc BQ is 2:3Calculate BOQ∠ in degrees
[5m]
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]
SPM 20011.
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.
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 π2636 ++[5m]
38
SPM 20021.
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.
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]
SPM 2003paper 1
1. Diagram 1 shows a sector ROS with centre O
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
Diagram 1
Calculate(a) the value of θ , in rad, [3m](b) the area of the shaded region , in
cm2. [4m]
39
SPM 2004 paper 11. Diagram 1 shows a circle with centre O
Given that the length of the major arc AB is
45.51 cm, find the length, in cm, of the radius.(use π = 3.142)
[3m]
paper 2(section B)
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]
the length, in cm, of the arc JKL[4m]
the area, in cm2, of the shaded region[4m]
SPM 2005paper 11. Diagram 1 shows a circle with centre O
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
6
π=∠POQ radian
40
It is given that OA : OP = 4 : 7(Use 142.3=π )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
Diagram 7
Given that OB = 10 cm, AY = 4 cm, 1.1=∠XAY 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
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.
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 956.1=∠COQ radians[use ]142.3=πCalculate
(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
41
[4m]
SPM 2007 Paper 1
1. Diagram 4 shows a sector BOC of a circle with centre O
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]
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 ]142.3=π
Calculate(a) the length, in cm, of the arc AB
[5 m](b) the area in cm 2 , of shaded region
[5 m]
SPM 2008 paper 11. Diagram 18 shows a circle with centre O
and radius 10 cm.
Given that P, Q and R are points such that OP = PQ and ∠ OPR = 900, [Use ]142.3=π Find (a) ∠ QOR, in radians
(b) the area, in cm2 of the coloured Region
42
[4m]
SPM 2008 paper 21. 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 ]142.3=πGiven that θ=∠PXR radian,(a) show that 37.1=θ (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]
43
SPM 1993
1. Given that 34
21)(
2
−−=x
xxf , find f '(x)
SPM 1994
1. (a) Given that 53 2 += xy , find dx
dy using
the first principle
(c) Find
+12
1
xdx
d
2. Given 4
16
xy = , find
dx
dy if 2=x . Hence,
estimate the value of ( ) 498.1
16
SPM 1995
1. Given 1
21)(
3
−−=x
xxf find f ' (x)
2. Given )3( xxy −= , express
122
2
++dx
dyx
dx
ydy in terms of x.
Hence, find the value of x that satisfy the
equation
122
2
++dx
dyx
dx
ydy
3. Find the coordinates at the curve 2)52( −= xy where the gradient of the normal for the curve
is 4
1
SPM 1996
1. Differentiate 74 )31( xx + with respect to x.
2. The gradient of the curve 2x
khxy += at
the point
−−
2
7,1 is 2. Find the values of h
and k
3. Given 32 −= xp and 2
3
py −= . Find
(a) the approximate change in x if thE rate of change in p is 3 units per second
dx
dyin terms of x
the small change in y, when x decreases from 2 to 1.98
SPM 1997
(a) Find the value of
−−
→ 2
4
2
lim 2
n
n
n
(b) Given 5)32()( −= xxf find f ′′ (x)
2. Differentiate 34 −=x
y using the
first principle
44
CHAPTER 9: DIFFERENTATION
3. (a)
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 4p2 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
)64(4
3 3hV −=
(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 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
ππ96
4
2
+−= xx
A
(b) Calculate the value of x so that the area of the shaded region is a
minimum
[5m]
SPM 19981. Given that ,)12(4)( 5−= xxxf find )(' xf
2.
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 24π cm 3
a) Prove that the total surface area of the block, A cm 2 , is given by
A =
+
xx
163 2π [3m]
b)Calculate the minimum surface area of the block [3m]
c) Given the surface area of the block changes at a rate of 42π cm 2 s 1− . Find the
45
of change of its radius when its radius is 4 cm. [2m]
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
1. Given ( )x
xxf
31
2)(
52
−−= , find
)0('f [4m]
2. Given 22tty −= and 14 += tx
(a)Find dx
dy, in terms of x
(b) If x increases from 3 to 3.01, find the corresponding small increase in t. [2m]3 (a)
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 248300 xxV −=
(ii) Calculate(b) the value of x that makes
V a maximum(c) the maximum value of V
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 142.3=π ) (i) Calculate the rate of change
in the radius of the circle(ii) Hence, calculate the radius of
the circle after 4 second
SPM 20001. Differentiate the following expressions with respect to x
(a) 431 x+ [2m]
(b) 3
524 +
+x
x[2m]
Given 643 2 +−= xxy . When 5=x , x increases by 2%. Find the corresponding rate of change of y.
Find the equation of the tangent to the curve rxy += 22 at the point kx = . If the tangent passes through the point (1,0), find r in terms of k
4.(a) The straight line kxy =+4 is the
normal to the curve ( ) 312 2 −−= xy at point A.
Find (i) the coordinates of point A and the
value of k(ii) the equation of the tangent at
point A
46
4.(b) The diagram shows a toy inthe 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
cm, find an expressions for dx
dL in
terms of x and hence, find the maximum area of triangle ABC
SPM 2001
1. Given r
rrf
25
34)(
−+= find limited value
of )(rf when ∞→r
2. Given that graph of function
23)(
x
khxxf += has gradient function
32 96
3)('x
xxf −= where h and k are
constants, Find
a. the values of h and kb. x-coordinate of the
turning point of the graph of the function
3. (a)
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 =
2
4
420 yy
+− π
b. Using 142.3=π , find the length and width of the rectangle that make the area of the shaded region a maximum
Given 752 23 +−= xxy , find
the value of dx
dy at the point (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
1. Given 23
2
5)1( ttp ++=
Find dt
dp and hence find the values of t
where 7=dt
dp
2.
47
y = 2x – x2
(a) The diagram shows the curve 23 xxy −= 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]
(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
SPM 2003 paper2(section A)
1. Given that )5(14 xxy −= , calculate (a) the value of x when y is a
maximum(c) the maximum value of y
[3m]
2. Given that xxy 52 += , use differentiation to find the small change in y when x increases from 3 to 3.01
[3m]
3. (a) Given that 22 += xdx
dy and
6=y when 1−=x , find y in terms of x [3m]
(b) Hence, find the value of x if
8)1(2
22 =+−+ y
dx
dyx
dx
ydx
[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
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 = hr 2
3
1π ) [4m]
SPM 20041. Differentiate 42 )52(3 −xx with respect to x [3m]
2. Two variables x and y are related by the
equation x
xy2
3 += .
Given that y increases at a constant rate of 4 units per second, find the rate of change of x when x = 2 [3m]
paper 2(section B)3. The gradient function of a curve which passes through A(1, -12) is xx 63 2 − . 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]
SPM 2005
1. Given that ( ) 253
1)(
−=
xxh , evaluate
h" )1( [4m]
2. The volume of water, V cm3, in a
container is given by hhV 83
1 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]
paper2(sectionA)3. A curve has a gradient function xpx 42 − ,
where p is a constant. The tangent to the curve at the point (1,3) is parallel to the straight line 05 =−+ xy . Find
(a) the value of p (b) the equation of the curve
SPM 2006 Paper 11. The point P lies on the curve 2)5( −= xy . It is given that the
gradient of the normal at P is 4
1−
Find the coordinates of P[3m]
2. It is given that 7
3
2uy = , where
53 −= xu . Find dx
dy in terms of x
[4m]
3. Given that 43 2 −+= xxy
(a) find the value of dx
dy when x =1
(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 2007Paper 2
1. A curve with gradient function 2
22
xx −
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 2007Paper 1
1. The curve )(xfy = is such that
dx
dy= 53 +kx , where k is a constant.
The gradient of the curve at 2=x is 9 Find the value of k
[2 m]2. The curve 64322 +−= xxy has a minimum point at px = , where p is a constant. Find the value of p
[3 m]
SPM 2008Paper 11. Two variables x and y are related by the
equation 2
16
xy = .
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]
49
2. The normal to the curve xxy 52 −= at point P is parallel to the straight line
12+−= xy . Find the equation of the normal to the curve at point P.
[4m]
SPM 19931.
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]
(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.
The diagram shows a land form triangle, ABC, divide by three parts. ADB, BFC, and AEGC is a straight line
Given that sin13
12=∠BAC
(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
SPM 19941.
50
CHAPTER 10: SOLUTION OF TRIANGLE
In the diagram, BCD is a straight line, calculate the length of CD
2.
The diagram shows a pyramid with ABC∆ as the horizontal base. Given that AB = 3 cm, BC = 4 cm and 090=∠ABC and vertex D is 4 cm vertically above B, calculate the area of the slanting face.
[5m]
SPM 19951.
In the diagram, sin5
4=∠ADC where
ADC∠ is an obtuse angle. Calculate
(a) the length of AC correct to two decimal places [3m](b) ABC∠ [2m]
SPM 1996 1.
The diagram shows a cuboid. Calculate(a) JQL∠ [4m](b) the area of JQL∆ [2m]
2.
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 1997
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]
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
2.
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 19991.
The diagram shows a trapezium ABCDCalculate(a) CBD∠(b) the length of straight line AC
2. JKL is a triangle with side JK = 10 cm. Given that sin 456.0=∠KJL and sin 36.0=∠JKL , Calculate (a) JLK∠ (b) the area of JKL∆
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
Hence, show that cos29
11=α
2.
In the diagram, PQR is a straight line. Calculate the length of PS
SPM 20011.
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
080=∠VQR Calculate(a) the length of QR(b) the area of the slanting face
SPM 20021.
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
2.
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
[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
SPM 2004
1. The diagram shows a quadrilateral ABCD such that ABC∠ is acute
(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]
54
SPM 2005
1. The diagram shows triangle ABC
(a) Calculate the length, in cm, of AC [2m]
(b) A quadrilateral ABCD is now formed so that AC is a diagonal, 040=∠ACD 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
kitaranvertical = mencancanghorizontal = mengufukobtuse angle = sudut cakahslanting face = permukaan condongacute = tirusformed = dibentuk
diagonal = pepenjuruSPM 200 6
1. Diagram 5 shows a 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]
55
SPM 2006
1. Diagram 7 shows quadrilateral ABCD
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]
SPM 19931. The table below shows the monthly expenses of Ali’s family
YearExpenses
1998 1992
Food RM 320 RM 384Transportation RM 80 RM 38Rental RM 280 RM 322Electricity & water RM 40 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]
SPM 19941. 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
56
CHAPTER 11: INDEX NUMBER
Monthly expenses
Price Index
Food 130House rental 115Entertainment 110Clothing 115Others 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
SPM 19951. 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
SPM 19961. (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]
(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
Index1994
Changes toPrice 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
year 1996[3m]
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.
Item Price Index WeightageShirt 100 NTrousers 110 6Bag 140 2Shoes 100 4
Food item
Price index Weightages
Fish 140 4Prawn 120 2Chicken 125 4Beef 115 3Cuttlefish 130 X
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Item Price index WeightageP 120 2Q 150 nR m 3
Calculate(a) the value of m [2m](b) the value n [2m]
SPM 19981. 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
1. The composite index number of the data in the below table is 108
Find the value of x [4m]
SPM 2000
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.
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]
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 Price index WeightageA x 5B 98 yC 123 14 - y
SPM 2002
1. The table below shows the prices, price indices and the number of three items
Item
Pric
e (RM) Price Index Number
Year 1999
Year 2000
(Base year 1999) of items
A 55 66 120 200B 40 x 150 500C 80 100 125 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
Index number, Ii 105 94 120Weightages, Wi 5 - x x 4
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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
(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]
SPM 2003
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.
Items Price in 1990
Price in 1995
Price Index in 1995 based on 1990
P x RM 0.70 175
Q RM 2.00 RM 2.50 125R RM 4.00 RM 5.50 yS RM 6.00 RM 9.00 150T RM 2.50 z 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
Type of item
Price (RM) in 1996
Price (RM) in 1998
Weightage%
A 70 105 Y
B 80 100 X
C 60 67.50 2x
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05
101520253035
P Q R S T
ITEMS
WE
EK
LY
CO
ST
(R
M)
( 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 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
(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]
(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]
SPM 20051. 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 Price per kg Price index for the
year 2004 based on the year 2001
Year 2001
Year 2004
P 0.80 1.00 xQ 2.00 y 140R 0.40 0.60 150S z 0.40 80
(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]
Item Price index for the year 1995 based on the year 1993
Percentage of usage (%)
P 135 40Q x 30R 105 10S 130 20
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(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 20061. A particular kind of cake is made by
using four ingredients P, Q, R and S. Table shows the prices of the ingredients
(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]
(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 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]
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
Ingredient Price per kilogram (RM)
Year 2004 Year 2005P 5.00 WQ 2.50 4.00R x YS 4.00 4.40
Component Price (RM) for the year
Price index for the year 2006 based on
the year 2004P 1.20 1.50 125Q x 2.20 110R 4.00 6.00 150S 3.00 2.70 yT 2.00 2.80 140
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[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]
62