Spm Past Questions Add Maths

CHAPTER 1: FUNCTIONS SPM 1993

SPM 1995

1. Given the function f : x → 3 – 4x and function g : x → x2 – 1, find (a) f -1 (b) f -1g(3) [5 marks]

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

2. Given the functions f, g and h as a f : x → 2x

2. Given the function f : x → mx + n, g : x → (x + 1)2 – 4 and fg : x → 2(x + 1)2 – 5. Find (i) g2(1) (ii) the values of m and n (iii) gf -1 [5 marks]

h : x → 6x2 – 2

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

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

SPM 1996 hx + k , x≠2 x −2 2x − 5 and inverse function f -1 : x → , x≠3 x −3

1. Given the function f : x →

SPM 1994 1. Given the functions f(x) = 2 – x and function g(x) = kx2 + n. If the composite function gf(x) = 3x2 – 12x + 8, find (a) the values of k and n [3 marks] (b) the value of g2(0) [2 marks]

Find (a) the values of h and k [3 marks] (b) the values of x where f(x) = 2x [3 marks] 2. Given the function f : x → 2x + 5 and fg : x →13 – 2x, Find (i) function gf (ii) the values of c if gf(c2 + 1) = 5c - 6 [5 marks]

2. The function f is defined as f:x→

[3 marks] [3 marks]

3 g:x→ ,x≠2 x −2

(i) (ii)

4 . Find 3

p +x , for all value of x except 3 + 2x

x = h and p is a constant. (i) determine the value of h (ii) the value of 2 maps by itself under function f. Find (a) the value of p (b) the value of another x which is mapped onto itself (c) f -1(-1) [7 marks]

SPM 1997 1. Given the functions g: x → px + q and g2 : x→ 25x + 48 (a) Find the value of p and q (b) Assume that p>0, find the value of x so that 2g(x) = g(3x + 1) b\

1

SPM 1998

SPM 2001

1. Given the functions h(t) = 2t + 5t2 and v(t) = 2 + 9t Find (a) the value of h(t) when v(t) = 110 (b) the values of t so that h(t) = v-1(2) (c) function hv

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

2. Given the function f -1(x) = p − x , x ≠ p and g(x) = 3 + x. Find (a) f(x) [2 marks] -1 2 (b) the value of p if ff (p –1) = g[(2-p)2] ( c) range of value of p so that fg-1(x) = x no real roots [5 marks]

1. Given the functions f(x) = 6x + 5 and g(x) = 2x + 3 , find (a) f g-1(x) (b) the value of x so that gf(-x) = 25 SPM 1999 1. Given the function f : x → k – mx. Find (a) f -1(x) in terms of k and m [2 marks] (b) the values of k and m, if f -1(14) = - 4 and f(5) = -13 [4 marks]

SPM 2002 1. Given the function f(x) = 4x -2 and g(x) = 5x +3. Find (i) fg -1(x)

2. (a) The function g is defined as g : x → x + 3. Given the function fg : x → x2 +6x + 7. Find (i) function f(x) (ii) the value of k if f(2k) = 5k [7 marks]

(ii)

f(x) = 3x2 – 5. Find (a) g(x)

x 2 )= 2 5

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

SPM 2000 1. Given the function g -1(x) =

the value of x so that fg-1(

5 − kx and 3

[2 marks]

(b) the value of k when g(x2) = 2f(-x) [3 marks] 2. Given the function f : x → 4 – 3x. (a) Find (i) f2(x) (ii) (f2)-1(x) (iii) (f -1)2 [6 marks]

2

SPM 2003

3. Given the function h(x) = P = {1, 2, 3} Q = {2, 4, 6, 8, 10}

6 , x ≠ 0 and x

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

1. Based on the above information, the relation between P and Q is defined by set of ordered pairs {(1,2), (1,4), (2,6), (2,8)}. State (a) the image of 1 (b) the object of 2 [2 marks]

SPM 2005 1. In Diagram 1, the function h maps x to y and the function g maps y to z

2. Given that g : x → 5x + 1 and h : x → x2 – 2x +3, find (a) g-1(3) (b) hg(x) [4 marks] SPM 2004 1. Diagram 1 shows the relation between set P and set Q

d∙

∙ ∙ ∙ ∙

e∙ f∙ Set P

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

w x y z

[2 marks]

2. The function w is defined as w(x) =

5 , x ≠ 2. Find 2 −x

(a) w-1(x) (b) w-1(4)

Set Q Diagram 1

[3 marks]

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

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

h : x → 2x – 3 g : x → 4x - 1

2. Given the function h : x → 4x + m and h-1 : x → 2hk +

5 , where m and k are 8

Find gh-1 [3 marks]

constants, find the value of m and of k. [3 marks]

3

SPM 2006 Paper 1 1. In diagram 1, set B shows the image of certain elements of set A

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

g:x→

[3 m] SPM 2007 Paper 1 1. Diagram 1 shows the linear function h. DIAGRAM 1 (a) State the type of relation between set A and set B (b) Using the function notation, write a relation between set A and set B [2 marks] 2. Diagram shows the function h:x →

m−x , x ≠ 0 , where m is a constant x

(a) State the value of m (b) Using the function notation, express h in terms of x [2 m] −

2.

1 2

Given the function , find the value of x such that f ( x) = 5 [2m] f : x →x −3

DIAGRAM 2 Find the value of m [2 marks]

4

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

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

h2

2. Given the function g : x → 5 x + 2 and h : x → x 2 − 4 x + 3 , find

, where a and b are constants and

a) g −1 (6) b) hg (x ) [4m]

3. Given the functions f ( x) = x −1 and g ( x ) = kx + 2 , find a) f(5) b) the value of k such that gf(5)=14 [3m]

Find the value of a and b [3m] SPM 2008 Paper 1 1. Diagram 1 shows the graph of the function f ( x) = 2 x −1 , for the domain 0 ≤ x ≤ 5 .

State (a) the value of t

5

(2x – 3)(x + 4) + k = 0 and m = 4n, find the value of k [5 marks] 2. Find the values of λ so that (3 – λ)x2 – 2(λ + 1)x + λ + 1 = 0 has two equal real roots. [2 marks] CHAPTER 2: QUADRATIC EQUATIONS SPM 1994

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

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

SPM 1998

3

[4 marks]

1. The equation of px2 + px + 3q = 1 + 2x 1

SPM 1995

have the roots p and q (a) Find the value of p and q (b) Next, by using the value of p and q in (a) form the quadratic equation with roots p and -2q

1. One of the roots of the equation x2 + px + 12 = 0 is one third of the other root. Find the possible values of p. [5 marks] 2. Given that

SPM 1999

1 and -5 are the roots of the 2

1. One of the roots of the equation 2x2 + 6x = 2k - 1 is double of the other root, where k is a constant. Find the roots and the possible values of k. [4 marks]

quadratic equation. Write a quadratic equation in a form ax2 + bx + c = 0 [2 marks] 3. Find the range of value of k if the equation x 2 + kx + 2k − 3 = 0 has no real roots [3 marks]

2. Given the equation x2 – 6x + 7 = h(2x – 3) have two equal real roots. Find the values of h. [4 marks]

4. Prove that the roots of the equation (1 – p)x2 + x + p = 0 has a real and negative roots if 0 < p < 1 [5 marks]

3. Given that α and β are the roots of the equation x2 – 2x + k = 0, while 2α and 2β are the roots of the equation x2 +mx +9=0. Find the possible values of k and m. [6 marks]

SPM 1996 1. Given that a and b are the roots of the equation x2 – (a + b)x + ab = 0. If m and n are the roots of the equation

SPM 2000

6

1. The equation 2x2 + px + q = 0 has the roots -6 and 3. Find (a) the values of p and q [3 marks] (b) the range of values of k if the Equation 2x2 + px + q = k has no real roots [2 marks]

has two distinct roots. Find the range of values of p [3 marks] SPM 2004 1. Form the quadratic equation which has the roots -3 and

the form ax2 + bx + c =0, where a, b and c are constants [2 marks] SPM 2005

SPM 2001 1. Given that 2 and m are the roots of the equation (2x -1)(x + 3) = k(x – 1), where k is a constant. Find the values of m and k [4 marks]

1. Solve the quadratic equation x(2x – 5) = 2x – 1. Give your answer correct to three decimal places. [3 marks]

2. If α and β are the roots of the quadratic equation 2 x 2 + 3x − 1 = 0 , form another quadratic equation with roots 3α + 2 and 3β + 2. [5 marks]

SPM 2006 1. A quadratic equation x 2 + px + 9 = 2 x has two equal roots. Find the possibles values of p. [3 marks]

SPM 2002 1. Given the equation x2 + 3 = k(x + 1) has the roots p and q, where k is a constant, find the range of value of k if the equation has two different real roots. [5 marks] 2. Given that

1 . Give your answer in 2

SPM 2007 1. (a) Solve the following quadratic equation: 3x 2 + 5 x − 2 = 0

α β and are the roots of the 2 2

(c) The quadratic equation hx 2 + kx + 3 = 0, where h and k are constants, has two equal roots Express h in terms of k [4 marks]

equation kx(x – 1) = 2m – x. If α + β = 6 and αβ = 3, find the values of k and m. [5 marks]

SPM 2008 1. It is given that -1 is one of the roots of the quadratic equation

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

x 2 − 4x − p = 0

Find the value of p [2 marks]

2. The quadratic equation x(x + 1) = px - 4

7

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

CHAPTER 3: QUADRATIC FUNCTIONS SPM 1993

SPM 1995

1. Given the quadratic equation f(x) = 6x – 1 – 3x2. (a) Express quadratic equation f(x) in the form k + m(x + n)2, where k, m and n are constants. Determine whether the function f(x) has the minimum or maximum value and state the value of the minimum or maximum value.

1. Without using differentiation method or drawing graph, find the minimum or maximum value of the function y = 2(3x – 1)(x + 1) – 12x – 1. Then sketch the graph for the function y. [5 marks] 2. Given that 3x + 2y – 1 = 0, find the range of values of x if y < 5. [5 marks]

(b) Sketch the graph of function f(x) (c ) Find the range of value of p so that the equation 6x – 4 - 3x2 = p has two different real roots. [10 marks] SPM 1994

3. Find the range of values of n if 2n2 + n ≥ 1 [2 marks] SPM 1996 1. f(x) = 0 is a quadratic equation which has the roots -3 and p. (a) write f(x) in the form ax2 + bx + c [2 marks] (b) Curve y = kf(x) cut y-axis at the point (0,60). Given that p = 5, Find (i) the value of k (ii) the minimum point [4 marks]

1. In the diagram 1, the minimum point is (2, 3) of the function y = p(x + h)2 + k. Find (a) the values of p, h and k (b) the equation of the curve when the graph is reflected on the x-axis [2 marks]

2. Find the range of values of x if (a) x(x + 1) < 2 [2 marks] (b)

2. (a) Find the range of value of x if 5x ≥ x2 [2 marks]

−3 ≥x 1 −2x

SPM 1997

8

[3 marks]

(b) Find the range of value of x that satisfy inequality (x – 2)2 < (x – 2)

2

1. Quadratic function f(x) = 2[(x – m) + n], with m and n are constants, have a minimum point p(6t,3t2). (a) state the value of m and n in terms of t (b) if t = 1, find the range of value of k so that the equation f(x) = k has a distinct roots

SPM 1999 1. (a) Find the range of value of x so that 9 + 2x > 3 and 19 > 3x + 4 (b) Given that 2x + 3y = 6, find the range of value of x when y < 5 2. Find the range of value of x if (x – 2)(2x + 3) > (x – 2)(x + 2)

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

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

3. Given that y = x2 + 2kx + 3k has a minimum value 2. (a) Without using differentiation method, find two possible value of k. (b) By using the value of k, sketch the graph y = x2 + 2kx + 3k in the same axis (c) State the coordinate of minimum point for the graph y = x2 + 2kx + 3k

2. The straight line y = 2x + k does not intersect the curve x2 + y2 – 6 =0 . Find the range of values of k [5 marks]

SPM 1998 1.

SPM 2001 1.(a) State the range of value of x for 5x > 2x2 – 3 (b) Given that the straight line 3y = 4 – 2x and curve 4x2 + 3y2 – k = 0. Show that the straight line and the curve does not intersect if k < 4 The graph show two curve y = 3(x-2)2 + 2p and y = x2 + 2x – qx + 3 that intersect in the two point at x-axis. Find (a) the value of p and q (b) the minimum value for the both curve

1

2. Given that f-1 (x) = p − x , x ≠ p and g(x) = 3 + x. Find the range of value of p so that f-1g(x) = x has no real roots SPM 2002 1. Given the quadratic equation x2 + 3 = k(x + 1), where k is a constant, which has the roots p and q. find the range of values of k if p and q

2

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

9

has two distinct roots. SPM 2005 (paper 1) 2. Given that y = p + qx – x2 = k – (x + h)2 for all values of x (a) Find (i) h (ii) k in terms of p and/or q (b) the straight line y = 3 touches the curve y = p + qx – x2 (i) state p in terms of q (ii) if q = 2, state the equation of the axis of symmetry for the curve. Next, sketch the graph for the curve

1. The straight line y = 5x – 1 does not intersect the curve y = 2x2 + x + p. Find the range of values of p [3 marks] 2. Diagram 2 shows the graph of a quadratic functions f(x) = 3(x + p)2 + 2, where p is a constant.

SPM 2003 (paper 2) 1. The function f(x) = x2 – 4kx + 5k2 + 1 has a minimum value of r2 + 2k, where r and k are constants. (a) By using the method of completing square, show that r = k -1 [4 marks] (b) Hence, or otherwise, find the values of k and r if the graph of the function is symmetrical about x = r2 - 1 [4 marks]

The y = f(x) has the minimum point (1, q), where q is a constant. State

curve

(a) the value of p (b) the value of q (c ) the equation of the axis of symmetry SPM 2005 (paper 1) 1. Diagram 2 shows the graph of a quadratic function f(x)=3(x + p)2 + 2, where p is a constant

SPM 2004 (paper 1) 1. Find the range of values of x for which x(x – 4) ≤ 12 [3 marks] 2. Diagram 2 shows the graph of the function y = -(x – k)2 – 2, where k is a constant.

Diagram 2 The curve y = f(x) has the minimum point (1,q), where q is a constant. State a) the value of p b) the value of q

Find Diagram 2 (a) the value of k (b) the equation of the axis of symmetry (c) the coordinates of the maximum point [3 marks]

10

c)

the equation of the axis of symmetry [3 m]

SPM 2008 (paper 1)

SPM 2006 1. The quadratic function f ( x) = p ( x + q ) 2 + r , where p, q and r are constants, has a minimum value of -4. The equation of the axis of symmetry is x = 3 State a) the range of values of p b) the value of q c) the value of r [3 m]

1. Diagram 3 shows the graph of quadratic function y = f (x) . The straight line y = −4 is a tangent to the curve y = f (x )

a)

write the equation of the axis of

2. Find the range of the value of x for ( x − 3) 2 < 5 − x . [3 m] SPM 2008 (paper 2) 1. Diagram 2 shows the curve of a quadratic function f ( x) = −x 2 + kx − 5 . The curve has a maximum point at B(2,p) and intersects the f(x)-axis at point A

symmetry of the curve b) express f ( x) in the form ( x + b) 2 + c , where b and c are constants. [3 marks] 3. Find the range of the values of x for (2 x −1)( x + 4) > 4 + x

[2 marks] SPM 2007(paper 1) 1. Find the range of values of x for which 2 x 2 ≤ 1 + x [3 marks]

Diagram 2 a) State the coordinates of A

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

[1m] b) By using the method of completing square, find the value of k and of p. [4m] c) determine the range of values of x, if f ( x ) ≥ −5

[2m]

11

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

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

SPM 1994 1. Solve the following simultaneous equation and give your answer correct to two decimal places 2x + 3y + 1 = 0, x2 + 6xy + 6 = 0

2. Diagram 2 shows a rectangular pond JKMN and a quarter part of a circle KLM with centre M. If the area of the pond is 10 π m2 and the length JK is longer than the length of the curve KL by π m, Find the value of x.

2. Diagram 2 shows a rectangular room. shaded region is covered by perimeter of a rectangular carpet which is placed 1 m away from the walls of the room. If the area and the perimeter of the carpet are 8

3 2 m and 12 m, find the measurements 4

of the room. 1m 1m

1m SPM 1998 1. Solve the simultaneous equation:

1m

x 2 + = 4 , x + 6y = 3 3 y

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

2. Diagram 2 shows the net of an opened box with cuboids shape. If perimeter of the net box is 48 cm and the total surface area is 135 cm3, Calculate the possible values of v and w.

2. A cuboids aquarium measured u cm × w cm × u cm has a rectangular base. The top part of it is uncovered whilst other parts are made of glass. Given the total length of the aquarium is 440 cm and the total area of the glass used to make the aquarium is 6300 cm2. Find the value of u and w SPM 1996 1. Given that (-1, 2k) is a solution for the equation x2 + py – 29 = 4 = px – xy , where k and p are constants. Determine the value of k and p

12

SPM 1999 1. Given the curve y2 = 8(1 – x) and the

1. Given that x + y – 3 = 0 is a straight line cut the curve x2 + y2 – xy = 21 at two different point. Find the coordinates of the point

y straight line = 4. Without drawing the x

graph, calculate the coordinates of the intersection for the curve and the straight line. 2. Solve the simultaneous equation 2x + 3y = 9 and

2.

x 6y − y = −1 x

yam

SPM 2000 1. Solve the simultaneous equation 3x – 5 = 2y , y(x + y) = x(x + y) – 5 2. Solve the simultaneous equation y y x 3 1 − + 3 = 0 and + − =0 3 2 x 2 2

Pak Amin has a rectangular shapes of land. He planted padi and yam on the areas as shown in the above diagram. The yam is planted on a rectangular shape area. Given the area of the land planted with padi is 115 m2 and the perimeter of land planted with yam is 24 m. Find the area of land planted with yam.

SPM 2001 1. Given the following equation: M = 2x − y N = 3x + 1 R = xy − 8 Find the values of x and y so that 2M = N = R 4. Diagram 2 shows, ABCD is a piece of paper in a rectangular shape. Its area is 28 cm2. ABE is a semi-circle shape cut off from the paper. the perimeter left is 26 cm. Find the integer values of x and y

SPM 2003 1. Solve the simultaneous equation 4x + y = −8 and x2 + x − y = 2 SPM 2004 1. Solve the simultaneous equations p − m = 2 and p2 + 2m = 8. Give your answers correct to three decimal places. SPM 2005 1. Solve the simultaneous equation

[use

π

=

x+

22 ] 7

1 y = 1 and y2 − 10 = 2x 2

SPM 2006

SPM 2002

13

1. Solve the simultaneous equations 2 x 2 + y = 1 and 2 x 2 + y 2 + xy = 5 Give your answer correct to three decimal places [5 m] SPM 2007 1. Solve the following simultaneous equations: 2 x − y − 3 = 0 , 2 x 2 −10 x + y + 9 = 0 [5 m] SPM 2008 1. Solve the following simultaneous equations : x −3 y + 4 = 0 x 2 + xy − 40 = 0

[5m]

CHAPTER 5: INDICES AND LOGARITHMS SPM 1993 1. If 3 − log 10 x = 2log 10 y, state x in terms of y

14

equation T = 30(1.2) x when the metal is heated for x seconds. Calculate (i) the temperature of the metal when heated for 10.4 seconds (ii) time, in second, to increase the temperature of the metal from 30 0 C to 1500 0 C

2. (a) If h = log m 2 and k = log m 3, state in terms of h and /or k (i) log m 9 (ii) log 6 24 (b)Solve the following equations: (i)

4 2x =

1 32

(ii)

log

16 − log

x

x

SPM 1996 1. (a) Express 2 n +2 − 2 n + 10(2 n −1 ) in a simplify terms (c) Solve the equation 3 x +2 − 5 = 0

2=3

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

2. (a)Solve the following equations: (i) 4 log 2 x =5 (ii) 2 x . 3 x = 5 x +1 (b) Given that log 5 3 = 0.683 and log 5 7 = 1.209. without using a calculator scientific or four-figure table , calculate (i) log 5 1.4 (ii) log 7 75

1 , find the 3

value of n (b) Given that 2 r = 3 s = 6 t . Express t in terms of r and s ( c) Given that y = kx m where k and m are constants. y = 4 when x = 2 and y = 8 when x = 5. Find the values of k and m

SPM 1997 1. Show that log 3 xy = 2 log 9 x + 2 log 9 y. Hence or otherwise, find the value of x and y which satisfies the equation log 9 xy 3 log 3 xy = 10 and = log 9 y 2

SPM 1995 1. Solve the following equations: (a) 81(27 2 x ) = 1 (b) 5 t = 26.3

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

2. (a) Given that m = 2 r and n = 2 t , state in terms of r and/or t  mn 3  32

(b) Solve the equation 5 log x 3 + 2 log x 2 - log x 324 = 4 and give your answer correct to four significant figures.

  , 

(i)

log 2 

(ii)

log 8 m − log 4 n

b) The temperature of a metal increased from 30 0 C to T 0 C according to the

3. (a) Given that 15

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

2 log 3 (x + y) = 2 + log 3 x + log 3 y, show that x 2 + y 2 = 7xy (b) Without using scientific calculator or four-figure mathematical tables, solve the equation log 9 [log 3 (4x – 5)] = log 4 2

SPM 2000 1. (a) Solve 3 log 2 x = 81 (b) If 3 2 x = 8(2 3 x ), prove that

(c ) After n year a car was bought the 7  8 

n

9  8 

price of the car is RM 60 000   .

x log a   = log a 8

Calculate after how many years will the car cost less than RM 20 000 for the first time

 log12 49 × log 64 12   2. (a)Simplify  log16 7   Without using scientific calculator or four-figure mathematical tables

SPM 1998 1. Given that log x 4 = u and log y 5 = y State log 4 x 3 y in terms of u and/or w

(b) Given that 3 lg xy 2 = 4 + 2lgy lgx

2. (a) Given that log a 3 = x and log a 5 = y.  45  Express log a  3  in terms of x and a  y (b) Find the value of log 4 8 + log r r (c ) Two experiments have been conducted to get relationship between two variables x and y. The equation x 3(9 ) = 27 y and log 2 y = 2 + log 2 (x – 2) were obtain from the first and second experiment respectively

with the condition x and y is a positive integer. Show that xy = 10 (c) The total savings of a cooperation after n years is given as 2000(1 + 0.07) n . Calculate the minimum number of years required for the savings to exceed RM 4 000. SPM 2001 1. Given that log 2 k = p and log 3 k = r Find log k 18 in terms of p and r

SPM 1999 1. Given that log 2 3 = 1.585 and log 2 5 = 2.322. Without using scientific calculator or four-figure mathematical tables, Find (a) log 2 45

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

9  5 

(b) log 4  

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

2. (a) Given that x = log 2 3, find the value of 4 x . Hence find the value of 4 y if y=1+x (b) Given that log a 3 = 0.7924. Without

SPM 2002 16

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

2 x −3 = 1. Solve the equation 8

2. Given that

log 2 xy = 2 + 3 log 2 x − log 2 y ,

2. (a) Given that 2 log 4 x − 4 log 16 y = 3 State x in terms of y

express y in terms of x [3 marks] 3. Solve the equation

(b) Solve the simultaneous equation 2 m −1 × 32 k +2 = 16 and 5

×125

3−k

4 x +2

[3 marks]

(b) Solve the equation log 2 ( 7t − 2 ) − log 2 2t = −1

−3 m

1

2 + log 3 ( x + 1) = log 3 x

=1

[3 marks]

where m and k are constants

SPM 2007

SPM 2003,P1 1. Given that log 2 T − log 4 v = 3 , express T in terms of V [4 marks] 2. Solve the equation 4 2 x −1 = 7 x [4 marks]

1. Given that log 2 = x and log 2 c = y ,  8b   in terms of x and y  c 

express log 4 

[4 marks] 2. Given that 9(3 n −1) = 27 n [3 marks]

SPM 2004,P1 1. Solve the equation 32 4 x = 4 8 x +6 [3 marks]

SPM 2008(paper 1) 1. Solve the equation

2. Given that log 5 2 = m and log 5 7 = p , express log 5 4.9 in terms of m and p

16 2 x −3 = 8 4 x

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

SPM 2005,P1 1. Solve the equation 2 x + 4 − 2 x +3 = 1 [3 marks] 2. Solve the equation

log 3 4 x − log 3 ( 2 x − 1) = 1

[3 marks] 3.Given that log m 2 = p and log m 3 = r ,  27m   in terms of p and r  4 

express log m 

CHAPTER 6: COORDINATE GEOMETRY SPM 2006

SPM 1993 17

1. Solutions to this question by scale drawing will not be accepted Point P and point Q have a coordinate of (4,1) and (2, 4). The straight line QR is perpendicular to PQ cutting x-axis at point R. Find (a) the gradient of PQ (b) the equation of straight line QR ( c) the coordinates of R SPM 1993 2. The above diagram show, a parallelogram KLMN. (a) Find the value of T. Hence write down the equation of KL in the form of intercepts (b) ML is extended to point P so that L divides the line MP in the ratio 2 : 3. Find the coordinates of P SPM 1994 1. From the above diagram, point K(1, 0) and point L(-2, 0) are the two fixed points. Point P moves such that PK:PL = 1:2 (a) Show that the equation of locus P is x 2 + y 2 − 4x = 0

(b) Show that the point M(2, 2) is on the locus P. Find the equation of the straight line KM (c ) If the straight line KM intersects again locus P at N, Find the coordinates of N (d) Calculate the area of triangle OMN

2. (a)The above diagram, P, Q and R are three points are on a line 2 y − x = 4 where PQ : QR = 1:4 Find (i) the coordinates of point P (ii) the equation of straight line passing through the point Q and perpendicular with PR (iii) the coordinates of point R

SPM 1994 1. Solutions to this question by scale drawing will not be accepted. Points A, B, C and D have a coordinates (2, 2), (5, 3), (4, -1) and (p, q) respectively. Given that ABCD is a parallelogram, find (a) the value of p and q (b) area of ABCD

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

SPM 1993

18

in the ratio 2SE = SF Find (i) the equation of the locus of S (ii) the coordinates of point when locus S intersect y-axis SPM 1995 1. Solutions to this question by scale drawing will not be accepted. 1. In the diagram, the straight line y = 2 x + 3 is the perpendicular bisector of straight line which relates point P(5, 7) and point Q(n, t) (a) Find the midpoint of PQ in terms of n and t (b) Write two equations which relates t and n ( c) Hence, find the distance of PQ

Graph on the above show that the straight line LMN Find (a) the value of r (b) the equation of the straight line passing through point L and perpendicular with straight line LMN 2. The straight line y = 4 x − 6 cutting the curve y = x 2 − x − 2 at point P and point Q (a) calculate (i) the coordinates of point P and point Q (ii) the coordinates of midpoint of PQ (iii) area of triangle OPQ where Q is a origin

2. The diagram shows the vertices of a rectangle TUVW on the Cartesian plane (a) Find the equation that relates p and q by using the gradient of VW (b) show that the area of ∆TVW can

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

5 2

be expressed as p − q + 10 ( c) Hence, calculate the coordinates of point V, given that the area of rectangular TUVW is 58 units2 (d) Fine the equation of the straight line TU in the intercept form

SPM 1996

SPM 1997

19

(c) A point move such that its distance from point S is point T. (i) (ii)

1 of its distance from 2

Find the equation of the locus of the point Hence, determine whether the locus intersects the x-axis or not

SPM 1998

1. In the diagram, AB and BC are two straight lines that perpendicular to each other at point B. Point A and point B lie on x-axis and y-axis respectively. Given the equation of the straight line AB is 3y + 2x −9 = 0

(a) Find the equation of BC [3m] (b) If CB is produced, it will intersect the xaxis at point R where RB = BC. Find the coordinates of point C [3m] 1. In the diagram, ACD and BCE are straight lines. Given C is the midpoint of AD, and BC : CE = 1:4 Find (a) the coordinates of point C (b) the coordinates of point E (c ) the coordinates of the point of intersection between lines AB and ED produced [3m] 2. Point P move such that distance from point Q(0, 1) is the same as its distance from point R(3, 0). Point S move so that its distance from point T(3, 2) is 3 units. Locus of the point P and S intersects at two points. (a) Find the equation of the locus of P (b) Show that the equation of the locus of point S is x 2 + y 2 − 6 x − 4 y + 4 = 0 ( c) Calculate the coordinates of the point of intersection of the two locus (d) Prove that the midpoint of the straight line QT is not lie at locus of point S

2. The diagram shows the straight line graphs of PQS and QRT on the Cartesian plane. Point P and point S lie on the x-axis and y-axis respectively. Q is the midpoint of PS (a) Find (i) the coordinates of point Q (ii) the area of quadrilateral OPQR [4m] (b)Given QR:RT = 1:3, calculate the coordinates of point T

20

2. The diagram shows the curve y 2 = 16 − 8 x that intersects the xaxis at point B and the y-axis at point A and D. Straight line BC, which is perpendicular to the straight line AB, intersects the curve at point C. Find (a) the equation of the straight line AB [3m] (b) the equation of the straight line BC [3m] (c) the coordinates of point C [4m]

3. In the  1  2

 

diagram, P(2, 9), Q(5, 7) and R  4 ,3  are midpoints of straight lines JK, KL and LJ respectively, where JPQR forms a parallelogram. (a) Find (i) the equation of the straight line JK (ii) the equation of the perpendicular bisector of straight line LJ [5m] (b) Straight line KJ is produced until it intersects with the perpendicular bisector of straight line LJ at point S. Find the coordinates of point S [2m] (c ) Calculate the area of ∆PQR and hence, find the area of ∆JKL [3m]

SPM 2000

SPM 1999 1. Given point A( −2,−4) and point B ( 4,8) . Point P divides the line segment AB in the ratio 2 : 3. Find (a) the coordinates of point P (b) the equation of straight line that is perpendicular to AB and passes through P.

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

produced=diperpanjangkan

21

gradient of line BC is 1 and straight line AC is

perpendicular to the straight line AB. Find (a) the coordinates of points A and B [1m] (b) the equation of the straight lines AC and BC [5m] (c) the coordinates of point C [2m] (d) the area of triangle ABC [2m]

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

3. In the diagram, the equation of BDC is y = −6 . A point P moves such that its distance from A is always

1 the 2

distance of A from the straight line BC. Find (a) the equation of the locus of P (b) the x-coordinates of the point of intersection of the locus and the x-axis [5m]

2. Solutions to this question by scale drawing will not be accepted. Straight line x − 2 y = 6 intersects the x-axis and y-axis at point A and point B respectively. Fixed point C is such that the

22

SPM 2002

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

1. The diagram shows a triangle ABC with an area 18 units2 . the equation of the straight line CB is y − x +1 = 0. Point D lies on the x-axis and divides the straight line CB in the ratio m : n. Find (a) the coordinates of point B (b) m : n

SPM 2003(P1) 1. The points A(2h, h) , B ( p, t ) and C ( 2 p,3t ) are on a straight line. B divides AC internally in the ratio 2 : 3 Express p in terms of t [3m] 2. The equations of two straight lines are y x + = 1 and 5 y = 3 x + 24 . 5 3

Determine whether the lines are perpendicular to each other

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

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

23

y against x, as shown in x

Given that y = 6 x − x 2 , calculate the value of k and of h [3m]

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

2. Diagram 4 shows a straight line PQ with the equation

x y + = 1 . The point P lies 2 3

on the x-axis and the point Q lies on the yaxis

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

Find the equation of the straight line perpendicular to PQ and passing through the point Q [3m] 3. The point A is (-1, 3) and the point B is (4, 6). The point P moves such that PA : PB = 2 : 3. Find the equation of the locus of P [3m] P2(section A) 4. Digram 1 shows a straight line CD which meets a straight line AB at the point D . The point C lies on the y-axis

SPM 2004(P1) 1. Diagram 3 shows a straight line graph of y against x x y x

24

P2(section B) 2. Solutions to this question by scale drawing will not accepted.

(a) write down the equation of AB in the form of intercepts [1m] (b) Given that 2AD = DB, find the coordinates of D [2m] (c) Given that CD is perpendicular to AB, find the y-intercepts of CD [3m] SPM 2005(P1) 1. The following information refers to the equations of two straight lines, JK and RT, which are perpendicular to each other. JK RT

: :

(a) Find (i)

the equation of the straight line AB (ii) the coordinates of B [5m] (b) The straight line AB is extended to a point D such that AB : BD = 2 : 3 Find the coordinates of D [2m] (c) A point P moves such that its distance from point A is always 5 units. Find the equation of the locus of P [3m] SPM 2006(P1) 1. Diagram 5 shows the straight line AB which is perpendicular to the straight line CB at the point B

y = px + k

y = ( k − 2) x + p

where p and k are constant Express p in terms of k

[2m]

The equation of the straight line CB is y = 2 x −1

Find the coordinates of B [3 marks]

25

P2(section B) 1. Solutions to this question by scale drawing will not be accepted

SPM 2007 Section A (paper 2) 1. solutions by scale drawing will not be accepted

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

(a) (b) (c)

In diagram 1, the straight line AB has an equation y + 2 x + 8 = 0 . AB intersects the x-axis at point A and intersects the y-axis at point B

Calculate the area, in unit2, of triangle AOB Given that AC:CB = 3:2, find the coordinates of C A point P moves such that its distance from point A is always twice its distance from point B (i) Find the equation of the locus of P (ii) Hence, determine whether or not this locus intercepts the y-axis

y + 2x + 8 = 0

Diagram 1 Point P lies on AB such that AP:PB = 1:3 Find (a) the coordinates of P [3 m] (b) the equations of the straight line that passes through P and perpendicular to AB [3 m] SPM 2007 (paper 1) 1. The straight line

x y + = 1 has a 6 h

y- intercept of 2 and is parallel to the straight line y + kx = 0 .Determine the value of h and of k [3 marks] 2. The vertices of a triangle are A(5,2), B(4,6) and C(p,-2). Given that the area of the triangle is 30 unit 2 , find the values of p. [3 marks] SPM 2008(paper 1)

26

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

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

(a)

Diagram 13 Write down the equation of the straight line ST in the form

(a) A point W moves such that its

x y + =1 a b

(b)

distance from point S is always 2

A point P(x,y) moves such that PS = PT. Find the equation of the locus of P [4 m]

1 2

units. Find the equation of the locus of W [3m] (b) It is given that point P and point Q lie on the locus of W. Calculate (i) the value of k, (ii) the coordinates of Q [5m] (c) Hence, find the area, in unit2, of triangle OPQ [2m]

2. The points (0,3), (2,t) and (-2,-1) are the vertices of a triangle. Given that the area of the triangle is 4 unit2, find the values of t. [3 m]

27

CHAPTER 7: STATISTICS SPM 1993 1. The mean for the numbers 6, 2, 6, 2, 2, 10, x, y is 5 (a) show that x + y = 12 (b) hence, find the mode for the numbers when (i) x = y (ii) x ≠ y (c) if standard deviation is

1 2

37 , find

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

1-20

21-40

41-60

61-80

5

8

12

11

81-100 4

students

(a) On a graph paper, draw a histogram and use it to estimate the modal mark (b) By calculating the cumulative frequency, find the median mark, without drawing an ogive (c) Calculate the mean mark SPM 1994 1. The below table shows the marks obtained by a group of students in a monthly test .

28

Marks

1

2

3

4

5

Number of students

4

6

2

x

1

Find (a) the maximum value of x if modal mark is 2 (b) the minimum value of x if mean mark more than 3 (c) the range of value of x if median mark is 2 2. Set A is a set that consist of 10 numbers. The sum of these numbers is 150 whereas the sum of the squares of these numbers is 2890.

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

Length (mm) Numbers of fish 20-29 2 30-39 3 40-49 7 50-59 12 60-69 14 70-79 9 80-89 3 (a) Find the mean and variance of the numbers in set A (b) If another number is added to the 10 numbers in set A, the mean does not change. Find the standard deviation of these numbers. [6m] SPM 1995 1. (a) Given a list of numbers 3, 6, 3, 8. Find the standard deviation of these number (b) Find a possible set of five integers where its mode is 3, median is 4 and mean is 5.

SPM 1996 1. The list of numbers x −2, x + 4, 2 x + 5, 2 x −1, x + 7 and x − 3 has a mean of 7.Find (a) the value of x (b) the variance [6m] 2. The table shows the length of numbers of 50 fish (in mm) (a) calculate the mean length (in mm) of the fish (b) draw an ogive to show the distribution of the length of the fish Numbers of classes Numbers of pupils 6 35 5 36 4 30 (c) from your graph, find the percentage of the numbers of fish which has a length more than 55 mm

2. (a) The table shows the results of a survey of the number of pupils in several classes in a school. Find (i) the mean (ii) the standard deviation, of the number of pupils in each class (b)

Age 1-20 21-40 41-60 61-80 81-100

Numbers of villagers 50 79 47 14 10

SPM 1997 1. The table shows a set of numbers which has been arranged in an ascending order where m is a positive integer 29

Set numbers Frequency

1

m-1

5

m+3

8

10

1

3

1

2

2

1

SPM 1998 1. The mean of the data 2, k, 3k, 8, 12 and 18 which has been arranged in an ascending order, is m. If each element of the data is reduced by 2, the new median

(a) express median for the set number in terms of m (b) Find the possible values f m (c) By using the values of m from (b), find the possible values of mode

is

Find (a) the values of m and k (b) the variance of the new data

(ii)

[4m] [2m]

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

2. (a) The following data shows the number of pins knocked down by two players in a preliminary round of bowling competition. Player A: 8, 9, 8, 9, 8, 6 Player B: 7, 8, 8, 9, 7, 9 Using the mean and the standard deviation, determine the better player to represent the state based on their consistency [3m] (b) use a graph paper to answer this question The data in the table shows the monthly salary of 100 workers in a company.

(i)

5m . 8

Monthly Salary Numbers of (RM) workers 500-1 000 10 1 001-1 500 12 1 501-2 000 16 2 001-2 500 22 2 501-3 000 20 3 001-3 500 12 3 501-4 000 6 4 001-4 500 2 a mean of 6 and the sum of the squares of these scores of 1 200, is taken out from set X. Calculate the mean and variance of the remaining scores in set X. [7m] SPM 1999 1. The set of numbers integer positive 2, 3, 6, 7, 9, x, y has a mean of 5 and a standard deviation of 6. Find the possible values of x and y

Based on the data, draw an ogive to show distribution of the workers’ monthly salary From your graph, estimate the number of workers who earn more than RM 3 200

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

30

Frequency 6

49-59 60-79 80-99

5 14 5

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

(a) By using a graph paper, draw a histogram and estimate the modal mark [4m] (b) Without drawing an ogive, calculate the median mark [3m] (c) Find the mean mark [3m]

Marks Number of students