Transcript

SPM 19931. 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 , 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 , 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 + . 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 , x2

and inverse function f -1 : x , x3 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\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) = 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) = , x p and g(x) = 3 + x. Find (a) f(x)

[2 marks]

(b) the value of p if ff -1(p21) = 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() =

[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 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 1State(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 + , where m and k are

constants, find the value of m and of k.

[3 marks]

3. Given the function h(x) = , 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

EMBED Equation.3 Determine

(a) h-1(5)

(b) gh(2)

[2 marks]

2. The function w is defined as

w(x) = , 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]

SPM 2006

Paper 1

1. 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 , , where m is a constant

DIAGRAM 2

Find the value of m

[2 marks] Paper 2 1. Given that and

, find (a)

[1 m]

(b)

[2 m]

( c) such that

[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 , find the value of x such that

[2m]

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

Find the value of a and b

[3m]SPM 2008

Paper 1

1. Diagram 1 shows the graph of the function , for the domain .

State (a) the value of t

(b) the range of f(x) corresponding to the given domain

[3 m]

2. Given the function and , find

a)

b)

[4m]3. Given the functions and , find

a) f(5)b) the value of k such that gf(5)=14

[3m]SPM 19941. If and are the roots of the quadratic

equation 2x2 3x 6 = 0, form another

quadratic equation with roots and

[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 and -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 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 so that (3 )x2 2( + 1)x + + 1 = 0 has two

equal real roots.

[2 marks]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. SPM 1998

1. The equation of px2 + px + 3q = 1 + 2x have the roots 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 -2qSPM 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 and are the roots of the

equation x2 2x + k = 0, while 2 and 2

are the roots of the equation x2 +mx +9=0. Find the possible values of k and m.

[6 marks]

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 and are the roots of the quadratic

equation , form another quadratic equation with roots

3 + 2 and 3 + 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 and are the roots of the

equation kx(x 1) = 2m x.

If + = 6 and = 3, find the values

of k and m.

[5 marks]

SPM 2003

1. 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 2004

1. Form the quadratic equation which has

the roots -3 and . 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 has two equal roots. Find the possibles values of p.

[3 marks]

SPM 20071. (a) Solve the following quadratic equation:

(c) The quadratic equation where h and k are constants, has two equal roots

Express h in terms of k

[4 marks]

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

roots of the quadratic equation

Find the value of p

[2 marks]

SPM 19931. 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 19961. 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)

EMBED Equation.3 x

[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 + x3. 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 + 3kSPM 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 19991. (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 20011.(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) = , x p and

g(x) = 3 + x. Find the range of value of p

so that f-1g(x) = x has no real rootsSPM 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

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 symmetrySPM 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

(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 [3 m]SPM 20061. Diagram 3 shows the graph of quadratic

function . The straight line

is a tangent to the curve

a) write the equation of the axis of symmetry of the curve

b) express in the form , where b and c are constants.

[3 marks]

3. Find the range of the values of x for

[2 marks]

SPM 2007(paper 1)1. Find the range of values of x for which

[3 marks]

2. The quadratic function

can be expressed

in the form ,

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 , 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 pb) the value of qc) the value of r

[3 m]

2. Find the range of the value of x for .

[3 m]

SPM 2008 (paper 2)1. Diagram 2 shows the curve of a quadratic function . 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

[2m]

SPM 1993

1. 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

8m2 and 12 m, find the measurements

of the room.

Diagram 2SPM 1995

1. 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 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 pSPM 1997

1. Given that (3k, -2p) is a solution for the

simultaneous equation x 2y = 4 and

+=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

10 m2 and the length JK is longer than

the length of the curve KL by m, Find

the value of x.

SPM 19981. Solve the simultaneous equation:

+= 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.

SPM 1999

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

straight line = 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 = 1 SPM 20001. Solve the simultaneous equation

3x 5 = 2y , y(x + y) = x(x + y) 5

2. Solve the simultaneous equation

+ 3 = 0 and + = 0

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

[use = ]

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 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

x + y = 1 and y2 10 = 2xSPM 20061. Solve the simultaneous equations and

Give your answer correct to three decimal places [5 m]SPM 2007

1. Solve the following simultaneous equations:

,

[5 m]SPM 2008

1. Solve the following simultaneous equations :

[5m]SPM 19931. If 3 log x = 2log y, state x in terms of y2. (a) If h = log 2 and k = log 3, state in

terms of h and /or k (i) log 9

(ii) log 24

(b)Solve the following equations:

(i) 4 =

(ii) log 16 log 2 = 3SPM 1994

1. Solve the following equations:

(a) log x + log 3x = 1

(b) =

2. (a) Given that log n = , find the value

of n (b) Given that 2 = 3 = 6. Express t in

terms of r and s

( c) Given that y = kx where k and m

are constants. y = 4 when x = 2 and

y = 8 when x = 5. Find the values of k

and mSPM 1995

1. Solve the following equations: (a) 81(27) = 1

(b) 5= 26.3

2. (a) Given that m = 2 and n = 2, state in

terms of r and/or t(i) log

EMBED Equation.3 ,

(ii) log m log nb) The temperature of a metal increased

from 30C to TC according to the

equation T = 30(1.2)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 30C

to 1500C

SPM 1996

1. (a) Express 2 2+ 10(2) in a

simplify terms

(c) Solve the equation 3 5 = 0

2. (a)Solve the following equations:

(i) 4=5

(ii) 2. 3 = 5

(b) Given that log3 = 0.683 and

log 7 = 1.209. without using a

calculator scientific or four-figure

table ,

calculate (i) log 1.4

(ii) log75

SPM 1997

1. Show that log xy = 2 log x + 2 log y.

Hence or otherwise, find the value of x

and y which satisfies the equation

log xy = 10 and =

2.(a) Find the value of 3without using a

scientific calculator or four figure table. (b) Solve the equation

5 log3 + 2 log2 - log324 = 4 and

give your answer correct to four

significant figures. 3. (a) Given that 2 log(x + y) = 2 + logx + log y,

show that x+ y= 7xy

(b) Without using scientific calculator or

four-figure mathematical tables, solve

the equation

log[log(4x 5)] = log2 (c ) After n year a car was bought the

price of the car is RM 60 000.

Calculate after how many years will

the car cost less than RM 20 000 for

the first time

SPM 1998

1. Given that log4 = u and log5 = y State log

in terms of u and/or w2. (a) Given that log3 = x and log5 = y. Express log

EMBED Equation.3 in terms of x and y (b) Find the value of log8 + log

EMBED Equation.3 (c ) Two experiments have been

conducted to get relationship between

two variables x and y. The equation

3(9) = 27 and logy = 2 + log (x 2)

were obtain from the first and second

experiment respectively SPM 1999

1. Given that log3 = 1.585 and log5 =

2.322. Without using scientific calculator

or four-figure mathematical tables, Find

(a) log45

(b) log

EMBED Equation.3 2. (a) Given that x = log3, find the value of

4. Hence find the value of 4 if

y = 1 + x (b) Given that log3 = 0.7924. Without

using scientific calculator or four-

figure mathematical tables

(i) prove that log27a = 3.3772(ii) solve the equation a = 3 SPM 2000

1. (a) Solve 3 = 81

(b) If 3= 8(2), prove that

x log

= log8 2. (a)Simplify

Without using scientific calculator

or four-figure mathematical tables

(b) Given that 3 lg xy= 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).

Calculate the minimum number

of years required for the

savings to exceed RM 4 000.

SPM 2001

1. Given that logk = p and log = r Find log18 in terms of p and r 2. (a) Given that log = 2 and

log = -1, show that

xy 100y= 9 (b) Solve the equation

(i) 3 = 24 + 3

(ii) log=log

EMBED Equation.3 SPM 2002 1. (a) Given that log = k. If 5 = 15,

Find in terms of k (b) Solve the equation

log

EMBED Equation.3 log

2. (a) Given that

State x in terms of y (b) Solve the simultaneous equation

and where m and k are constants

SPM 2003,P11. Given that , express T

in terms of V

[4 marks]2. Solve the equation

[4 marks]

SPM 2004,P1

1. Solve the equation

[3 marks]

2. Given that and ,

express in terms of m and pSPM 2005,P1

1. Solve the equation

[3 marks]

2. Solve the equation

[3 marks]

3.Given that and , express in terms of p and rSPM 2006

1. Solve the equation [3 marks]

2. Given that , express y in terms of x[3 marks]3. Solve the equation

[3 marks] SPM 2007 1. Given that log and log,

express log in terms of x and y

[4 marks]

2. Given that

[3 marks] SPM 2008(paper 1)1. Solve the equation

[3 m]

2. Given that , find the value of x.

[3 m]SPM 19931. 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 RSPM 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

(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 OMNSPM 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

2. (a)The above diagram, P, Q and R

are three points are on a line

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 LMN2. The straight line cutting the

curve 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

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

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 can

be expressed as

( 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 formSPM 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

(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 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 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

( 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 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 and hence, find the area of

[3m]

SPM 19991. Given point and point . 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

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]

(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 and

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

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 , where O is the origin.

[2m]

2. Solutions to this question by scale

drawing will not be accepted.Straight line 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 [2m]

3. In the diagram, the equation of BDC is . A point P moves such that its distance from A is always the 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 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 . This straight line

is tangent to curve 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

[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]SPM 2003(P1)1. The points , and

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

and .

Determine whether the lines are

perpendicular to each other

[3m]

3. x and y are related by the equation

, where p and q are

constants. A straight line is obtained

by plotting 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

against x

Given that , calculate the value of k and of h

[3m]

2. Diagram 4 shows a straight line PQ with

the equation . The point P lies

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

axis

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 The equation of the straight line CB is

Find the coordinates of B

[3 marks]P2(section B)1. Solutions to this question by scale drawing will not be acceptedDiagram 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 2007

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

In diagram 1, the straight line AB has an equation .

AB intersects the x-axis at point A and intersects the y-axis at point B

Diagram 1Point 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 has a

y- intercept of 2 and is parallel to the straight line .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, 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

(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 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]

SPM 19931. The mean for the numbers 6, 2, 6, 2, 2,

10, x, y is 5 (a) show that

(b) hence, find the mode for the numbers

when

(i)

(ii)

(c) if standard deviation is , find

the values of x2. The below table shows the marks obtained by a group of students in a monthly test . Marks1-2021-4041-6061-8081-100

Number of students5812114

(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 19941. The below table shows the marks obtained by a group of students in a monthly test .

Marks12345

Number of students462x1

Find

(a) the maximum value of x if modal mark is 2(b) the minimum value of x if mean mark more than 3

(c) the range of value of x if median mark is 2

2. Set A is a set that consist of 10 numbers. The sum of these numbers is 150 whereas the sum of the squares of these numbers is 2890.(a) Find the mean and variance of the

numbers in set A

(b) If another number is added to the 10

numbers in set A, the mean does not

change. Find the standard deviation

of these numbers.

[6m]

SPM 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.

Numbers of classesNumbers of pupils

635

536

430

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)

AgeNumbers of villagers

1-2050

21-4079

41-6047

61-8014

81-10010

The table shows the age distribution of 200 villagers. Without drawing a graph, calculate

(i) the median

(ii) the third quartile

of their agesSPM 19961. The list of numbers and has a mean of 7.Find

(a) the value of x(b) the variance

[6m]

2. Length (mm)Numbers of fish

20-292

30-393

40-497

50-5912

60-6914

70-799

80-893

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 mmSPM 19971. The table shows a set of numbers which

has been arranged in an ascending order

where m is a positive integer Set numbers1m-15m+3810

Frequency131221

(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, 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.

Monthly Salary

(RM) Numbers of workers

500-1 00010

1 001-1 50012

1 501-2 00016

2 001-2 50022

2 501-3 00020

3 001-3 50012

3 501-4 0006

4 001-4 5002

(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 . Find

(a) the values of m and k

[4m]

(b) the variance of the new data[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 and

(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 y2. The frequency distribution of marks for

30 pupils who took a additional mathematics test is shown in the table

MarksFrequency

20-396

49-595

60-7914

80-995

(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 testMarks