Learning to score

Additional Mathematics Learning To Score 2007

TOPIC: FUNCTION

PAPER 1

P = { 1, 2, 3}
Q = { 2, 4, 6, 8, 10}

YEAR 2003
1. Based on the above information, the relation between P and Q is defined by the 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 → 5 x + 1 and h : x → x 2 − 2 x + 3 , find
(a) g −1 (3) ,
(b) hg ( x) .
[4
marks]

YEAR 2004
3. Diagram 1 shows the relation between set P and set Q.

w
d
x
e
y
f
z

Set P Set Q
Diagram 1
State
(a) the range of the relation,
(b) the type of the relation.
[2 marks]

−1 5
4. Given the functions h : x → 4 x + m and h : x → 2kx + , where m and k are constants, find
8
the values of m and k.
[3 marks]

LTS 2007 1

6
5. Given the function h ( x ) = , x ≠ 0 and the composite function hg ( x) = 3 x , find
x
(a) g ( x)
(b) the value of x when gh ( x) = 5 .
[4
marks]

YEAR 2005
6. In Diagram 2, the function h maps x to y and the function g maps y to z.
h g
x y z
8

5

2
Diagram 2

Determine
(a) h −1 (5) ,
(b) gh (2) .
[2 marks]

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

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

h : x → 2x − 3
g : x → 4x −1

Find gh −1 ( x) .
[3 marks]

LTS 2007 2


YEAR 2006
9. In Diagram 3, set B shows the images of certain elements of set A.

5
2
4 5
-4
-5 1

Set A Set B
Diagram 3

(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]

m−x
10. Diagram 4 shows the function h : x → , x ≠ 0, where m is a constant.
x
h m−x
x
x

8

1
−
2

Diagram 4

Find the value of m.
[2 marks]

PAPER 2

YEAR 2006
x
1. Given that f : x → 3 x + 2 and g : x → + 1 , find
5
−1
(a) f ( x) , [1 marks]
(b) f −1 g ( x) , [2 marks]
(c) h (x ) such that h g ( x) = 2 x + 6 . [3 marks]

LTS 2007 3


ANSWERS (FUNCTION)

PAPER 1

1.
(a) 2 or 4
(b) 1

2.
2
(a ) g −1 (3) =
5
(a) hg ( x) = 25x2 + 2

3.
(a) range = {x, y}
(b) many to one relation.

4.
1 5
k= m=−
8 2

5.
2
(a) g ( x) = , x≠0
x

(b) x = 15 .

6.
(a) h −1 (5) = 2

(b) gh (2) = 8

7.
2x − 5
(a) w −1 ( x ) = , x≠0
x
3
(b) w−1 (4) =
4

8.
gh −1 ( x) = 2 x + 5

9.
(a) Many to one relation
(b) f : x  x 2

LTS 2007 4

10. m=4

PAPER 2

1.
x−2
(a) f −1 ( x ) =
3
x −5
(b) f −1 g ( x) =
15
(c) h ( x ) = 10 x − 4

LTS 2007 5


TOPIC: QUADARTIC EQUATION

PAPER 1

YEAR 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]

YEAR 2004
1
3. From the quadratic equation which has the roots -3 and . Give your answer in the form
2
ax2 + bx + c = 0, where a, b and c are constants. [2 marks]

YEAR 2005

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

YEAR 2006

5. A quadratic equation x2 + px + 9 = 2x has two equal roots. Find the possible values of p.
[3 marks]

PAPER 2

YEAR 2003

1. Solve the simultaneous equations 4x + y = -8 and x2 + x – y = 2. [5
marks]

YEAR 2004

2. Solve the simultaneous equations p - m = 2 and p2 + 2m = 8.
LTS 2007 6

Give your answers correct to three decimal places. [5 marks]

YEAR 2005

1
3. Solve the simultaneous equations x + y = 1 and y2 - 10 = 2x.
2
[5 marks]

YEAR 2006

4. Solve the simultaneous equations 2x + y = 1 and 2x2 + y2 + xy = 5.
Give your answers correct to three decimal places. [5
marks]

ANSWERS (QUADRATIC EQUATION)

PAPER 1

1. x = 2.591 or -0.2573
2. p < -3 or p > 5
3. 2x2 + 5x – 3 = 0
4. x = 8.153 or 0.149
5. p = 8 or -4

PAPER 2

1. x = -2 or -3
y = 0 or 4

2. m = 0.606 or -6.606
p = 2.606 or -4.606
1
3. x = 3 or -
2
y = -4 or 3

4. x = 1.443 or -0.693
y = -1.886 or 2.386

LTS 2007 7


TOPIC: QUADARTIC FUNCTION

PAPER 1

YEAR 2003
1. The quadratic equation x( x + 1) = px − 4 has two distinct roots. Find the range of values of p .
[3
marks]

YEAR 2004
2. Find the range of values of x for which x( x − 4) ≤ 12 . [3
marks]

3. Diagram below shows the graph of the function y = −( x − k ) 2 − 2 , where k is a constant.

y
0 x

-3 ● (2,-3)

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

YEAR 2005
4. Diagram below shows the graph of a quadratic functions f ( x ) = 3( x + p ) 2 + 2 , where p is a
constant.

y y=f(x)

0 ●
(1,q)
The curve y = f (x) has a minimum point (1,q) where q is a constant. State
x
(a) the value of p ,
(b) the value of q ,
(c) the equation of the axis of symmetry.
LTS 2007 8

[3 marks]

YEAR 2006
5. Diagram below shows the graph of a quadratic function y = f (x) . The straight line y = −4
is a tangent to the curve y = f (x) .

y
y = f(x)

(a) Write the equation of the axis of symmetry of the curve.
(b) Express f(x) in the form of ( x + b) 2 + c , where b and c are constants.
[3 marks]
O 1 5 x

y = -4
6. Find the range of the values of (2 x − 1)( x + 4) > 4 + x . [2 marks]

ANSWERS (QUADRATIC FUNCTIONS)

PAPER 1

1. p < 3 or p > 5
2. − 2 ≤ x ≤ 6
3. (a) k = 1
(b) x = 1
(c) Maximum point (1,-2)
4. (a) p = −1
(b) q = 2
(c) x = 1
5. (a) x = 3
(b) f ( x) = ( x − 3) 2 − 4
6. x < −4 or x > 1

LTS 2007 9


TOPIC: SIMULTANEOUS EQUATION

PAPER 1

YEAR 2003

1. Solve the simultaneous equations 4x + y = -8 and x2 + x – y = 2. [5
marks]

YEAR 2004

2. Solve the simultaneous equations p - m = 2 and p2 + 2m = 8.
marks]

YEAR 2005

1
3. Solve the simultaneous equations x + y = 1 and y2 - 10 = 2x. [5
2
marks]

YEAR 2006
4. Solve the simultaneous equations 2x + y = 1 and 2x2 + y2 + xy = 5.
marks]

YEAR 2007
5. Solve the simultaneous equations 2 x − y − 2 = 0 and 2 x 2 − 10 x + y + 9 = 0 . [5
marks ]

ANSWERS (SIMULTANEOUS EQUATIONS)

PAPER 1

1. x = -2 or -3
y = 0 or 4

2. m = 0.606 or -6.606
p = 2.606 or -4.606

LTS 2007 10

1
3. x = 3 or -
2
y = - 4 or 3

4. x = 1.443 or -0.693
y = -1.886 or 2.386

5. x = 2.707 , y = 3.404
x = 1.293 , y = 5858

TOPIC: INDICES & LOGARITHM

PAPER 1

YEAR 2003

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]

YEAR 2004
3. Solve the equation 32 4 x = 4 8 x +6 . [3
marks]

4. Given that log 5 2 = m and log 5 7 = p , express log 5 4.9 in terms of m and p. [4
marks]

YEAR 2005
5. Solve the equation 2 x + 4 − 2 x +3 = 1 . [3
marks]

6. Solve the equation log 3 4 x − log 3 (2 x − 1) = 1 . [3
marks]

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

YEAR 2006

LTS 2007 11

2 x −3 1
8. Solve the equation 8 = . [3 marks]
4 x+2

9. Given that log 2 xy = 2 + 3 log 2 x − log 2 , express y in terms of x . [4 marks]

10. Solve the equation 2 + log 3 ( x − 1) = log 3 x. [3 marks]

ANSWERS (INDICES & LOGARITHMS)

PAPER 1

1. T =8 V
2. x = 1.677
3. x=3
4. 2 p − m −1
5. x = −3

3
6. x =
2
7. 3r − 2 p + 1
8. x = 1
9. y = 4 x
1
10. x = 1
8

LTS 2007 12


TOPIC: COORDINATE GEOMETRY

PAPER 1

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

y x
2. The equation of two straight lines are + = 1 and 5 y = 3 x + 24 . Determine whether the lines
5 3
are perpendicular to each other.
[3 marks]

YEAR 2004
x y
3. Diagram 3 shows a straight line PQ with the equation + = 1.
2 3
The point P lies on the x-axis and the point Q lies on the y-axis.
y

Q•

•
O P x
Diagram 3

Find the equation of the straight line perpendicular to PQ and passing through the point Q.
[3 marks]
LTS 2007 13


4. 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.
[3 marks]

YEAR 2005
5. The following information refers to the equations of two straight lines, JK and RT, which are
perpendicular to each other.

JK : y = px + k
RT : y = (k − 2) x + p

where p and q are constants.

Express p in terms of k.
[2 marks]

YEAR 2006
6. Diagram 6 shows the straight line AB which is perpendicular to the straight line CB at the point
B.
y

• A(0, 4)

• B

O x
•C

Diagram 6

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

PAPER 2

YEAR 2003
1. Solutions to this question by scale drawing will not be 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

LTS 2007 14

(i) the equation of the locus of the point P,
(ii) the value of k.
[6 marks]

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

YEAR 2004
2. Diagram 7 shows a straight line CD which meet straight line AB at the point D. The point D
lies on the y-axis.
y

C

O B(9, 0) x

D

A(0, –6)
Diagram 7

(a) Write down the equation of AB in the form of intercepts.
[1 marks]
(b) Given that 2AD = DB, find the coordinates of D.
[2 marks]
(c) Given that CD is perpendicular to AB, find the y-intercept of CD.
[3 marks]

YEAR 2005
y
A(–4, 9 )

LTS 2007 15


B

O x
2y + x + 6 = 0

C
Diagram 8

(a) Find
(i) the equation of the straight line AB.
(ii) the coordinates of B.
[5 marks]

(b) The straight line AB is extended to a point D such that AB : BD = 2 : 3.
Find the coordinates of D.
[2 marks]
(c) A point P moves such that its distance from point A is always 5 units.
Find the equation of the locus of P.
[3 marks]

YEAR 2006

Diagram 9 shows the triangle AOB where O is the origin.
Point P lies on the straight line AB.

y

A(–3, 4 )

•C

O x

B(6, –2)

Diagram 3

(a) Calculate the area, in unit2, of triangle AOB.
[2 marks]
(b) Given that AC : CB = 3 : 2, find the coordinates of C.
[2 marks]
(c) A point P moves such that its distance from point A is always twice its distance from
point B.

LTS 2007 16

(i) Find the equation of the locus of P.
(ii) Hence, determine whether or not this locus intercepts the y-axis.
[6 marks]

ANSWERS (COORDINATE GEOMETRY)

PAPER 1

1. p = -2 t
y x
+ =1 , 5 y = 3 x + 24
5 3
5 3 24
m1 = − , y= x+
3 5 5
3
2. m2 = .
5
5 3
m1 × m2 = − ×
3 5
= −1
∴ the lines are perpendicular to each other

3.
2
y= x+3
3

4.
5 x 2 + 5 y 2 + 50 x − 6 y − 118 = 0

5.
−1 1
p= or p =
k −2 2−k

6. B ( 2, 3)

PAPER 2

1.
(i) x 2 + y 2 − 4 x − 6 y − 12 = 0
(a)
(ii) k = −1 or 7
8
(b) Area of ∆OQT = − unit 2
3

2.
x y
(a) + =1
9 6

LTS 2007 17


(b)
D = (3, − 4)

1
(c) y - intercept = .
2

3.
(a)(i) Equation of line AB, y = 2 x + 17

(ii) B (−8, 1)

(b) D(−14, − 11)

(c) x 2 + y 2 + 8 x − 18 y + 72 = 0

4. (a) area = 9 unit 2 .

(b)
 12 2 
Coordinates of C =  , 
 5 5

(c)
(i) locus of P : 3 x 2 − 54 x − 3 y 2 + 24 y + 135 = 0

(ii) when x = 0, 3 y 2 − 24 y − 135 = 0
y 2 − 8 y − 45 = 0
a = 1, b = −8, c = −45
b 2 − 4ac = 64 + 180
= 244
2
Q b − 4ac > 0,
∴ locus of P intercepts the y -axis

LTS 2007 18


TOPIC: STATISTICS

PAPER 1

YEAR 2005
1. The mean of four numbers is m . The sum of the squares of the numbers is 100 and the
standard deviation is 3k. Express m in terms of k.
[3 marks]

PAPER 2

YEAR 2003
1. A set of examination marks x1, x2, x3, x4, x5, x6 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, ∑ x 2 . [3
marks]

(b) Each mark is multiplied by 2 and then 3 is added to it.
Find, for the new set of marks,
(i) the mean,
(ii) the variance. [4 marks]

YEAR 2004
2. A set of data consists of 10 numbers. The sum of the numbers is 150 and the sum of the
squares of the numbers is 2472.

a. Find the mean and variance of the 10 numbers,
[3 marks]

b. Another number is added to the set of data and the mean is increased by 1.
Find
(i) the value of this number,
(ii) the standard deviation of the set of 11 numbers.
[4 marks]

YEAR 2005

LTS 2007 19

3. Diagram below shows a histogram which represents the distribution of the marks obtained by
40 pupils in a test.

14

9
7
6

4

0.5 10.5 20.5 30.5 40.5 50.5

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

YEAR 2006
4. Table below shows the frequency distribution of the scores of a group of pupils in a game.

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

(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]

LTS 2007 20


(c) What is the mode score if the score of each pupil is increased by 5
[1 marks]

ANSWER (STATISTICS)

PAPER 1

1. m = 25 – 9k2

PAPER 2

1. (a) ∑ x = 30 , ∑ x 2 . =163.5
(b) mean = 13, variance = 9

2. (a) mean = 15, variance = 22.2
(b) k = 26, standard deviation = 5.494

3. (a) median = 24.07 (b) standard deviation= 11.74

4. (a) k = 4 (b) mode = 43 (c) mode score = 48

LTS 2007 21


TOPIC: CIRCULAR MEASURE

PAPER 1

YEAR 2003
1. Diagram 1 shows a sector ROS with centre O .

R

θ
O S

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 radian. [ 3 marks
]

YEAR 2004
2. Diagram 2 shows a circle with centre O .

A

O 0.354 rad

B

DIAGRAM 2
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 ) [3
marks ]

LTS 2007 22


YEAR 2005
3. Diagram 3 shows a circle with centre O .

A

O θ

B

DIAGRAM 3

The length of the minor arc is 16 cm and the angle of the major sector AOB is 290o .
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 . [ 3 marks ]

YEAR 2006
4. Diagram 3 shows sector OAB with centre O and sector AXY with centre A .

A

Y
X
θ
O
B
DIAGRAM 4
Given that OB = 10 cm , AY = 4 cm , ∠ XAY = 1.1 radians and the length of arc AB = 7cm ,
calculate

( a) the value of θ , in radian ,

( b) the area, in cm2 , of the shaded region . [ 4 marks ]

PAPER 2

YEAR 2003
1. Diagram 1 shows the sectors POQ, centre O with radius 10 cm. The point R on OP is such
that OR : OP = 3 : 5 .

LTS 2007 23


P

R

Q
θ
O
DIAGRAM 1

Calculate

(a) the value of θ , in radian . [ 3 marks ]

(b) the area of the shaded region , in cm2 . [ 4 marks ]

YEAR 2004
2. Diagram 2 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 andR respectively. OPQR is a
rhombus . JLK is an arc of a circle , centre O .
L

J K
Q
P R
α rad
O

T
DIAGRAM 2

Calculate
(a) the angle α , in terms of π , [ 2 marks ]

(b) the length , in cm , of the arc JLK , [ 4 marks ]
P
(c) the area , in cm2 , of the shaded region. A [ 4 marks ]

YEAR 2005
3. π
Diagram 3 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. rad
6
O B Q
DIAGRAM 3

8 cm
LTS 2007 24


It is given that OA: OP= 4 : 7 .
( Using π = 3.142 )

Calculate
(a) the length , in cm , of AP , [ 1 mark
]

(b) the perimeter , in cm , of the shaded region , [5
marks ]

( c) the area , in cm2 , of the shaded region . [ 4 marks
]

YEAR 2006
4. Diagram 4 shows the plan of a garden. PCQ is a semicircle with centre O and has a radius
of 8 m. RAQ is a sector of a circle with centre A and has a radius of 14 m .

R

C

P A O Q

DIAGRAM 4

Sector COQ is a lawn . The shaded region is a flower bed and has to be fenced . It is given
that AC = 8 m and ∠ COQ = 1.956 radians . [ use π = 3.142 ]

Calculate
(a) the area , in m2 , of the lawn . [ 2 marks ]

(b) the length , in m , of the fence required for fencing the flower bed , [ 4 marks ]

(c ) the area , in m2 , of the flower bed . [ 4 marks ]

LTS 2007 25

ANSWERS (CIRCULAR MEASURE)

PAPER 1

1. θ = 0.8153 rad.

2. r = 7.675

3. (a) θ = 1.222 rad

(b) r = 13.09
7
4. (a) θ = = 0.7
10
(b) A = 26.2

PAPER 2

1. (a) θ = 0.9273
(b) Area of the shaded region = 22.37

2
2. (a) ∠ POR = π
3
(b) The length of arc JLK = 20.94
(c) Area of the shaded region = 61.40

3. (a) AP = 6
(b) Perimeter of ehe shaded region = 24.40

4. (a) Area of COQ = 62.59
(b) The perimeter = 38.25

LTS 2007 26


TOPIC: DIFFERENTIATION

PAPER 1

YEAR 2003
1. Given that y = 14 x (5 − x) , calculate
(a) the value of x when y is maximum,
(b) the maximum value of y. [3 marks]

2. Given that y = x 2 + 5 x , use differentiation to find the small change in y when x increases from 3
to 3.01. [3 marks]

YEAR 2004
3. Differentiate 3 x 2 (2 x − 5) 4 with respect to x. [3 marks]

2
4. Two variables, x and y are related by the equation y = 3 x + . Given that y increases at a
x
constant rate of 4 units per second, find the rate of change of x when x = 2.
[3 marks]

YEAR 2005
1
5. Given that h( x) = , evaluate h”(1). [4 marks]
(3 x − 5) 2

1 3
6. The volume of water, V cm3, in a container is given by V = h + 8h , when h cm is the height
3
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.
[3 marks]

LTS 2007 27


YEAR 2006
1
7. The point P lies on the curve y = ( x − 5) 2 . It is given that the gradient of the normal at P is −
4
. Find the coordinates of P. [3 marks]

2 7 dy
8. It is given that y = u , when u = 3 x − 5. Find in terms of x. [4 marks]
3 dx

9. Given that y = 3 x 2 + x − 4,
dy
(a) find the value of when x = 1,
dx
(b) express the approximate change in y, in terms of p, when x changes from 1 to 1 + p,
where p is a small value. [4 marks]

PAPER 2
YEAR 2007

dy
11. The curve y = f (x) is such that = 3kx + 5 , where k is a constant. The gradient of the curve
dx
at x = 2 is 9 .

Find the value of k . [ 2 marks ]

12. The curve y = x 2 − 32 x + 64 has a minimum point at x = p , where p is a constant.

Find the value of p . [ 3 marks ]

YEAR 2003
dy
1. (a) Given that = 2 x + 2 and y = 6 when x = −1, find y in terms of x.
dx
[3 marks]
2
d y dy
(b) Hence, find the value of x if x 2 2
+ ( x − 1) + y = 8. [4 marks]
dx dx

2. (a) Diagram 2 shows a conical container of diameter 0.6 m and height 0.5 m. Water is poured
into the container at a constant rate of 0.2 m3 s−1.

0.6 m

0.5 m
LTS 2007 28

water

Diagram 2
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.
1 2
(Use π = 3.142; Volume of a cone = πr h ) [4 marks]
3

YEAR 2004
3. The gradient function of a curve which passes through A(1, −12) is 3 x 2 − 6 x. Find

(a) the equation of the curve, [3 marks]

(b) the coordinates of the turning points of the curve and determine whether each of the turning
points is a maximum or a minimum. [5 marks]

3
4. Diagram 5 shows part of the curve y = which passes through A(1, 3).
(2 x − 1) 2
y

• A(1, 3)

3
y=
(2 x − 1) 2
x
O

(a) Find the equation of the tangent to the curve at the point A. [4 marks]

YEAR 2007

2
5. A curve with the gradient function 2 x − has a turning point at ( k , 8 ) .
x2
(a) Find the value of k . [ 3 marks ]

(b) Determine whether the turning point is a maximum or a minimum point .
[ 2 marks ]

(c) Find the equation of the curve . [ 3 marks ]

LTS 2007 29


ANSWERS (DIFFERENTIATION)

PAPER 1

5
1. (a) x =
2
175
(b) y =
2

2. x = 0.11

3. 6 x(6 x − 5)(2 x − 5) 3

8
4. unit second−1
5

27
5.
8

6. 0.8333 cm s−1

7. (7, 4)

8. 14(3 x − 5) 6

9. (a) 7 (b) 7p

PAPER 2

1. (a) y = x 2 + 2 x + 7
3
(b) x = or x = −1
5

2. (a) y = 3 x 2 − 6 x − 10
(b) (2, −10)

3. (a) p = 3
(b) f ( x) = x 3 − 2 x 2 + 4

4. y = −12 x + 15

LTS 2007 30


TOPIC: SOLUTION OF TRIANGLE

PAPER 2

YEAR 2003
1. Diagram 5 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
50°.
V

A C

B
Diagram 5

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, [3 marks]
(b) the length of AV if the angle between AV and the base is 25°, [3 marks]
(c) the area of triangle VAB. [4 marks]

YEAR 2004
2. Diagram 6 shows a quadrilateral ABCD such that ∠ABC is acute.

D
5.2 cm
9.8 cm 12.3 cm C

LTS 2007 31


A 40.5° 9.5 cm

Diagram 6 B

(a) Calculate
(i) ∠ABC,
(ii) ∠ADC,
(iii) the area, in cm2, of quadrilateral ABCD. [8 marks]

(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’ = 40.5°, but which is different in shape to
triangle ABC.
(i) Sketch the triangle A’B’C’,
(ii) State the size of ∠A’B’C’. [2 marks]

YEAR 2005
3. Diagram 7 shows triangle ABC.
A

20 cm

B
15 cm
C
Diagram 7

(a) Calculate the length , in cm, of AC. [2 marks]
(b) A quadrilateral ABCD is now formed so that AC is a diagonal, ∠ACD = 40° and AD = 16
cm. Calculate the two possible values of ∠ADC. [2 marks]
(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. [6 marks]

YEAR 2006
4. Diagram 5 shows a quadrilateral ABCD.

D 5 cm
C
40°
6 cm

LTS 2007 32


B
9 cm
A Diagram 5

The area of triangle BCD is 13 cm2 and ∠BCD is acute.
Calculate
(a) ∠BCD, [2 marks]
(b) the length, in cm, of BD, [2 marks]
(c) ∠ABD, [3 marks]
(d) the area, in cm2, quadrilateral ABCD. [3 marks]

ANSWERS (SOLUTION OF TRIANGLE)

PAPER 2

1. (a) 2.70 cm
(b) 3.149 cm
(c) 2.829 cm2

2. (a) (i) 57.23°
(ii) 106.07°
(iii) 80.96 cm2

(b) (i) C’

A’ B
B’

(ii) 122.77°

3. (a) 19.27 cm
(b) ∠AD1C = 129.27°, ∠AD2C = 50.73°
(c) (i) 24.89 cm
(ii) 290.1 cm2

4. (a) 60.07° or 60° 4’
(b) 5.573 cm
(c) 116.55° or 116° 33’
(d) 35.43 cm2

LTS 2007 33


TOPIC: INDEX NUMBER

PAPER 2

YEAR 2003
1. Diagram 1 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.

Weekly cost ( RM )

33
30
24
15
12

0
P Q R S T Items

DIAGRAM 1

Price Index in 1995 based
Items Price in 1900 Price in 1995 on 1990
P x RM 0.70 175
Q RM 2.00 RM 2.50 125
R RM 4.00 RM 5.50 y
S RM 6.00 RM 9.00 150
T RM 2.50 z 120

TABLE 1
LTS 2007 34


(a) Find the value of
(i) x
(ii) y
(iii) z [ 3 marks ]

(b) Calculate the composite index for items in the year 1995 based on the year 1990 .
[ 2 marks ]
(c) The total monthly cost of the items in the year 1990 is RM 456 . Calculate the
corresponding total monthly cost for the year 1995 .
[ 2 marks ]
(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.
[ 3 marks ]

YEAR 2004
2. Table 2 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 biscuit.

Price index for the year 1995 Percentage of usage
Item based on the year 1993 (%)
P 135 40
Q x 30
R 105 10
S 130 20
TABLE 2

(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.
[ 5 marks ]

(b) The composite index number of the cost of biscuit 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 biscuit in the year 1993 if the corresponding price in the
year 1995 is RM 32 .
[ 5 marks ]

YEAR 2005

LTS 2007 35

3. Table 3 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 2 is a pie chart which represents the
relative amount of the ingredients P , Q , R and S , used in making biscuits .

Price per kg
Ingredients ( RM ) Price index for the
Year Year year 2004 based on
2001 2004 the year 2001
P 0.80 1.00 x
Q 2.00 y 140
R 0.40 0.60 150
S z 0.40 80

TABLE 3

P
Q
120o
60o S
o
100

R

DIAGRAM 2
(a) Find the value of x , y and z . [ 3 marks ]

(b) (i) Calculate the composite index for 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 .
[ 5 marks ]
(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.
[ 2 marks ]

YEAR 2006
4. A particular kind of cake is made by using four ingredients , P , Q , R and S . Table 4 shows
the prices of the ingredients .

Price per kilogram ( RM )
Ingredient Year 2004 Year 2005
P 5.00 w

LTS 2007 36

Q 2.50 4.00
R x y
S 4.00 4.40

TABLE 4

(a) The index number of ingredient P in the year 2005 based on the year 2004 is 120 .
Calculate the value of w. [ 2 marks ]

(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 . [ 3 marks ]

(c ) The composite index for the cost of making the cake in the year 2005 based on the
year 2004 is 127.5 .

YEAR 2007
5. 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.

Price ( RM ) for the
Component year Price index for the
Year Year year 2006 based on
2004 2006 the year 2004
P 1.20 1.50 125
Q x 2.20 110
R 4.00 6.00 150
S 3.00 2.70 y
T 2.00 2.80 1.40

TABLE 4

S R
72 o 90o

T 36o
Q
o
144

P

DIAGRAM 6

(a) Find the value of x and y . [ 3 marks ]

(b) (i) Calculate the composite index for the production cost of the toys in the year
2006 based 2004 .

LTS 2007 37

[ 3 marks ]

(c) The price of each component increase 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.
[ 4 marks ]

ANSWERS (INDEX NUMBER)

PAPER 2

1. a) i) x = 0.40
ii) y = 137.5
iii) z = 3.00

b) I = 140.9

c) RM 642.5

d) 169.10

2. a) i) P93 = RM 29.00
ii) I = 162

b) i) x = 125
ii) P93 = RM 25

3. a) x = 125 , y = 2.80, z = 0.50

b) i) I = 129.4
ii) P01 = 2306.80

c) Expected composite index = 194.1

4. a) w = 6.00

b) x = 8.00
y = 10.00

c) i) P04 = 24.00
ii) m=4

TOPIC: PROGRESSIONS

LTS 2007 38

PAPER 1

YEAR 2003
1. The first three terms of an arithmetic progression are k – 3, k + 3, 2k + 2.
Find
(a) the value of k,
(b) the sum of the first 9 terms of the progression.
[3 marks]

2. In a geometric progression, the first term is 64 and the fourth term is 27. Calculate
(a) the common ratio,
(b) the sum to infinity of the geometric progression.
[4 marks]

YEAR 2004
4
3. Given a geometric progression y , 2, , p  , express p in terms of y.
y
[2 marks]

4. Given an arithmetic progression –7, –3, 1, …, state three consecutive terms in this
progression which sum up to 75.
[3 marks]

5. The volume of water in a tank is 450 litres on the first day. Subsequently, 10 litres of water is
added to the tank everyday.
Calculate the volume, in litres, of water in the tank at the end of the 7th day.
[2 marks]

6. Express the recurring decimal 0.969696 … as a fraction in its simplest form.
[4 marks]

YEAR 2005
7. The first three terms of a sequence are 2, x, 8.
Find the positive value of x so that the sequence is
(a) an arithmetic progression,
(b) a geometric progression.
[2 marks]

8. The first three terms of an arithmetic progression are 5, 9 13.
Find
(a) the common difference of the progression,
(b) the sum of the first 20 terms after the third term.
[3 marks]

9. The sum of the first n terms of the geometric progression 8, 24, 72, … is 8744.

LTS 2007 39

Find
(a) the common ratio of the progression,
(b) the value of n.
[4 marks]

YEAR 2006
10. The 9th term of an arithmetic progression is 4 + 5p and the sum of the first four terms of the
progression is 7p – 10, where p is a constant.
Given that the common difference of the progression is 5, find the value of p.
[3 marks]

11. The third term of a geometric progression is 16. The sum of the third term and the fourth term
is 8.
Find
(a) the first term and the common ratio of the progression.
(b) the sum of infinity of the progression.
[4 marks]

PAPER 2

YEAR 2006
1. Two companies, Delta and Omega, start to sell cars at the same time.

(a) Delta sells k cars in the first month and its sales increase constantly by m cars every
subsequent month. It sells 240 cars in the 8th month and the total sales for the first 10
months are 1900 cars.
Find the value of k and of m.
[5 marks]
(b) Omega sells 80 cars in the first month and its sales increase constantly by 22 cars every
subsequent month.
If both companies sell the same number of cars in the nth month, find the value of n.
[2 marks]

ANSWERS (PROGRESSIONS)

PAPER 1

1. (a) k = 7
(b) S9 = 252 .

2.
3
(a) r =
4
(b) Sn = 256 .

LTS 2007 40

8
3. p= 2 .
y

4. 21, 25, 29

5. T7 = 510

32
6. 33

7. (a) x = 5
(b) x = 4 .

8.
(a) d = 9 − 5 = 4
(b) S20 = 1100 .

9.
24
(a) r = =3
8
(b) n = 7

10. p=8

1
(a) r=−
2
11. a = 64

2
(b) S∞ = 42
3

PAPER 2

1.
(a) m = 20
k = 100
(b) n = 11

LTS 2007 41

TOPIC: LINEAR LAW

PAPER 1

YEAR 2003
1. x and y are related by the equation y = px2 + qx, where p and q are constants. A straight line is
y
obtained by plotting against x, as shown in Diagram 1.
x
y
x
• (2 , 9)

• (6 , 1)

0 x

Diagram 1

Calculate the values of p and q. (4
marks)

YEAR 2004
y
2. Diagram 3 shows a straight line graph of against x.
x

y
x
• (2 , k)

• (h , 3)

0 x

Diagram 3

Given that y = 6x – x2, calculate the value of k and of h. (3
marks)

LTS 2007 42


YEAR 2005

3. The variables x and y are related by the equation y = kx4, where k is a constant.
(a) Convert the equation y = kx4 to linear form.
(b) Diagram 3 shows the straight line obtained by plotting log10 y against log10 x.

log10 y
• (2 , h)

• (0, 3)

0 log10 x

Diagram 3

Find the value of
(i) log10 k,
(ii) h.

YEAR 2006
4. Diagram 4(a) shows the curve y = -3x2 + 5. Diagram 4(b) shows the straight line graph
obtained when y = -3x2 + 5 is expressed in the linear form Y = 5X + c.

y Y

y = -3x2 + 5

0 x 0 X
• -3

DIAGRAM 4(a) DIAGRAM 4(b)

Express X and Y in terms of x and/or y.

YEAR 2007
5. The variables x and y are related by the equation y 2 = 2 x(10 − x) . A straight line graph is
y2
obtained by plotting against x , as shown in Diagram 2 .
x

y2
x
• (3 , q)

LTS 2007 43


•
0 (p,0) x

Diagram 2

Find the value of p and of q. [ 3 marks ]

PAPER 2

YEAR 2003
1. Use graph paper to answer this question.

Table 1 shows the values of two variables, x and y, obtained from an experiment. It is known
that x and y are related by the equation y = pkx2 , where p and k are constants.

x 1.5 2.0 2.5 3.0 3.5 4.0
y 1.59 1.86 2.40 3.17 4.36 6.76
Table 1

(a) Plot log y against x2.
Hence, draw the line of best fit. [5
marks]
(b) Use the graph in (a) to find the value of
(i) p,
(ii) k. [5
marks]

YEAR 2004
2. Use graph paper to answer this question.

Table 1 shows the values of two variables, x and y, obtained from an experiment. Variables x
and y are related by the equation y = pkx , where p and k are constants.

x 2 4 6 8 10 12
y 3.16 5.50 9.12 16.22 28.84 46.77

Table 1

(a) Plot log10 y against x by using a scale of 2 cm to 2 units on the x-axis and 2 cm to 0.2 unit
on the log10 y-axis. Hence, draw the line of best fit. [4 marks]
(b) Use your graph from (a) to find the value of
(i) p,
(ii) k. [6
marks]

YEAR 2005
3. Table 1 shows the values of two variables, x and y, obtained from an experiment. Variables x

LTS 2007 44

r
and y are related by the equation y = px + , where p and r are constants.
px

x 1.0 2.0 3.0 4.0 5.0 5.5
y 5.5 4.7 5.0 6.5 7.7 8.4

Table 1

(a) Plot xy against x2 by using a scale of 2 cm to 5 units on both axes. Hence, draw the line of best
fit. [5 marks]
(b) Use the graph from (a) to find the value of
(i) p,
(ii) r. [5 marks]

YEAR 2006
4. Use graph paper provided by the invigilator to answer this question.

Table 2 shows the values of two variables, x and y, obtained from an experiment. Variables x
and y are related by the equation y = pkx+1 , where p and k are constants.

x 1 2 3 4 5 6
y 4.0 5.7 8.7 13.2 20.0 28.8
Table 1

(a) Plot log y against (x + 1) by using a scale of 2 cm to 1 units on the (x + 1)-axis and 2 cm to 0.2
unit on the log y-axis. Hence, draw the line of best fit. [5
marks]
(b) Use your graph from 7(a) to find the value of
(i) p,
(ii) k. [5
marks]

YEAR 2007
4. Table 3 shows the values of two variables, x and y, obtained from an experiment. Variables x
p
and y are related by the equation y = 2kx2 + , where p and k are constants.
kx

x 2 3 4 5 6 7
y 8 13.2 20 27.5 36.6 45.5

Table 1

LTS 2007 45


y
(a) Plot against x , using a scale of 2 cm to 1 units on both axes.
x
Hence, draw the line of best fit. [4 marks]

(b) Use your graph in 7(a) to find the value of
(i) p,
(ii) k.
(iii) y when x = 1.2 . [5
marks]

ANSWERS (LINEAR LAW)

PAPER 1

1. p = - 2, q = 13
2. h = 3, k = 4
3. (a) log10 y = 4 log10 x + log10 k
(a) (i) log10 k = 1000
(ii) h = 11
1
4. X= 2
x

y
Y=
x2

PAPER 2

1. (a)
x2 2.25 4.0 6.25 9.0 12.25 16.0
Log10 y 0.20 0.27 0.38 0.50 0.64 0.83

(b) (i) p = 1.259
(ii) k = 1.109

2. (a)
x 2 4 6 8 10 12
Log10 y 0.50 0.74 0.96 1.21 1.46 1.67

(b) (i) p = 1.820

(ii) k = 1.309

3. (a)
x2 1 4 9 16 25 30.25
xy 5.5 9.4 15.0 26.0 38.5 46.2

LTS 2007 46

(b) (i) p = 1.37

(ii) r = 5.48

4. (a)
x+1 2 3 4 5 6 7
Log y 0.60 0.76 0.94 1.12 1.30 1.46

(b) (i) p = 1.738

(ii) k = 1.495

TOPIC: INTEGRATION

PAPER 1

YEAR 2003

LTS 2007 47

5
1. Given that ∫ dx = k (1 + x ) + c , find the values of k and n
n
[3marks]
(1 + x) 4

2. Diagram below shows the curve y = 3x2 and the straight line x = k.

y y = 3x2

O x=k x

If the area of the shaded region is 64 unit2, find the value of k. [3marks]

YEAR 2004
k

3. Given that ∫ ( 2 x − 3) dx = 6,
−1
where k > -1 , find the value of k. [4marks]

YEAR 2005
6 6

4. Given that ∫ f ( x)dx = 7
2
and ∫ ( 2 f ( x) − kx ) dx = 10 ,
2
find the value of k. [4marks]

YEAR 2006
5. Diagram below shows the curve y = f(x) cutting the x – axis at x = q and x = b

y = f(x)

O a b

b

Given that the area of the shaded region is 5 unit2, find the value of ∫ 2 f ( x)dx
a

[2marks]

5

6. Given that ∫ g ( x)dx = 8 ,
1
find

LTS 2007 48

1

(a) the value of ∫ g ( x)dx
5
5

(b) the value of k if ∫ [kx − g ( x)]dx = 10
1
[4marks]
YEAR 2007

7

7. Given that ∫ h( x)dx = 3 , find
2
2

(a) ∫ h( x)dx = 3
7

7

(b) ∫ [5 − h( x)dx
2
[ 4 marks]

PAPER 2

YEAR 2003
1. Diagram below shows a curve x = y2 – 1 which intersects the straight line 3y =2x at point A.
y
3y =2x
A
x = y2 – 1
1

-1 O x

Calculate the volume generated when the shaded region is revolved 360o about
the y-axis.
[6marks]

YEAR 2004

LTS 2007 49

3
2. Diagram below shows part of the curve y = which passes through
( 2 x − 1) 2
A(1,3).

A(1,3)

3
y=
( 2 x − 1) 2

(a) Find the equation of the tangent to the curve at the point A. [4marks]

(b) A region is bounded by the curve, the x-axis and the straight lines x = 2 and
x = 3.

(i) Find the area of the region
(ii) The region is revolved through 360o about the x –axis. Find the volume
generated, in terms of π
[6marks]

3. The gradient function of a curve which passes through A(1, -12) is 3x2 – 6x.
Find
(a) the equation of the curve [3marks]

YEAR 2005
4. A curve has a gradient function px2 – 4x, where p is a constant. The tangent to
the curve at the point (1,3 ) is parallel to the straight line y + x – 5 = 0.
Find
(b) the value of p, [3marks]
(c) the equation of the curve. [3marks]

1 2
5. In Diagram below, the straight line PQ is normal to the curve y = x + 1 at
2
A(2,3). The straight line AR is parallel to the y – axis.
y
LTS 2007 50

1 2
y= x +1
2
P

A(2,3)

O R Q(k,0)

Find
(a) the value of k, [3marks]
(b) the area of the shaded region, [4marks]
(c) the volume generated, in terms of π, when the region bounded by the curve, the y – axis
and the straight line y = 3 is revolved through 360o about the y-axis.
[3marks]

YEAR 2006
6. Diagram below shows the straight line y = x + 4 intersecting the curve
y = (x – 2 )2 at the points A and B.

y

y = ( x − 2) 2 y = x+4
B

A Q
P
P

Q
O k x

Find,
(a) the value of k [2marks]
(b) the are of the shaded region P [5marks]
(c) the volume generated, in terms of π, when the shaded region Q is revolved 360o about the
x – axis. [3marks]

ANSWERS (INTEGRATION)

PAPER 1

1. k = - 5/3 , n = -3

LTS 2007 51

2. k=4
3. k=4
4. k=¼
5. -10
6. (a) - 8 (b) k = 3/2

PAPER 2

52
1. volume = π
15
49
2. (a) y = -12x + 15 (b) area = 1/5 , volume = π
1125
3. y = 3x2 – 6x – 10
4. p = 3 , y = x3 – 2x2 = 4
1
5. (a) k = 8 (b) area = 12 (c) Volume = 4 π
3
32
6. (a) k = 5 (b) area = 20.83 (c) volume = π
5

TOPIC: VECTORS

PAPER 1

YEAR 2003
1. Diagram below shows two vectors, OP and QO

Tak de tanda anak panah
Q(-8,4)

P(5,3)

Express

x
(a) OP in the form  ,
 y
 
(b) QO in the form x i + y j [2marks]

p = 2a + 3b
q = 4a – b
r = ha + ( h – k ) b, where h and k are constants

2. Use the above information to find the values of h and k when r = 3p – 2q.

[3marks]
LTS 2007 52


3. Diagram below shows a parallelogram ABCD with BED as a straight line.

D C

E
A B

Given that AB = 6p , AD = 4q and DE = 2EB, express, in terms of p and q

(a) BD
(b) EC
[4marks]

YEAR 2004
4. Given that O(0,0), A(-3,4) and B(2, 16), find in terms of the unit vectors, i and j,
(a) AB
(b) the unit vector in the direction of AB
[4marks]

5. Given that A(-2, 6), B(4, 2) and C(m, p), find the value of m and of p such that

AB + 2 BC = 10i – 12j. [4marks]

YEAR 2005
6. Diagram below shows vector OA drawn on a Cartesian plane.
y
6 A

4

2

0 2 4 6 8 10 12 x

x
(a) Express OA in the form  
 y
 
(b) Find the unit vector in the direction of OA [2marks]

LTS 2007 53


7. Diagram below shows a parallelogram, OPQR, drawn on a Cartesian plane.

y

Q

R P

O x

It is given that OP = 6i + 4j and PQ = - 4i + 5j. Find PR .

YEAR 2006
8. Diagram below shows two vectors, OA and AB .

y
A(4,3)

O x

-5

Express

x
(a) OA in the form  
 y
 
(b) AB in the form xi + yj [2marks]

9. The points P, Q and R are collinear. It is given that PQ = 4a – 2a and

QR = 3a + (1 + k )b , where k is a constant. Find

(a) the value of k
(b) the ratio of PQ : QR
[4marks]

LTS 2007 54


PAPER 2

YEAR 2003
5   2 k 
1. Give that AB =   , OB =   and CD =   , find
7 3  5 
     

(a) the coordinates of A, [2marks]
(b) the unit vector in the direction of OA , [2marks]
(c) the value of k, if CD is parallel to AB [2marks]

YEAR 2004
2. Diagram below shows triangle OAB. The straight line AP intersects the straight line OQ at
R. It is given that OP = 1/3 OB, AQ = ¼ AB, OP = 6 x and OA = 2 y.

A

Q
R

O P B

(a) Express in terms of x and/or y:
(i) AP
(ii) OQ [4marks]
(b) (i) Given that AR = h AP, state AR in terms of h, x and y.
(ii) Given that RQ = k OQ, state RQ in terms of k, x and y.
[2marks]

(c) Using AR and RQ from (b), find the value of h and of k.
[4marks]

YEAR 2005
3. In diagram below, ABCD is a quadrilateral. AED and EFC are straight lines.

LTS 2007 55

D

E F C

A B

It is given that AB = 20x, AE = 8y, DC = 25x – 24y, AE = ¼ AD
3
and EF = EC.
5

(a) Express in terms of x and/or y:

(i) BD

(ii) EC [3marks]

(b) Show that the points B, F and D are collinear. [3marks]

(c) If | x | = 2 and | y | = 3, find | BD |. [2marks]

YEAR 2006
4. Diagram below shows a trapezium ABCD.

B C
F
•

•
A E D

uuu
r 2 5
It is given that AB =2y, AD = 6x, AE = AD and BC = AD
3 6

(a) Express AC in terms of x and y [2marks]

(b) Point F lies inside the trapezium ABCD such that 2 EF = m AB , and m is a
constant.

(i) Express AF in terms of m , x and y
(j) Hence, if the points A, F and C are collinear, find the value of m.

LTS 2007 56

[5marks]

ANSWERS (VECTORS)

PAPER 1

 5
1. (a)  ,
 3 (b) -8 i + 4j
 
2. h = -2 , k = - 13
3. (a) = - 6p + 4q (b) 2p + 8/3 q
5  1 5 
4. (a) AB =  ,
12  (b) vector in direction AB =  ,
  13 12 
 
5. m = 6, p = -2
12  1 12 
6. (a) OA =   (b) vector in direction OA =
5   ,
  13  5 
 
7. PR = - 10 i + j
 4
8. (a) OA =  
3 (b) AB = -4i – 8j
 

9. (a) k = - 5/2 (b) 4 : 3

PAPER 2

1 − 3 25
1. (a) A( -3, -4 ) (b) OA =   (c) k =
5  − 4
  7
3 9
2. (a) AP = - 2y + 6x OQ = y + x
2 2

9 3 
(b) AR = h(6 x − 2 y ), RQ = k  x + y 
2 2 

(d) k = 1/3 , h = ½

3. (a) BD = -20x + 32y , EC = 25x

(b) BF = - 5x + 8y BD = 4 ( - 5 x + 8y )

(c) | BD | = 104
8
4. (a) AC = 5x + 2y (b) AF = 4x+ my , m=
5

LTS 2007 57


TOPIC: TRIGONOMETRIC FUNCTION

PAPER 1

YEAR 2003
1. Given that tan θ = t , 0o < θ < 90o , express , in terms of t :

(a) cot θ
(b) sin ( 90 - θ ) [ 3 marks ]

2. Solve the equation 6 sec2 A – 13 tan A = 0 , 0o ≤ A ≤ 360o. [ 4 marks ]

YEAR 2004
3. Solve the equation cos2 x – sin2 x = sin x for 0o ≤ x ≤ 360o . [ 4 marks ]

YEAR 2005
4. Solve the equation 3cos 2x = 8 sin x – 5 for 0o ≤ x ≤ 360o . [ 4 marks ]

YEAR 2006
5. Solve the equation 15 sin2 x = sin x + 4 sin 30o for 0o ≤ x ≤ 360o . [ 4 marks ]

PAPER 2

YEAR 2003
LTS 2007 58

1. (a) Prove that tan θ + cot θ = 2 cosec 2θ . [ 4 marks ]

3
(b) (i) Sketch the graph y = 2 cos x for 0o ≤ x ≤ 2π .
2
(ii) Find the equation of a suitable straight line for solving the equation
3 3
cos x = x −1 .
2 4π
Hence , using the same axes , sketch the straight line and state the number of
3 3
solutions to the equation cos x = x − 1 for 0o ≤ x ≤ 2π.
2 4π
[ 6 marks ]

YEAR 2004
2. (a) Sketch the graph of y = cos 2x for 0o ≤ x ≤ 180o. [ 3 marks ]

(b) Hence , by drawing a suitable straight line on the same axes , find the number of
x
solutions satisfying the equation 2 sin2 x = 2 - for 0o ≤ x ≤ 180o.
180
[ 3 marks ]

YEAR 2005
3. (a) Prove that cosec2 x – 2 sin2 x – cot2 x = cos 2x. [ 2 marks ]

(b) (i) Sketch the graph of y = cos 2x for 0 ≤ x ≤ 2π .

(ii) Hence , using the same axes , draw a suitable straight line to find the number
x
of solutions to the equation 3(cosec2 x – 2 sin2 x – cot2 x ) = - 1 for
π
0 ≤ x ≤ 2π . State the number of solutions . [ 6 marks ]

YEAR 2006
4. (a) Sketch the graph of y = - 2 cos x for 0 ≤ x ≤ 2π . [ 4 marks ]

(b) Hence , using the same axis , sketch a suitable graph to find the number of solutions
π
to the equation + 2 cos x = 0 for 0 ≤ x ≤ 2π . State the number of solutions.
x
[ 3 marks ]

ANSWERS (TRIGONOMETRIC FUNCTION)

PAPER 1

1
1. a) cot θ =
tan θ
1
= t2 +1
t
sin (90 − θ ) = kos θ tθ
b)
θ

LTS 2007 59

1
= 1θ
t +1
2

2. A = 33.69 , 213.69 or 56.31 , 236.31

3. x = 30o , 50o , 270o

4. x = 41.81o , 138.19o

5. x = 23.58o , 156.42o , 199.47o , 340.53o

PAPER 2

sin θ kosθ
1. a) tan θ + kot θ = +
kosθ sin θ
sin 2 θ + kos 2θ
=
sin θkosθ
2
=
2 sin θkosθ
2
=
sin 2θ
= 2cosec θ.

b) (i) & (ii)

2
3
y= x−2
1 2π
⊗
O π 2π
π/3 5π/42
⊗
-2
⊗
3
-2 y = 2 cos x
2

LTS 2007 60

Number of solution = 3

2. a) & b)

y = cos 2x
1

x
y= −1
180
O 0.25π 0.5π 0.75π π
⊗
⊗
-1


3. a) ( )
cosec2 x – 2 sin2 x – cot2 x = 1 + cot 2 x − 2 sin 2 x − cot 2 x
= 1 − 2 sin 2 x
= cos 2x.

b)
y = cos 2 x
1
x 1
y= −
⊗ 3π 3
O ⊗
⊗
⊗ 0.5π π 1.5π 2π

2

π
y=
x
4.
a) & b) 2
⊗ y = cos x
⊗

O 0.5π π 1.5π 2π
LTS 2007 61

-2



TOPIC: PERMUTATIONS & COMBINATIONS

PAPER 1

YEAR 2003

1. Diagram 6 shows 5 letters and 3 digits.

A B C D E 6 7 8

Diagram 6

A code is to be formed using those letters and digits. The code must consists of 3 letters
followed by 2 digits. How many codes can be formed if no letter or digit is repeated in each
code?
[3 marks]

2. A badminton team consists of 7 students. The team will be chosen from a group of 8 boys and 5
girls. Find the number of teams that can be formed such that each team consists of
(a) 4 boys,
(b) not more than 2 girls. [4 marks]

YEAR 2004

3. Diagram 6 shows five cards of different letters.

H E B A T
LTS 2007 62


Diagram 6

(a) Find the number of possible arrangements, in a row, of all the cards.
(b) Find the number of these arrangements in which the letters E and A are side by side.
[4 marks]

YEAR 2005

4. A debating team consists of 5 students. These 5 students are chosen from 4 monitors, 2 assistant
monitors and 6 prefects.
Calculate the number of different ways the team can be formed if
(a) there is no restriction,
(b) the team contains only 1 monitor and exactly 3 prefects. [4 marks]

YEAR 2006

5. Diagram 9 shows seven letters cards.

U N I F O R M
Diagram 9

A four-letter code is to be formed using four of these cards. Find
(a) the number of different four-letter codes that can be formed,
(b) the number of different four-letter codes which end with a consonant.
[4 marks]

ANSWERS (PERMUTATIONS & COMBINATIONS)

PAPER 1

1. 360

2. (a) 700 (b) 708

3. (a) 120 (b) 48

4. (a) 792 (b) 160

5. (a) 840 (b) 480

LTS 2007 63


TOPIC: PROBABILITY

PAPER 1

YEAR 2004
1. A box contains 6 white marbles and k black marbles. If a marble is picked randomly from the
3
box, the probability of getting a black marble is .
5
Find the value of k. [3 marks]

YEAR 2005
2. The following table shows the number of coloured cards in a box
Colour Number of Cards
Black 5
Blue 4
Yellow 3

Two cards are drawn at random from the box.
Find the probability that both cards are of the same colour. [3 marks]

YEAR 2006
2
3. The probability that Hamid qualifies for the final of a track event is while the probability that
5
1
Mohan qualifies is .
3
Find the probability that
(a) both of them qualify for the final,

LTS 2007 64

(b) only one of them qualifies for the final. [ 3 marks]

ANSWERS (PROBABILITY)

PAPER 1

1. k = 9

19
2.
66

2 7
3. (a) (b)
15 15

TOPIC: PROBABILITY DISTRIBUTION

PAPER 1

YEAR 2003
1. The following diagram shows a standard normal distribution graph.

f(z)

0 k z

If P(0 < z < k) = 0.3128, find P(z > k). [2 marks]

2. In an examination, 70% of the students passed. If a sample of 8 students is randomly selected,
find the probability that 6 students from the sample passed the examination. [3 marks]

YEAR 2004
3. X is a random variable of a normal distribution with a mean of 5.2 and a variance of 1.44.
Find
(a) the Z score if X=6.7
(b) P(5.2≤ X ≤6.7) [4 marks]

YEAR 2005

LTS 2007 65

4. The mass of students in a school has a normal distribution with a mean of 54 kg and a standard
deviation of 12 kg. Find
(a) the mass of the students which give a standard score of 0.5,
(b) the percentage of students with mass greater than 48 kg. [4 marks]

YEAR 2006
5. The diagram below shows a standard normal distribution graph.

f(z)
0.3485

0 k z

The probability represented by the area of the shaded region is 0.3485 .
(a) Find the value of k.
(b) X is a continuous random variable which is normally distributed with a mean of 79 and a
standard deviation of 3.
Find the value of X when z-score is k. [4 marks]

PAPER 2

YEAR 2003
1. (a) Senior citizens make up 20% of the population of a settlement.
(i) If 7 people are randomly selected from the settlement, find the probability that at least two
of them are senior citizens.
(ii) If the variance of the senior citizens is 128, what is the population of the settlement?
[5 marks]

(b) The mass of the workers in a factory is normally distributed with a mean of 67.86 kg and a
variance of 42.25 kg2. 200 of the workers in the factory weigh between 50 kg and 70 kg.
Find the total number of workers in the factory. [5 marks]

YEAR 2004
2. (a) A club organises a practice session for trainees on scoring goals from penalty kicks. Each
trainee takes 8 penalty kicks. The probability that a trainee scores a goal from a penalty kick
is p. After the session, it is found that the mean number of goals for a trainee is 4.8
(i) Find the value of p.
(ii) If a trainee is chosen at random, find the probability that he scores at least one goal.
[5 marks]

(b) A survey on body-mass is done on a group of students. The mass of a student has a normal
distribution with a mean of 50 kg and a standard deviation of 15 kg.
(i) If a student is chosen at random, calculate the probability that his mass is less than 41 kg.
(ii) Given that 12% of the students have a mass of more than m kg, find the value of m.

LTS 2007 66

[5 marks]

YEAR 2005
3. For this question, give your answer correct to three significant figures.
(a) The result of a study shows that 20% of pupils in a city cycle to school.
If 8 pupils from the city are chosen at random, calculate the probability that
(i) exactly 2 of them cycle to school,
(ii) less than 3 of them cycle to school. [4 marks]

(b) The mass of water-melons produced from an orchard follows a normal distribution with a
mean of 3.2 kg and a standard deviation of 0.5 kg.
Find
(i) the probability that a water-melon chosen randomly from the orchard has a mass of not
more than 4.0 kg,
(ii) the value of m if 60% of the water-melons from the orchard has a mass of more than
m kg. [6 marks]

YEAR 2006
4. An orchard produces lemons.
Only lemons with diameter, x greater than k cm are graded and marketed.
Table below shows the grades of the lemons based on their diameters.
Grade A B C
Diameter, x (cm) x>7 7≥x>5 5≥x>k
It is given that the diameter of lemons has a normal distribution with a mean of 5.8 cm and a
standard deviation of 1.5 cm.
(a) If one lemon is picked at random, calculate the probability that it is of grade A. [2 marks]
(b) In a basket of 500 lemons, estimate the number of grade B lemons. [4 marks]
(c) If 85.7% of the lemons is marketed, find the value of k. [4 marks]

ANSWERS (PROBABILITY DISTRIBUTION)

PAPER 1

1. 0.1872

2. 0.2965

3. (a) 1.25 (b) 0.3944

4. (a) X = 60 (b) 69.146%

5. (a) 1.03 (b) 82.09

PAPER 2

LTS 2007 67


1. (a) (i) 0.4232832 (ii) 800

(b) 319

2. (a) (i) p = 0.6 (ii) 0.9993

(b) (i) 0.2743 (ii) m = 67.625 kg

3. (a) (i) 0.2936 (ii) 0.79691

(b) (i) 0.9452 (ii) m = 3.0735

4. (a) 0.2119 (b) 245 (c) k = 4.1965

TOPIC: MOTION ALONG A STRAIGHT LINE

PAPER 2

YEAR 2003
1. A particle moves in a straight line and passes through a fixed point O, with a velocity of
24 m s −1 . Its acceleration, a m s −2 , t s after passing through O is given by a = 10 − 2t. The
particle stops after k s.
(a) Find
(i) the maximum velocity of the particle,
(ii) the value of k.
(b) Sketch a velocity-time graph for 0 ≤ t ≤ k . [6 marks]
Hence, or otherwise, calculate the total distance traveled during that period.
[4 marks]

YEAR 2004
2. A particle moves along a straight line from a fixed point P. Its velocity, V m s −1 , is given by
V = 2t (6 − t ) , where t is the time, in seconds, after leaving the point P.
(Assume motion to the right is positive)
Find
(a) the maximum velocity of the particle, [3 marks]
(b) the distance traveled during the third second, [3 marks]
(c) the value of t when the particle passes the points P again, [2 marks]
(d) the time between leaving P and when the particle reverses its direction of motion.
[2 marks]
YEAR 2005
3. Diagram 9 shows the positions and directions of motion of two objects, P and Q, moving in a
straight line passing two fixed points, A and B, respectively. Object P passes the fixed point A
LTS 2007 68

and object Q passes the fixed point B simultaneously. The distance AB is 28 m.

P Q

A C B
28 m

Diagram 9

The velocity of P, v p m s −1 , is given v p = 6 + 4t − 2t , where t is the time, in seconds A, after
2

it passes A while Q travels with a constant velocity of -2 m s −1 . Object P stops instantaneously
at point C.
(Assume that the positive direction of motion is towards the right.)
Find
(a) the maximum velocity , in, m s −1 , of P, [3 marks]
(b) the distance, in m, of C from A, [4 marks]
(c) the distance, in m, between P and Q when P is at the points C. [3 marks]

YEAR 2006

4. A particle moves in a straight line and passes through a fixed point O.
Its velocity, v ms −1 , is given by v = t 2 − 6t + 5 , where t is the time, in seconds, after leaving
O.
[Assume motion to the right is positive.]

(a) Find
(i) the initial velocity of the particle,
(ii) the time interval during which the particle moves towards the left,
(iii) the time interval during which the acceleration of the particle is positive.
[5 marks]

(b) Sketch the velocity-time graph of the motion of the particle for 0 ≤ t ≤ 5 .
[2 marks]

(c) Calculate the total distance traveled during the first 5 seconds after leaving O.
[3 marks]

ANSWERS (MOTION ALONG A STRAIGHT LINE)

PAPER 1

1. (a) (i) 49 (ii) k = 12
(b)

LTS 2007 69


y

49

24

0 5 12 x

432 m
2. (a) 18
1
(b) 17 m
3
(c) t = 9
(d) t = 6

3. (a) 8
(b) 18
(c) 4

4. (a) (i) v= 5
(ii) 1 < t < 5
(iii) t > 3

(b)

vy

5

0 1 5 t

(c) 13 m

LTS 2007 70


TOPIC: LINEAR PROGRAMMING

PAPER 2

1. Yahya has an allocation of RM 225 to buy x kg of prawns and y kg of fish. The total mass of the
commodities is not less than 15 kg. The mass of prawns is at most three times that of fish. The
price of 1 kg of prawns is RM 9 and price of 1 kg of fish is RM 5.
(a) Write down three inequalities, other than x≥0 and y≥0, that satisfy all of the above
conditions. [3 marks]

(b) Hence, using a scale of 2 cm to 5 kg for axes, construct and shade the region R that satisfies
all the above conditions. [4 marks]

(c) If Yahya buys 10 kg of fish, what is the maximum amount of money that could remain from
his allocation? [3 marks]

LTS 2007 71

2. A district education office intends to organise a course on the teaching of Mathematics and
Science in English.
The course will be attended by x Mathematics participants and y Science participants.
The selection of participants is based on the following constraints:
I : The total number of participants is at least 40.
II : The number of Science participant is at most twice that of Mathematics.
III : The maximum allocation for the course is RM7200. The expenditure for a
Mathematics participant is RM120 and for Science participant is RM80.

(a) Write down three inequalities, other than x≥0 and y≥0, that satisfy all of the above
constraints. [3 marks]

(b) Hence, using a scale of 2 cm to 10 participants on axes, construct and shade the region R
which satisfies all the above constraints. [3 marks]

(c) Using your graph from (b), find
(i) the maximum and minimum number of Mathematics participants when the number of
Science participant is 10,
(ii) the minimum cost to run the course. [4 marks]

3. An institution offers two computer courses, P and Q. The number of participants for course P is
x and for course Q is y.
The enrolment of the participants is based on the following constraints:
I : The total number of participants is not more than 100.
II : The number of participants for course Q is not more than four times the number of
participants for course P.
III : The number of participants for course Q must exceed the number of participants for
course P by at least 5.

(a) Write down three inequalities, other than x≥0 and y≥0, which satisfy all of the above

(b) By using a scale of 2 cm to 10 participants for axes, construct and shade the region R that
satisfies all the above constraints. [3 marks]

(c) By using your graph from (b), find
(i) the range of the number of participants for course Q if the number of participants for
course P is 30, [3 marks]
(ii) the maximum total fees per month that can be collected if the fees per month for course P
and Q are RM50 and RM60 respectively. [4 marks]

4. A workshop produces two types of rack, P and Q.
The production of each type of rack involves two processes, making and painting.
Table below shows the time taken to make and paint a rack of type P and a rack of type Q.

Time taken (minutes)
Rack
Making Painting
P 60 30
Q 20 40

LTS 2007 72

The workshop produces x racks of type P and y racks of type Q per day.
The production of the racks per day is based on the following constraints:
I: The maximum total time for making both racks is 720 minutes.
II: The total time for painting both racks is at least 360 minutes.
III: The ratio of number of racks of type P and type Q is at least 1:3.

(a) Write down three inequalities, other than x≥0 and y≥0, which satisfy all of the above
(b) Using a scale of 2 cm to 2 racks on axes, construct and shade the region R which satisfies all
the above constraints. [3 marks]
(c) By using your graph from (b), find
(i) the minimum number of racks of type Q if 7 racks of type P are produced per day,
(ii) the maximum total profit per day if the profit from one rack of type P is RM24 and from
one rack of type Q is RM32. [4 marks]

ANSWERS (LINEAR PROGRAMMING)

Paper 2

1. (a) x + y ≥ 15 x ≤ 3y 9x + 5y ≤ 225

(b)

55

50

45

40

35

30

25

90
20
R
15

80

10

70
5

60
10 20 30 40 50 60 70 80 90 100

(c) y =10 50
x =19 RM 130

2. (a) I: x + y ≥ 40
40 II: y ≤ 3x III: 3x + 2y ≤ 180

(b) 30

20

LTS 2007 10 R 73

20 40 60 80 100 120 140 160


(c) (i) xminimum= 30 xmaximum= 53

(ii) RM 3760

3. (a) I: x + y ≤ 100 II: y ≤ 4x III: y ≥ x +5

(b)
90

80

70

60

50 R

40

30

20

10

20 40 60 80 100 120 140 160

(c) (i) 35 ≤ y ≤ 70 (ii) Maximum total fees = RM 5800

LTS 2007 74

4. (a) I: 3x + y ≤ 36 II: 3x + 4y ≥36 III: 3x ≥ y

(b)
22

20

18

16

14

12

10

R
8

6

4

2

5 10 15 20 25 30 35 40

(c) (i) 4 (ii) RM 720

*IF ANY DOUBT ARISES, PLEASE REFER TO THE ORIGINAL SPM PAPERS (2003-2006)

LTS 2007 75

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