Add Maths Perfect Score Module Form 4 Topical

PERFECT SCORE 2010 F4

1

CHAPTER 1 : FUNCTIONS

1. Given that mxxf 4: and 4

3:1 nxxf , find the values of m and n.

Answer:- m = – 3 ; n = 1

4

2. Given that 12: xxf , xxg 4: and baxxfg : , find the values of a and b .

Answer:- a = 8 ; b = –1

3. Given that 3: xxf , 2: bxaxg and 56366: 2 xxxgf , find the

values of a and b .

Answer:- a = 2 ; b = 6

4. Given that xmxg 3: and 3

42:1 kxxg , find the values of m and k.

Answer:- k = 1

6 ; m = 4

5. Given the inverse function 2

32)(1 x

xf , find

(a) the value of f(4),

(b) the value of k if f –1

(2k) = – k – 3 . Answer:-(a) 11

2(b)

1

2

6. Given the function : 2 1f x x and : 23

xg x , find

(a) f –1

(x) ,

(b) f – 1

g(x) ,

(c) h(x) such that hg(x) = 6x – 3 .

Answer:-(a) 1

2

x (b) 1 1

6 2x (c) 18x + 33

7. Diagram 1 shows the function 3

:2

p xg x

x

, 2x , where p is a constant.

Diagram 1

Find the value of p. Answer:- p = 4

7

3

2

p x

x

x

g

5


2

8.

x y z

4 4 4

2 2 2

0 0 0

1

2 2 2

Diagram 2

Diagram 2 shows the mapping of y to x by the function g : y ay + b and mapping

to z by the function h : y 6

2y b, y

b

2. Find the,

(a) value of a and value of b,

(b) the function which maps x to y,

(c) the function which maps x to z.

Answer:- (a)a= –6, b=10 (b)10

6

y (c)

18

20y

9. In the Diagram 3, function h mapped x to y and function g mapped y to z.

x h y g z

8

5 Diagram 3

2

Determine the values of,

(a) h1

(5),

(b) gh(2)

Answer:- (a)2 (b)8

10. Given function f : x 2 x and function g : x kx2 + n. If composite function

gf is given as gf : 3x2 12x + 8, find

(a) the value of k and value of n,

(b) the value of g2(0).

Answer:-(a) k = 3 ,n = –4 (b)44


3

11. The following information refers to the functions f and g.

Find f (x). Answer:-23 2

3

x

12. (a) Function f, g and h are given as

f : x 2x

g : x 3

2x , x 2

h : x 6x2 2.

(i) Determine the function fh(x). At the same axis, sketch the graphs of y = f(x)

and y = fh(x). Hence, determine the number of solutions for f(x) = fh(x).

(ii) Find the value of g1

(2).

(b) Function m is defined as m : x 5 3x. If p is another function and mp is

defined as mp : x 1 3x2, determine function p.

Answer:-(a)(i)12x2 – 4 (b) 22p x x

13. Given function f : x 4 3x.

(a) Find (i) f 2(x),

(ii) (f 2)1

(x).

(b) Hence, or otherwise, find (f 1

)2(x) and show (f

2)1

(x) = (f 1

)2(x).

(c) Sketch the graph of f 2(x) for the domain 0 x 2 and find it’s corresponding

range. Answer:-(a)9x – 8 (b)8

9

x

14. A function f is defined as f : x p x

x

3 2, for all values of x except x = h and p

are constants.

(a) Determine the value h.

(b) Given value 2 is mapped to itself by the function f. Find the

(i) value p,

(ii) another value of x which is mapped to itself,

g (x) = 4 – 3 x

fg (x) = 2 x + 5


4

CHAPTER 2 : QUADRATIC EQUATIONS

1. One of the roots of the quadratic equation is twice the other root.

Find the possible values of p.

Answer ; 5 7,p

2. If one of the roots of the quadratic equation is two times the other root,

find an expression that relates .

Answer : 22 9b ac

3. Find the possible value of m , if the quadratic equation has two equal

roots.

Answer ;

4. Straight line y = mx + 1 is tangent to the curve x2 + y

2 2x + 4y = 0. Find the possible

values of m.

Answer : 1

2or 2

5. Given 2

and

2

are roots of the equation k x(x 1) = 2m x.

If + = 6 and = 3, find the value of k and of m.

Answer : k = 2

1, m =

16

3

6. Find the values of such that the equation (3 )x2 2( + 1)x + + 1 = 0 has equal

roots. Hence, find the roots of the equation base on the values of obtained.

Answer : = 1; roots: = 1, x = 1; = 1, x = 0

(iii) value of f 1

(1).

Answer:-(a)3

2h (b)(i)p =12(ii)x = –3 (iii)–5


5

CHAPTER 3 : QUADRATIC FUNCTIONS

1. Diagram 1 shows the graph of the function 2

2 5y x p , where p is constant.

Find,

(a) the value of p ,

(b) the equation of the axis of symmetry,

(c) the coordinate of the maximum point.

Answer:- (a) p = 2 (b) x = 2 (c) ( 2, 5 )

2.

Diagram 2 shows the graph of the function 21 2f x p x x q .

(a) State the value of q .

(b) Find the range of values of p .

Answer:-(a) q = – 2 (b) 1

2p

y

( 0, –3 ) ( 4 , –3 )

x 0

Diagram 1

( 0, –2 ) 21 2f x p x x q

Diagram 2

y

x 0


6

3.

Diagram 3 shows the graph of the function 2y x bx c that intersects the

y- axis at point ( 0, 9 ) and touches the x- axis at point K.

Find,

(a) the value of b and c ,

(b) the coordinates of point K.

Answer:-(a) b = – 6 , c = 9 (b) 3 0,

4. y

( 2, 3)

0 x

Diagram 4

In Diagram 4 above point ( 2, 3 ) is the turning point on the graph which has equation

of the form y = p(x + h)2 + k.

Find the,

(a) values of p, h and k,

(b) equation of the curve formed when the graph as shown is reflected at the x–axis.

(c) equation of the curve formed when the graph as shown is reflected at the y–axis.

Answer :- (a) p = 5 , h = 2, k = 3 (b) y = 5(x 2)2 3 (c) y = 5(x + 2)

2 + 3

y

( 0, 9)

0

K

2y x bx c

x

Diagram 3

( 0, 23)


7

5. Function 2 28 20 1f x x kx k has a minimum value of 2 4r k , where r and k

are constants.

(a) By using the method of completing the square, show that 2 1r k .

(b) Hence or otherwise, find the values of k and r if the graph of the function is

symmetrical about 2 13x r .

Answer:-(b) k = 3 , –1 and r = –3 , 5

6. The function 6 2f x x x h has a maximum value of 10 and h is a constant.

(a) Find the value of h.

(b) Sketch the graph of 6 2f x x x h for the value of h that is determined

in (a) above.

(c) Write the equation of the axis of symmetry.

Answer:- ( a) h = --6 (c) x = –2

7. Given y = x2 + 2kx + 3k has minimum value 2.

(a) Without using the method of differentiation, find the two possible values of k.

(b) With these values of k, sketch on the same axis, two graphs for

y = x2 + 2kx + 3k.

(c) State the coordinates of the minimum point for y = x2 + 2kx + 3k .

Answer:- (a) k =1 , 2 (c) (1, 2), (2, 2)

******************************************************************************

CHAPTER 4 : SIMULTANEOUS EQUATIONS

1. Solve the simultaneous equations 3x + 2y = 1 and 3x2 – y

2 = 5x + 3y.

Answer: x = -7/3 , y = 4 ; x = 1, y = -1

2. Solve the simultaneous equations

3 2 5

and 2 3 5

xy x yx y

Answer: x = 41/10, y = 16/3 ; x = 1, y = -5

3. Solve the simultaneous equations 2x - y = 4 and 2x2 + xy - 3x = 7. Give your answers

correct to three decimal places.

Answer: x = 2.461, y = 0.922; x = -0.711, y = -5.422


8

CHAPTER 5 : INDICES AND LOGARITHMS

1. Solve the equation 1

2

25.5

1625

xx

x Answer:-

5

2x

2. Solve the equation 3548.2 xxx

Answer:- 1x

3. Show that xxx 3533 12 is divisible by 13.

Answer : 1313 x

.4. Solve the equation 34264 32 x

Answer:- 23

12

5. Solve the equation 1023 12 xx

Answer: 0 6477.

6. Solve the equation 3log]32[loglog 924 x

Answer:3.5

7. Solve the equation x

xx1

log)14log()2log(

Answer:5.8284 or 0.1716

8. Solve the equation log2x - 4 logx16 = 0

Answer: 1

1616

,

9. Solve the equation xx 572 1

Answer: 7232.3x

10. Solve 813 2log

x

Answer : 16x

11. Solve the equation 32x+1

- 2 (3x+0.5

) - 3 = 0

Answer: 0.5

12. Solve the equation 102x+1

- 7 (10x) = 26

Answer:0.3010

13. Given that m5log3 and n2log9 , express 50log3 in terms of m and n

Answer:- 2 2m n

14. Given that kx 2log and hx 7log , express xx

5.3log in terms of k and h

Answer:- 2h-2k+1

15. Given that yxyx 222 loglog3log2 , show that xyyx 622

16. If log2a + log2b = 4, show that log4ab = 2 and that log8ab = 4/3. If log2a + log2b = 4,

show that log4ab = 2 and that log8ab = 4/3.


9

CHAPTER 6 :COORDINATE GEOMETRY

1. The following information refers to the equations of two straight lines, AB and CD which

are parallel to each other.

Express p in terms of q. Answer: p = )1(3

2q

2. The triangle with vertices A(4,3), B(-1,1) and C(t , -3) has an area 11 unit2.

Find the possible values of t.

Answer: t = 0 , -22

3. The points P(3, p), B(-1, 2) and C(9,7) lie on a straight line. If P divides BC

internally in the ratio m : n , find

(a) m : n ,

(b) the value of p. Answer:(a) 2 : 3 (b) p = 4

4. (a) A point P moves such that its distance from point A (1,– 4) is always 5 units.

Find the equation of the locus of P. Answer: 2 2 2 8 8 0x y x y

(b) The point A is (1, -3) and the point B is (4, 6). The point Q moves such that

QA : QB = 2 : 3. Find the equation of the locus of Q.

Answer : 5x2+5y

2+14x+102y-118 =0

(c) A point R moves along the arc of a circle with centre A(2, 3). The arc passes

through Q(-2, 0). Find the equation of the locus of R.

Answer : x2+y

2-4x-6y-12 =0

(d) A point S moves such that its distance from point A(–3,4) is always twice its

distance from point B(6,-2). Find the equation of the locus of S.

Answer: 2 2 18 8 45 0x y x y

(e) The point M is (2, –3) and N is (4, 5). The point T moves such that it is always

equidistance from M and from N. Find the equation of locus of T.

Answer : e) x+4y = 7

(f) Given point A (1,2) and point B (4, –5). Find the locus of point W which

moves such that AWB is always 900.

Answer: 2 2 5 3 6 0x y x y

AB : 2y = p x + q

CD : 3y = (q + 1) x + 2

Where p and q are constants


10

Solutions to question no 5, 6 and 7 by scale drawing will not be accepted.

5. In Diagram 1, the straight line PR cuts y-axis at Q such that PQ : QR = 1 : 3. The

equation of PS is 2y = x + 3.

(a) Find (i) the coordinates of R,

(ii) the equation of the straight line RS,

(iii) the area PRS.

(b) A point T moves such that its locus is a circle which passes through the points P, R and

S. Find the equation of the locus of T.

Answer: a)(i) R = (9 , 16) (ii) y = – 2x + 34 (iii) 80 unit 2 b) x

2 + y

2 – 6x – 16y – 27 = 0

6. Diagram 2 shows the straight line graphs PQS and QRT in a Cartesian plane.

Point P and point S lies on the x-axis and y-axis respectively. Q is the mid point of PS.

y

S y 3x = 4

Q

R(0, 1)

P

x

O

T

Diagram 2

P(–3, 0 )

Q( 0, 4 )

S

R y

x O

Diagram 1


11

(a) Find,

(i) coordinates of the point Q,

(ii) area of the quadrilateral OPQR.

(iii) The equation of the straight line which is parallel to QT and passes through S.

(b) Given 3QR = RT, calculate the coordinates of the point T.

(c) A point moves in such a way that it’s distance from S is 1

2it’s distance from the point T.

(i) Find the equation of locus of the point T.

(ii) Hence, determine whether the locus cuts the x-axis or not.

.

Answer: (a)(i) (2

3, 2) (ii)

5

3(iii)

34

2y x (b) (2, 2) (c)(i) 3x

2 + 3y

2 + 4x 36y +56 = 0

(ii) No

7. y

K

P

J

Q

R

x

O

L

Diagram 3

In Diagram 3, P(2, 9), Q(5, 7) and R(42

1, 3) are the mid point of the straight line JK, KL and

LJ such that JPQR form a parallelogram.

(a) Find,

(i) the equation of the straight line JK,

(ii) the equation of the perpendicular bisector of the straight line LJ.

(b) Straight line KJ is extended until it intersects the perpendicular bisector of the straight line

LJ at the point S. Find the coordinates of the point S.

(c) Calculate the area of PQR and consequently the area of JKL.

Answer: (a)(i) y = 8x 7 (ii) 4y = 6x 15 (b)

3,

2

1 (c) 6

2

1; 26


12

CHAPTER 7 :STATISTICS

1. Table 1 shows the results obtained by 100 pupils in a test.

Marks < 20 < 30 < 40 < 50 < 60 < 70 < 80 < 90

Number of pupils 3 8 20 41 65 85 96 100

Table 1

(a) Based on Table 1, complete the table below.

Marks 10 – 19

Frequency

(b) Without drawing an ogive, estimate the interquartile range.

Answer:-(b)Interquartile range = 22.62

2. The mean and standard deviation of a set of integers 2 , 4 , 8 , p and q are 5 and 2

respectively.

(a) Find the values of p and of q .

(b) State the mean and variance of the set integers 7, 11, 9 , 2p + 3 and 2q + 3

Answer:-(a) 5 6 or 6 5, ,p q p q (b) Mean =11 Variance = 8

3. The histogram in Diagram 1 shows the marks obtained by 40 students in Mathematics test.

(a) Without drawing an ogive , calculate the median mark.

(b) Calculate the standard deviation of the marks.

Answer:(a) 27.17 (b) 6.595

Diagram 1

Marks

Number of students

15.5 20.5 25.5 30.5 35.5 40.5

10

8

6

4

2

0


13

4. Table 2 shows the frequency distribution of the Chemistry marks of a group of students.

Marks Number of students

1 – 10 2

11 – 20 3

21 – 30 5

31 – 40 10

41 – 50 p

51 – 60 2

Table 2

(a) If the median mark is 34.5 , calculate the value of p .

(b) By using a scale of 2 cm to 10 marks on the horizontal axis and 2 cm to 2 students

on the vertical axis, draw a histogram to represent the frequency distribution of

the marks. Find the modal mark.

(c) What is the modal mark if the mark of each student is increased by 8 ?

Answer:- (a) p = 6 (b) Mode = 36.5 (c) 44.5

5. The scores, x , obtained by 32 students of Class 5 Alfa in a test are summarized as 22496 and 195488.x x The mean and the standard deviation of the scores, y ,

obtained by 40 students and Class 5 Beta in the test are 66 and 6 respectively.

(a) Find (i) y (ii) 2

y

(b) Calculate the mean and the standard deviation of the scores obtained by all the 72

students.

Answer:-(a)(i) 2640 (ii)175680 (b)Mean = 71.33 , S Deviation= 8.1652

6. A set of data consists of 10 numbers. The sum of the numbers is 120 and the sum of the

squares of the numbers is 1650.

(a) Find the mean and variance of the set of data,

(b) A number a is added to the set of data and the mean is increased by 2, find

(i) the value of a,

(ii) the standard deviation of the new set of data.

Answer:-(a) Mean = 12 , Variance = 21 (b)(i) a = 34 (ii) S Deviation = 7.687


14

CHAPTER 8 : CIRCULAR MEASURES

1.

Diagram 1

Diagram 1 shows a circle with centre O and OA = 10 cm. Straight line AT is a tangent to the

circle at point A, and AOT is a triangle. Given that the area of triangle OAT = 60 cm2, find the

area of sector OAB.

Answer:- 43.80 2cm

2.

Diagram 2

Given that the area of a sector OAB in Diagram 2 with centre O and radius 20 cm is 240 cm2.

Calculate

(a) the length of arc AB

(b) area of shaded region

Answer:- : (a) 24 cm (b) 53.60

3.

Diagram 3

In the Diagram 3, POQ is a circular sector with centre O and a radius of 17 cm. Point R is on

the straight line ORQ such that RQ = 5 cm. Calculate

(a) the value of in radian

(b) the area of the shaded region, in cm2

Answer:- (a) 0.7871 (b) 41.49

B

O

T

A

B

A

O

O

P

Q R


15

4.

Diagram 4

Diagram 4 shows a semicircle with centre O and radius of 10 cm. Given that QS is the length

of arc with centre P and rad 6

QPS . Find

(a) the length of OS.

(b) the area of 1S Answer:-(a) 7.32 cm (b) 35.23 2cm

5.

CC2r

Diagram 5

Two identical circles of radius 2r are drawn with their centres, C on the circumference of each

circle as shown in the Diagram 5. Show that the area of shaded region A 2cm , is given by

3343

2 2 r .

6.

Diagram 6 shows two circles with centres O and A. The respective radii are 8 cm and 3 cm. A

tangent touches the circles at the points Q and R. Given that QOP = 3

radians, find

(a) the length of QR

(b) the perimeter of the shaded region

(c) the area of the shaded region

Answer : a) 9.80 cm b) 24.46 cm c) 10.96 cm2

O S R

Q

P

1S

P 3cm8cm

R

AO

Q

Diagram 6


16

CHAPTER 9 : DIFFERENTIATION

1.(a) (i) Given y = 3x2 + 5, find

dy

dxby using the first principle.

(ii) Differentiate y = 4

3x with the first principle.

(b) (i) Find d

dx

1

2 1x

.

(ii) Given f(x) = 4x(2x 1)5, find f’(x).

(iii) Differentiate 3x2(2x 5)

4 with respect to x.

(iv) Given f(x) = (2x 3)5, find f (x).

(v) Given f(x) = 1

21 3

x

x, find f ‘(x).

(c) (i) Given h(x) = 2)53(

1

x, find the value h

’’(1).

(ii) Given f(x) =

x

x

31

252

, find f '(0).

(d) (i) Find the limit of 2

limit

n

n

n

2 4

2

.

(ii) Given f (r) = r

r

25

34

. Find the limit of f (r) when r 1.

(e) Given y = x(3 x), express yd y

dxx

dy

dx

2

212 in terms of x, in the simplest form.

Hence, find the value of x which satisfy the equation yd y

dxx

dy

dx

2

212 = 0.

Answer :

(a) (i) 6x

(ii) 2

4

x

(b)( i) 2)12(

2

x

(ii)[4(2x 1)4)(12x 1)]

3)52)(56(6) xxxiii

iv) 80(2x 3)3

v)2

23

)1(

164

x

xx

c) (i) 8

27

(ii) 96

d) i) 4

ii) 3

7

e) 12 3x;

x = 4


17

2. (a) Given the function of the graph f(x) = hx3 +

2x

k, which has a gradient function of

f ‘ (x) = 3x2

3

96

x, where h and k are constant. Find,

(i) the value of h and the value of k,

(ii) the coordinate x of the turning point of the graph.

(b) The point P lies on the curve 2)5( xy . It is given that the gradient of the

normal at P is 4

1 . Find

(i) the coordinates of P.

(ii) the equation of the normal to the curve at point P.

(c) A curve with the gradient function 2

22

xx has a turning point at ( k , 8 ) .

(i) Find the value of k .

(ii) Determine whether the turning point is a maximum or a minimum point .

Answer:

2(a) (i) h=1, k= 48

(ii) 2

(b) (i) (7, 4)

(ii)4y + x = 23

(c) i) k = 1,

ii) Minimum

3. (a) Two variables, x and y, are related by the equation y = 3x + x

2.

Given that y increases at a constant rate of 4 units per second, find the rate of

change of x when x = 2.

(b) On a certain day, the rate of increase of temperature, , with respect to time, t s, is

given by dt

d=

2

1(12 t).

(i) Find the value of t at the instant when is maximum.

(ii) Given = 4 when t = 6, find the maximum value of .

Answer: (a) 1.6 units-1

(b) i) 12 ii) 13

4. (a) Given y = t 2t2 and x = 4t + 1.

(i) Find dx

dyin terms of x.

(ii) If x increases from 3 to 3.01, find the corresponding small increment in t.


18

(b) Given y = 2x3 5x

2 + 7, find the value of

dx

dy at the point (2, 3).

Hence, find

(i) the small change in x which causes y to decrease from 3 to 2.98.

(ii) the rate of change of y when x = 2 if the rate of change of x is 0.6 unit per second.

(c) Given yx

16

4, find the value of

dy

dx when x = 2. Hence, find the approximate

value of 16

198 4( . ).

Answer: (a) (i) 4

2 x (ii) 0.0025 (b) (i) 0.005 (ii) 2.4 unit s

1 (c) 1.04

5. Diagram 1 shows a composite solid made up of a cone resting on a cylinder with radius x cm.

Diagram 1

The total surface area of the solid, A cm2, is given by the equation A = 3

xx

162.

(a) Calculate the minimum value of the surface area of the solid.

(b) Given the surface area of the solid is changing at a rate of 42 cm2

s1

. Find the

rate of change of radius at the instant when the radius is 4 cm.

(c) Given the radius of the cylinder increases from 4 cm to 4.003 cm. Find approximate

increment in the surface area of the solid

Answer: b) 36 (c) 2 (d) 0.063

x cm


19

CHAPTER 10 : SOLUTION OF TRIANGLES

1. P

8 cm

6.5 cm

50

Q R

Diagram 1 shows a PQR.

(a) Calculate the obtuse angle PRQ.

(b) Sketch and label another triangle different from PQR in the diagram above,

so that the lengths of PQ and PR and the angle PQR remain unchanged.

(c) If the length of PR is reduced whereas the length of PQ and angle PQR remain

unchanged, calculate the length of PR so that only one PQR can be formed

Answer: (a) 109 28’ or 109.47 (c) 6.128 cm

2.(a) Diagram 2 shows a pyramid VABCD with a square base ABCD. VA is

vertical and the base ABCD is horizontal. Calculate,

(i) VTU,

(ii) the area of the plane VTU.

Diagram 2

Answer: (a)(i) 84 58’ or 84.97 (ii) 22.72 cm2

Diagram 1

V

8 cm

A B

6 cm U

2 cm

D T 4 cm C


20

(b) .

Q R

4 cm

K

L Diagram 3

P S

6 cm

J 8 cm M

Diagram 3 shows a cuboid. Calculate, (i) JQL,

(ii) the area of JQL.

Answer: (i) 75 38’ or 75.640 (ii) 31.24 cm

2

3.

5 cm

D 50

4 cm

8 cm

C

Diagram 4

Diagram 4, ABCD is a cyclic quadrilateral of a circle centered O. Calculate

(a) the length of AC, correct to two decimal places,

(b) ACD,

(c) the area of quadrilateral ABCD.

Answer: (a) 11.85 cm (b) 115.01 (c)36.08cm2

4. Diagram 5 shows two triangles PQT and TRS. Given that PQ = 24 cm, TS = 12 cm,

032TPQ , PT = TQ and PTS and TRQ are straight lines.

.O

A

B

A

Diagram 5

32o 24 cm

12 cm

T

S

R Q

P


21

(a) Find the length, in cm, of PT,

(b) If the area of triangle PQT is three times the area of triangle TRS, find the length of TR.

(c) Find the length of RS.

(d) (i) Calculate the angle TSR.

(ii) Calculate the area of triangle QRS.

Answer: (a) 14.15 cm, (b) 5.563 cm, (c) 10.79 cm, (d)(i) 27.60o (ii) 46.31 cm

2

5. Diagram 6 shows a triangle PQR.

(a) Calculate the length of PR.

(b) A quadrilateral PQRS is now formed so that PR is the diagonal, PRS = 40o and PS = 8

cm. Calculate the two possible values of PSR .

(c) Using the obtuse PSR in (b), calculate

(i) the length of RS,

(ii) the area of the quadrilateral PQRS.

Answer: (a) 10.62 cm (b) 58.57o ; 121.43

o (c) (i) 3.964 cm ; (ii) 47.34 cm

2

6. Diagram 7 shows a camp of the shape of pyramid VABC. The camp is built on

a horizontal triangular base ABC. V is the vertex and the angle between the inclined

plane VBC with the base is 60.

C

A

Given VB = VC = 25 cm and AB = AC = 32 cm and BAC is an acute angle. Calculate

(a) BAC if the area of ABC is 400 cm2,

(b) the length of BC,

(c) the lengths of VT and AT, where T is the midpoint of BC,

(d) the length of VA,

(e) the area of VAB

Answer: (a) 51.38o (b) 27.74

o (c) 20.80 cm; 28.84 cm (d) 25.78 cm (e) 315.33 cm

2

B

V

R

75o

10 cm 7 cm

Q

P

Diagram 6

Diagram 7


22

CHAPTER 11 : INDEX NUMBER

1. Table 1 shows the price indices and percentage usage of four items, P, Q, R, and S, which are

the main ingredients of a type biscuits.

Item Price index for the year 1995 based on

the year 1993

Percentage of usage (%)

P 135 40

Q x 30

R 105 10

S 130 20

Table 1

Calculate,

(a) (i) the price of S in the year 1993 if its price in the year 1995 is RM37.70

(ii) the price index of P in the year 1995 based on the year 1991 if its price index in the

year 1993 based in the year 1991 is 120.

(b) The composite index number of the cost of biscuits production for the year 1995

based on the year 1993 is 128. Calculate,

(i) the value of x,

(ii) the price of a box of biscuit in the year1993 if the corresponding price in the

year 1995 is RM 32.

Answer: (a)(i) RM29 (ii) 162 (b)(i)125 (ii) RM25

2.

Diagram 1

A technology product consists of five components, V, W, X, Y and Z. Diagram 1 shows a bar

chart showing the daily usage of the components used to produce the technology product.

The following table shows the prices and the price indices of the components.

V W X Y Z

47

34

22

12

3

Daily Usage ( RM)

Component


23

Component Price in the year 2001

(RM)

Price in the year 2003

(RM)

Price index in 2003

based on 2001

V 13.00 16.25 y

W 12.50 17.25 138

X 2.50 x 106

Y 14.90 22.35 150

Z z 24.50 140

(a) Find the values of x, y and z.

(b) Calculate the composite index representing the cost of the technology product in

the year 2003 using the year 2001 as the base year.

(c) If the total monthly cost of the components in the year 2001 is RM1.5 million, find

the total monthly cost of the components in the year 2003.

(d) If the cost of each component rises by 23% from the year 2003 to 2004, find the composite

index representing the cost of the technology product in the year 2004 based on the year

2001.

Answer:

3. (a) In the year 1995, price and price index for one kilogram of certain grade of rice is

RM2.40 and 160 respectively. Based on the year 1990, calculate the price per

kilogram of rice in the year 1990.

Item Price index in

the year 1994

Change of price index from the

year 1994 to the year 1996

Weightage

Timber 180 Increased 10 % 5

Cement 116 Decreased 5 % 4

Iron 140 No change 2

Steel 124 No change 1

Table 2

(b) Table 2 shows the price index in the year 1994 based on the year 1992,

the change in price index from the year 1994 to the year 1996 and the weightage

respectively. Calculate the composite price index in the year 1996.

.

Answer : (a) 1.50 (b) ITimber = 198, ICement = 110.2; 152.9


24

4. Table 3 shows the price indices and the weightages of Azizan’s monthly expenses in the

year 2005 based in the year 2004.

Expenses Price index in 2005 based on 2004 Weightage

Rental 108 3

Food 120 4

Car installment 102 2

Miscellaneous 112 1

Table 3

(a) If the expenses for miscellaneous in the year 2005 was RM 1 456 , find the

miscellaneous expenses in the year 2004.

(b) If the rental increases by 10% from the year 2005 to the year 2006,find the price index for

the rental in the year 2006 based on the year 2004.

(c) Calculate the composite index for the expenses in the year 2005 based on the year 2004.

(d) The price index for food in the year 2006 based on the year 2205 is 105. If the expenses

on food in the year 2006 were RM3150, find the expenses on food in the year 2004.

Answers : a) 1300 b) 118.8 c) 112 d) 2500

Add Maths Perfect Score Module Form 4 Topical

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