STUDENT’S NAME: __________________________ CLASS:

FINAL EXAMINATION

2012

ADDITIONAL MATHEMATICS

PAPER 1

SENIOR 4

TWO HOURS

Instructions to candidate:

DO NOT OPEN THIS BOOKLET UNTIL YOU ARE TOLD TO DO SO

1. This question paper consists of 25 questions.

2. Answer ALL questions.

3. Show your working. It may help you to get marks.

4. The diagrams in the question are not drawn to scale unless stated.

5. You may use a non – programmable scientific calculator.

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This question paper consists of 16 printed pages.

The following formulae may be helpful in answering the questions. The symbols given are the ones commonly used.

ALGEBRA

1

2 am an = a m + n

3 am an = a m - n

4 (am) n = a nm

5 loga mn = log am + loga n

6 loga = log am - loga n7 log a mn = n log a m

8 logab =

9 Tn = a + (n-1)d

10 Sn = 11 Tn = ar n-1

12 Sn = , (r 1)

13 , <1

CALCULUS

1 y = uv ,

dydx

=udvdx

+vdudx

2 y=u

v ,

dxdy

=v

dudx

−udvdx

v2,

3

dydx

=dydu

×dudx

4 Area under a curve

= ∫a

b

y dx or

= ∫a

b

x dy

5 Volume generated

= ∫a

b

πy2

dx or

= ∫a

b

πx2

dy

GEOMETRY

1 Distance = √( x1−x2 )2+( y1− y2 )

2

2 Midpoint

(x , y) = (

x1+ x2

2 ,

y1+ y2

2)

3 |r|=√ x2+ y2

4

5 A point dividing a segment of a line

( x,y) = (

nx1+mx2

m+n,ny1+my2

m+n)

2 4

2

b b acx

a

n

m

a

b

c

c

log

log

])1(2[2

dnan

r

ra

r

ra nn

1

)1(

1

)1(

r

aS

1 r

6 Area of triangle =12|(x1 y2+ x2 y3+x3 y11

)−( x2 y1+x3 y2+x1 y3 )|

STATISTICS

1 =

2 =

3 = =

4 = =

5 m =

6

7

8

9

10 P (A B) = P (A) +P (B) - P (A B)

11 P (X = r) = , p + q = 1

12 Mean µ = np

13

14 z =

TRIGONOMETRY

1 Arc length, s = rθ

2 Area of sector , L = 3 sin 2A + cos 2A = 1

4 sec2A = 1 + tan2A

5 cosec2 A = 1 + cot2 A

6 sin 2A = 2 sinA cosA

7 cos 2A = cos2A – sin2 A = 2 cos2A - 1 = 1 - 2 sin2A

8 tan 2A =

2 tan A

1− tan2 A

9 sin (A±B) = sinA cosB ± cosA sinB

10 cos (A±B) = cosA cosB ∓ sinA sinB

11 tan (A±B) =

tan A±tan B1∓tan A tan B

12

asin A

= bsin B

= csin C

x N

x

x

f

fx

N

xx 2)( 2_2

xN

x

f

xxf 2)( 22

xf

fx

Cf

FNL

m

2

1

1

11

w

IwI

)!(

!

rn

nPr

n

!)!(

!

rrn

nCr

n

rnrr

n qpC

npq

x

13 a2 = b2 + c2 - 2bc cosA

14 Area of triangle =

12

ab sin C

Answer all questions.

1. Given set A = {9, 36, 49, 64} and set B = {-8, -6, 3, 4, 6, 7, 8}. The relation from set A to set B is "the square root of ", state (a) the range of the relation,

(b) the object of 8,

(c) the image of 49. [ 3 marks ]

Answer : (a) ……………………..

(b) ……………………...

(c)....................................

2. Given function f : x→ x+2

5 , find

(a) f (−4 ), (b) the value of x whenf ( x )=x .

[3 marks]

Answer : (a) ……………………..

(b) ……………………...

2

3

For examiner’s

use only

1

3

3. Diagram 1 shows a functionf : x→ax 2+bx .

Find the value of a and of b. [ 3 marks ]

Answer : a =……………………..

b =……………………...

4. Given the roots of the quadratic equation are 3 and p.[4 marks]

Answer : k = ……………………..

p = ……………………...

For examiner’s

use only

4

4

3

3

3

1

5

3

f

DIAGRAM 1

x

5. Solve the quadratic equation2 x ( x−5 )=(2−x ) ( x+3 ) . Give your answer correct to four significant figures.

[ 3 marks ]

Answer : .................................

6. Diagram 2 shows the graph of a quadratic function f(x) = 3(x + p)2 + 2, where p is a constant. The curve y = f(x) has the minimum point (4, 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 marks ]

Answer : (a) ……........................

5

3

For examiner’s

use only

6

3

( 4, q )

DIAGRAM 2

( 4, q )( 4, q )( 4, q )( 4, q )

(b) ……........................

(c)..................................

7. Find the range of values of x for which . [3 marks]

Answer : ..................................

8. Find the values of p if quadratic equation 3 x2+ px+12=0 has two equal roots.

[3 marks]

Answer : ...................................

9. Given and Express in terms of x and y.[3 marks]

7

3

8

3

For examiner’s

use only

4 5 3 12x x x

Answer : ......................................

10. Solve the equation log 3(3 t+9)−log32 t=1 . [3 marks]

Answer : ....……………...………..

11. Find the value of m given that the product of roots for ( x+1)2+m=0 is -5.[3 marks]

Answer : ....……………...………..

12. Solve the equation 81x+1−272 x−3=0 .[ 3 marks ]

9

3

10

3

For examiner’s

use only

11

3

12

3

Answer : ....……………...………..

13. The point A(-6, 6), B(6, -2) and M(-3, 4) are on a straight line. M divides ABinternally in the ratio m : n. Find the ratio m : n.

[4 marks]

Answer: m =…...…………..….......

n = ....................................

14. Diagram 3 shows a straight line PQ with equation

DIAGRAM 3Find (a) the value of h and k

(b) the equation of PQ in intercept form.

[3 marks]

Forexaminer’s

use only

13

4

Answer: a)…...….………..….......

b) ....................................

___________________________________________________________________________

15. The equations of two straight lines are

x5+ y

4=1

and4 y=5 x+2 . Determinewhether the lines are perpendicular to each other.

[4 marks]

Answer : .…………………

16. The coordinates of point P and Q are (-3, 1) and (6, 12) respectively. The point X moves such that XP: XQ = 2: 4. Find the equation of the locus of X.

[3 marks]

15

4

12

3

16

3

For examiner’s

use only

Answer : .…………………

17. The mean of a set of numbers, 6, 8, x, 12 and 13, is 10. Find(a) the value of x.(b) the variance and standard deviation.

[4 marks]

Answer : (a) ……........................

(b) ……........................

18. Table 1 shows the distribution of the weight of 40 pupils in form 4 Johai.

Weight (kg) Number of pupils31 – 35 736 – 40 441 – 45 846 – 50 751 – 55 656 – 60 461 – 65 4

Table 1

(a) Find the range of the weight.

(b) Without drawing an ogive, calculate the median of the distribution of weight

[4 marks]

For examiner’s

use only

17

5

Answer : .………………….

19. The mean of a set of numbers, 5, 7, 12, x and y, is 22. If x= y , find themode of the set of numbers.

[3 marks]

Answer: …….…………...

20. Diagram 4 shows a circle with centre O and AB is the diameter of the circle.

DIAGRAM 4

It is given that AB = 13cm and BC = 6.5cm. Find the perimeter of the shaded region,

giving your answer correct to two decimal places.

[3 marks]

18

4

19

3

For examiner’s

use only

Answer : ……………………..

21. Diagram 5 shows a sector OPQ of a circle with centre O and radius of 7cm. Given the

length of the arc PQ is 16.8cm.

Find

(a) the value of in radians,

(b) the area, in cm2 of the shaded region.[3 marks]

20

3

21

4

DIAGRAM 5

For examiner’s

use only

Answer: (a) = ……………………..

(b) = .…………………

4

22. The radius of circle decreases at the rate of . Find the rate of change of the area when the radius is 6 cm.

[3 marks]

Answer:………………………

23. Given that , find . [3 marks]

Answer:………………………

24. Given that and where k and m are constants, find the value of k and m.

[3 marks]

Answer: k =..…...…………..….......

22

3

For examiner’s

use only

23

3

24

3

m = ....................................

25. The straight line is a tangent to the curve at the point P. find the x-coordinate of the point P.

[3 marks]

Answer: …...…………..….......

END OF QUESTION PAPER

Prepared by: Checked by: Verified by: Received by:

…………….. ……………….. ………………. ………………Mr. Thiyaku Mr. Mahadevan Mdm. Tee Exam Unit

(Subject Teacher) (Subject Teacher) (HOD Mathematics)

For examiner’s

use only

25

3