Bowen 2009 Prelim Am P1

Class

Full Name

Index Number

Preliminary Examination 2009

O 4038/01

I believe, therefore I am

ADDITIONAL MATHEMATICS Paper I Secondary 4 Express / 5 Normal (A) 16/09/2009

2 hours

READ THESE INSTRUCTIONS FIRST 1. Write your name, class and register number in the spaces provided. 2. Answer all questions in the writing paper provided. 3. If working is needed for any question, it must be neatly and clearly shown in the space below that question. Omission of essential working will result in loss of marks. 4. The number of marks is given in brackets [ ] at the end of each question or part question. 5. You should not spend too much time on any one question. 6. The total mark for this paper is 80 You are expected to use a scientific calculator to evaluate explicit numerical expressions. If the degree of accuracy is not specified in the question, and if the answer is not exact, give the answer to three significant figures. Give answers in degrees to one decimal place. For  , use either your calculator value or 3.142.

DO NOT OPEN THIS PAPER UNTIL YOU ARE TOLD TO DO SO For Examiner’s Use

This document consists of 5 printed pages. Setter: Mr Andrew Yeo

Mathematical Formulae 1.

ALGEBRA

Quadratic Equation For the equation

ax 2  bx  c  0 , x

b  b 2  4ac , 2a

Binomial Theorem

n  n n (a  b) n  a n   a n1b    a n  2 b 2  .....   a n r b r  ....  b n 1  2 r n n! n(n  1)...(n  r  1) where n is a positive integer and     r  (n  r )! r! = r!   2.

TRIGONOMETRY

Identities

sin 2 A  cos 2 A  1 sec 2 A  1  tan 2 A cos ec 2 A  1  cot 2 A sin( A  B )  sin A cos B  cos A sin B cos( A  B )  cos A cos B  sin A sin B tan A  tan B 1  tan A tan B sin 2 A  2 sin A cos A

tan( A  B ) 

cos 2 A  cos 2 A  sin 2 A  2 cos 2 A  1  1  2 sin 2 A 2 tan A 1  tan 2 A sin A  sin B  2 sin 12 ( A  B ) cos 12 ( A  B ) tan 2 A 

sin A  sin B  2 cos 12 ( A  B ) sin 12 ( A  B ) cos A  cos B  2 cos 12 ( A  B ) cos 12 ( A  B ) cos A  cos B  2 sin 12 ( A  B ) sin 12 ( A  B ) Formulae for ABC

a b c   , sin A sin B sin C a 2  b 2  c 2  2bc cos A, 1   bc sin A. 2

1.

7  2x in partial fractions. ( x  1)( x  2)

(a)

Express

(b)

Hence find the gradient of the curve y 

[3]

7  2x at the point ( x  1)( x  2)

x = -1.

2.

[3]

The diagram shows a semicircle OABCD with diameter 8 cm and centre O. 5 ABCD is a trapezium with OAB = π radians 18 and AD parallel to BC.

B

5

C

π

18

3.

4.

5.

(a)

Find, in terms of , AOB and BOC.

(b)

Hence or otherwise, find the area of the shaded region.

A

D

O

[5]

The coordinates of the points A and B are A(0, 5) and B(9, 8). (a)

Find the equation of the perpendicular bisector of the line joining A and B.

(b)

If the perpendicular bisector of AB cuts the x and y-axes at point P and Q respectively, calculate the distance between P and Q.

(c)

Find the area of the quadrilateral AQBP.

[9]

(a)

If log 8 x  p , express log 2 x in terms of p.

[2]

(b)

Solve log 2 y 2  4  log 2 ( y  5)

[3]

(c)

Simplify

8 x 1 4 x  4 . 32 x 1  20(2 5 x 1 )

Solve the simultaneous equations 5x  25y  0.2 , log 2 ( y  x )  log 3 9  log 2 ( x  4) .

[5]

[5]

6.

dy k  x  2 , where k is a constant, is such that the tangent dx x at (2, 1) passes through the origin.

The curve for which

(a)

State the gradient of the tangent in term of k.

(b)

Find the equation of the tangent, hence, determine the value of k.

(c)

Determine the equation of the curve.

[7]

y

7.

y  ( x  2) 2  3 A B

yx 7 x O x The curve y  ( x  2)2  3 and the straight line y  x  7 intersect at the points A and B as shown in the diagram above. Find the coordinates of A and of B. Calculate the area of the shaded region. [8]

8.

9.

 1 2  , find A-1 and hence solve the simultaneous equations Given that A     3 4 a  2b  5  0 , [5] 4b  5  3a .

(a)

Given that y 

cos x dy k , express in the form , dx sin x  cos x ( sin x  cos x ) 2

where k is a constant. Hence, evaluate

(b)

Given that f (x) > 0 and evaluate

4

3 dx . ( sin x  cos x ) 2

6

6

1

4

 f x dx  12 and 

 4.5  1.8 f x dx . 1



 3 0

[6]

f x  dx  5 ,

[4]

10.

(a)

A particle P moves along a horizontal straight line so that t seconds after motion has begun, its displacement, s m, from a fixed point O, is given by s = t3 + 5t2 – 2t + 4. Express the velocity of P in terms of t.

(b)

[2]

A second particle Q starts its motion at the same instant as P and moves along the same horizontal line as P. Its acceleration, a m/s2, t seconds after motion has begun, is given by a = 6t + 4. Given that Q has a velocity of 4 m/s when t = 1 and an initial displacement of 6m, obtain an expressions, in terms of t, for the velocity and displacement of Q.

(c)

[4]

Find the value of t at the instant when P and Q collide and determine whether P and Q are traveling in the same direction at this instant. [4]

11.

Sketch the curve y  1  3sin 2 x for 0  x  , and state the corresponding range of values of y. By adding a suitable line to the same diagram, state, for the interval 0  x  , the number of solutions to the equation π 1  3sin 2 x  4  π  x  . [5]

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