Bowen 2009 Prelim Am P2

Class

Full Name

Index Number

Preliminary Examination 2009

O 4038/02

I believe, therefore I am

ADDITIONAL MATHEMATICS Paper II Secondary 4 Express / 5 Normal (A) 17/09/2009

2 hours 30 minutes

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 100 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

1

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

2

1.

(a)

(b)

2.

Find all the angles between 0o and 360o which satisfy the equation (i)

sin( 2 x  30  )  cos 30  ,

(ii)

1  3 cos 2 y  4 sin y .

[5]

Prove the following identities (i)

sec(90   A)   cosec A,

(ii)

sin 5 B  sin 3B  sin B  tan 3B . cos 5B  cos 3B  cos B

[5]

In the triangle DEF,  EDF =  BCF = 90o, EB = 7 cm and BF = 4 cm.

(a)

Given that  DEF =  CBF =  , 0o <  < 90o, show that P, the perimeter of the rectangle ABCD is given by P  14 sin   8 cos  . [2]

(b)

Express P in the form R sin(   ) and hence find the value of  for which P = 10 cm. [3]

(c)

Find the maximum value of P and the corresponding value of  .

[2]

3

3.

Given that y  x ln 2 x , find an expression for (a)

(b)

4.

(a)

dy . Hence, find dx

the coordinates of the stationary point of the curve y  x ln 2 x . Determine the nature of the point.

[5]

the rate of change of x at the instant when x = 1, given that y increases at the rate of 0.2 units/second.

[3]

Find the equation of the normal to the curve y  tan 2 x at x  Give your answer in terms of  .

(b)

5.

 . 8 [5]

x dy . Obtain an expression for and 2x  3 dx hence explain why the curve has no turning point. [3]

A curve has the equation y 

In the diagram, the tangent at P meets XY produced at Z.

(a)

Show that ∆PYZ is similar to ∆XPZ.

[2]

(b)

Prove that PZ  PY  PX  YZ .

[2]

(c)

Hence show that

PX 2 XZ  . PY 2 YZ

[3]

4

6.

7.

(a)

When x3  3x2  2x + k is divided by x  3, the remainder is half the remainder when it is divided by x + 2. Find the value of k. [3]

(b)

If 5x3 + px2 + x + 3  (x + 1)(qx2 + rx) + s, find the value of p, of q, of r and of s. [4]

(a)

Simplify the expression

(b)

6 2 3



2 3 2 3

, leaving your answer in

surd form.

[4]

Solve 3 2 x 1  3 x  2  6  0 .

[4]

8. 13x

10x 13x

h

An open box has a triangular base with sides 13x, 13x, 10x cm and a height of h cm. Given that the volume of the box is 3840cm 3 , express h in term of x. (a)

If the total external surface area of the box is A cm 2 , show that 2304 A  60 x 2 . x

(b)

Hence find the value of x for which A is a minimum.

(a)

1   Find the coefficient of x –5 in the expansion of  2 x 2  3  . 3x  

[4] [4]

10

9.

(b)

[4]

Find, in ascending powers of x, the first three terms in the expansion of (1+ax)6. Given that the first two non-zero terms in the expansion of 7 (1+bx) (1+ax)6 are 1 and  x 2 , find the values of a and b. [5] 3 5

10.

11.

(a)

Find the equation of the circle with the line segment AB as the diameter if the coordinates of A and B are (-2, -3) and (2, 1) respectively. [4]

(b)

Find the equation of the tangent to the circle x2 + y2  8x  2y + 12 = 0 at the point P(3, 3) on the circle. [4]

(a)

The table shows experimental values of the two quantities x and y, which are known to be related by a law of the form y  a(1  x ) n , where a and n are constants. x 1 2 3 4 5 y

3.48

4.82

6.06

7.25

8.39

Explain how a straight line graph can be drawn to represent the given data. On a piece of graph paper, draw this graph for the given data in the table. Use the graph to estimate the value of a and of n. [5] (b)

12.

13.

1 , a straight line is form with x2 x and y intercepts (-5, 0) and (0, -2) respectively. Express y in terms of x. [4]

When the graph of xy is plotted against

Find the range of values of k for which the line y = x + k and the curve y 2  2 y  1  4 x do not intersect.

[4]

Given that  and  are the roots of the equation x 2  6 x  2  0 . (a)

(b)

Find the numerical value of    (i) , 2 2  (ii) 3  3. the equation whose roots are

[2] [2]

  and . 2 2

[3]

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