2009 Rv Am P2

1

RIVER VALLEY HIGH SCHOOL 2009 PRELIMINARY EXAMINATION SECONDARY FOUR CANDIDATE NAME CLASS

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INDEX NUMBER

___________________________________________________________________________

ADDITIONAL MATHEMATICS

4038/02

Paper 2

22 September 2009 2 hours 30 minutes

Additional Materials:

Answer Paper

___________________________________________________________________________ READ THESE INSTRUCTIONS FIRST Write your class, index number and name on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use a soft pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all questions. Write your answers on the separate Answer Paper provided. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. The use of a scientific calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers. At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. The total of the marks for this paper is 100.

This paper consists of 6 printed pages (including this page) 2009 RVHS Preliminary Examination

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O Level A. Maths (4038) P2

2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation

ax2 + bx + c = 0,

 b  b 2  4ac 2a

x= Binomial Theorem

n

n

n

 

 

 

(a + b) n = a n +   a n  1b +   a n  2b2 + . . . +   a n rbr + . . . + b n , 1 2 r n! n( n  1)...( n  r  1) n  where n is a positive integer and    r r !( n  r )! r!  

2. TRIGONOMETRY Identities sin2 A + cos2 A = 1 sec2 A = 1 + tan2 A cosec2 A = 1 + cot2 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  B) =

tan A  tan B 1  tan A tan B

sin 2A = 2 sin A cos A cos 2A = cos 2A – sin 2A = 2 cos 2A – 1 = 1 – 2 sin 2A tan 2A =

2 tan A 1  tan 2 A

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

a b c   sin A sin B sin C a2 = b2 + c2  2bc cos A = 2009 RVHS Preliminary Examination

1 bc sin A 2

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O Level A. Maths (4038) P2

3 1

A cup of hot water was left to cool on the table so that, t minutes later, its temperature,

 C , is given by   23  45e



t 6

.

(a) Find (i) the initial temperature of the water,

[1]

(ii) the time taken for the water to reach a temperature of 28C.

[3]

(b) Determine the value for which  approaches as t becomes very large and explain your reasoning clearly. [2]

2

The roots of the quadratic equation 2x2  3x + 1 = 0 are  and . (a) State the value of  +  and of . (b) Show that  2   2 =

[2]

5 and hence, find the quadratic equation in x whose roots are 4

1 1 and 2 . 2  

3

[5]

(a) Prove the identity sin 3  sin   4cos 2  sin  .

[2]

(b) Find all the angles between 0 and 360 which satisfy the equation

4

5

(i) sin(2 x  30)  0.5 ,

[3]

(ii) 2(1  cos y)  3sin 2 y .

[4]

(a) Solve the equation lg( x  21)  2  lg x .

[4]

4 p  (b) Given that x  log 2 p and y  log q 2 , express log 2  in terms of x and y.  q   

[4]

(a) (i) Given that x 3  2 x 2  2 x  3  ( Ax  B)( x  1)( x  1)  Cx  1 for all real values of x, find the values of A, B and C. [3] (ii) Hence, or otherwise, find the remainder when x 3  2 x 2  2 x  3 is divided by [1] x2  1. (b) Solve the equation x 3  3x 2  54 x  112  0.

2009 RVHS Preliminary Examination

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[5]

O Level A. Maths (4038) P2

4 6

(a) Q

R

S

U

T

O

P

In the above figure, the lines OP, STU and RQ are parallel and OS = 2SR. UQ 1 TU 2 By considering similar triangles, show that  and  . PU 2 RQ 3

[4]

(b)

B

C

D

A

T

In the above diagram, the line TAD is a tangent to the centre at the point A. BCD is a straight line and the chord BA and CA intersect at the point A.

7

(i) Show that ACD is similar to BAD .

[2]

(ii) Prove that AD  AC = AB  CD.

[2]

The function f is defined, for all values of x, by f ( x )  sin 2 x  1 . (a) State the amplitude and period of f.

[2]

(b) Sketch the graph of y  sin 2 x  1 for 0  x  2 .

[2]

(c) Sketch, on the same diagram for 0  x  2 , the graph of y   cos x .

[2]

Hence state the number of roots of the equation cos x  sin 2 x  1 for 0  x  2 .[2]

2009 RVHS Preliminary Examination

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O Level A. Maths (4038) P2

5 8

y y=

1 x+4 2

y

P

1 8 x 2 x 1

Q

0

4

The diagram above shows part of the line y =

x

1 1 8 x + 4 and curve y  x  2 2 x 1

intersecting at a point P. The point Q is a minimum point of the curve. (a) Show that the x-coordinate of the point P and Q is 1 and 3 respectively.

[5]

(b) Hence, find the area of the shaded region.

[5]

9 D

C



12 cm 5 cm

A

E

B

The diagram shows a trapezium ABCD. The point E lies on the side AB such that DE = 12 cm, EC = 5 cm, DEC  90 and ADE    . (a) Show that AB  12 sin   5 cos  .

[2]

(b) Express AB in the form R sin(   ) , where R > 0 and 0 <  < 90.

[4]

(c) Find the value of  for which AB has a maximum length.

[2]

(d) Find the value of  for which AB = 10.5 cm.

[3]

2009 RVHS Preliminary Examination

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O Level A. Maths (4038) P2

6 10

(a) Given that the gradient

dy   of a curve is 3cos  2 x   and that this curve passes 2 dx 

 , 2). 4 (i) Find the equation of the curve. through the point P(

[4]

1  (ii) Show that the equation of the normal at P is given by y  x  2  . 3 12

[3]

(b) A point (x, y) moves along the curve y  ( x  3) x  2 , where x  2, in such a way that the x-coordinate increases at a constant rate of 0.5 units per second. (i) Show that

dy 3x  1  . dx 2 x  2

[3]

(ii) Find the rate of change of the y-coordinate as the point passes through the point where x = 3 units. [2]

11

In the diagram below, A (7,  6), B (9, 2) and C (1, 4) are points on a circle and D is the mid-point of the line AC. (a) Find the gradient of AB and BC and show that AC is the diameter of the circle. [3] (b) Find the point D and hence, determine the equation of the circle.

[5]

(c) The line x  y  2  0 intersects the circle at the point P and Q. Show that the x1 m 1 m and respectively, where m and n are n n positive integers to be found. [4]

coordinates of P and Q are

~ ~ ~ ~ ~ End of Paper ~ ~ ~ ~ ~

2009 RVHS Preliminary Examination

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O Level A. Maths (4038) P2