2009 Rv Am P1

1

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

4

INDEX NUMBER

___________________________________________________________________________

ADDITIONAL MATHEMATICS

4038/01

Paper 1

18 September 2009 2 hours

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 80.

This paper consists of 5 printed pages (including this page)

2009 RVHS Preliminary Examination

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O Level A. Maths (4038) P1

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!

 where n is a positive integer and     r  r!(n  r )!

n (n  1)...(n  r  1) 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) P1

3 1

4 2m

S 5m

P 105 108

W

53 45

71

Q Q 1.5 m R In the above diagram, PQR =  , PRQ = 90, PRS = 45, RPS = 105 and PS = 4 2 m (a) Show that the exact length of PR = 4 m.

[2]

(b) Given further that QR = ( 3  1) m, find tan  in the form a  b 3 where a and b are integers. [3]

2

Solve, for x and y , the simultaneous equations

(16) x (64) y  1 ( 27) y 

3

 3

x

 81 3

[5]

 4 3  and use this to solve the simultaneous Calculate the inverse of the matrix  7 6 equations 4x  3y  7  0 , 7 x  6 y  16  0 .

4(i) (a) Differentiate x e 3 x with respect to x. (b)

5

[2]

xe3 x dx .

[3]

5x 1 in partial fractions. ( x  3)( x  4)

[3]

Hence, evaluate

(a) Express

1 3 0

[5]



(b) Hence, or otherwise, find the gradient of the curve y  where x  2 . 2009 RVHS Preliminary Examination

5x  1 at the point ( x  3)( x  4) [3]

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O Level A. Maths (4038) P1

4 6

A particle moves in a straight line so that, t seconds after leaving a fixed point O , its velocity, v m s 1 , is given by v  3 t 2  15 t  18 . Find

7

(a) the values of t for which the particle is at instantaneous rest.

[2]

(b) the acceleration of the particle when the velocity is equal to its initial velocity.

[2]

(c) the distance travelled by the particle during the first 2 seconds after passing O.

[2]

The equation of the curve is y  where 0  x 

8

2 cos x dy . Find and hence, the x -coordinate, 3  2 sin x dx

 , of the point at which the tangent to the curve is parallel to the x -axis. 2 [6]

(a) Prove that 1  sin 2 x  cos 2 x  2(sin x  cos x) cos x.

[2]

(b) Find all the angles between 0 and 2 which satisfy the equation [4]

1  sin 2 x  cos 2 x  0

9

The two shorter sides of a right-angled triangle are of length ( x  y ) cm and ( x  y ) cm respectively. Given that the length of the hypotenuse is 68 cm and that the area of the triangle is 8 cm 2 , find the length of the two shorter sides, [6]

10

(a) Find the range of values of k for which 2 x( x  2k )  4  7k is always positive for all real values of x. [3] (b) Find the values of k for which the straight line 2x + y = k is a tangent to the curve 4x 2 + y 2 = 8. [3]

11

(a) Find the fifth and sixth terms, in ascending powers of x, in the binomial expansion of 9

 1 2 1  x  .  3 

[4] 9

 1  (b) Hence find the coefficient of x 10 in the expansion of 1  x 2  3 x 2  1 .  3 

2009 RVHS Preliminary Examination

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



[3]

O Level A. Maths (4038) P1

5 12

The diagram shows part of the straight line graph drawn to represent the equation 2 py  e qx . ln y

(2,8)

(0,2) x2

0 Given that the straight line passes through (0,2) and (2,8), find

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(a) the values of p and of q.

[6]

1 (b) the value of y when x  . 3

[2]

A rectangular block has a total surface area is 2700 cm 2 . The base of the block is 2x cm by 3x cm and the height is h cm. (a) Express h in terms of x .

[2] 3

36 x . [2] 5 (c) Hence find the height of the block which has the maximum volume and show that this volume is a maximum. [5] (b) Show that the volume, V cm 3 , of the block is given by V  1620 x 

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2009 RVHS Preliminary Examination

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O Level A. Maths (4038) P1