Tampines Prelim 2009 Am 2

TAMPINES SECONDARY SCHOOL PRELIMINARY EXAMINATION 2009 SECONDARY FOUR EXPRESS ADDITIONAL MATHEMATICS PAPER 2 15 September 2009

4038/02 2 hours 30 minutes

Additional Materials: Writing Paper

READ THESE INSTRUCTIONS FIRST

Write your name, class and register number on all the work you hand in. Write in dark blue or black pen. You may use a pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Write your calculator model on the top right hand corner of your answer script. Answer all questions. Write your answers on the separate writing papers 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 number of marks for this paper is 100.

This question paper consists of 6 printed pages. Setter: Mdm Loh M W

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Mathematical Formulae

1. ALGEBRA Quadratic Equation For the equation ax 2  bx  c  0 ,

x 

 b  b 2  4ac . 2a

Binomial expansion n  n 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!  r  n  r  ! r!

a  b n 

2. TRIGONOMETRY Identities

sin 2 A  cos 2 A  1 sec 2 A  1  tan 2 A cosec 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 tan( A  B )  1  tan A tan B sin 2 A  2 sin A cos A cos 2 A  cos 2 A  sin 2 A  2 cos 2 A  1  1  2 sin 2 A 2 tan A tan 2 A  1  tan 2 A 1 1 sin A  sin B  2 sin ( A  B ) cos ( A  B ) 2 2 1 1 sin A  sin B  2 cos ( A  B ) sin ( A  B ) 2 2 1 1 cos A  cos B  2 cos ( A  B ) cos ( A  B ) 2 2 1 1 cos A  cos B  2 sin ( A  B ) sin ( A  B ) 2 2

Formulae for ∆ABC

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

TPSS/PRELIM AM P2 4E 2009

1 bc sin A 2 2

1. The mass M, in grams, of a radioactive substance present t years after first being observed is given by the formula

1 M  24  ( ) 0 .03472 t 2 (i) Find the initial mass of the substance at the beginning of the observation.

[1]

(ii) Calculate the mass of the substance after 45 years.

[2]

(iii) How long will it take for the substance to reduce to 10 grams?

[3]

2. The roots of the equation 3 x 2  2 x  4  0 are α and β. Find the quadratic equation whose roots are 2α + β and α + 2β. [7]

cos A  sin A cos A  sin A   2 sec 2 A . cos A  sin A cos A  sin A

3. (i) Prove the identity

[4]

(ii) Find all the angles between 0o and 360o which satisfy the equation cos A  sin A cos A  sin A  5 cos A  sin A cos A  sin A

4. (a) Given that x – m is a factor of the expression x 2  (m  2) x  m 2  4m  8 , calculate the possible values of m.

[4]

[4]

(b) Given that 3 x 3  4 x 2  6 x  3  ( Ax  B )( x  1)( x  2)  C ( x  1)  D for all real values of x, find the values of A, B, C and D. Hence deduce the remainder when 3 x 3  4 x 2  6 x  3 is divided by x 2  x  2 . [5]

6 ) metres. 3 Express, in the form a  b 3 , where a and b are integers, the area of the rectangle.

5. (a) A rectangle has sides of length (3  4 3 ) metres and (5 

[3]

(b) Given that log 4 (3 x  5)  log 4 ( y  2)  2 , express y in terms of x.

[3]

log x 8  2 log 2 x  7 .

[5]

(c) Solve the equation

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6.

B N C

A

F E D

In the diagram, B, C, D, E and F are points on the circle. AB is a tangent to the circle at B. CFA, DFN and DEA are straight lines. BA is parallel to CE. (a)(i) Prove that triangles NFA and NAD are similar.

[4]

(ii) Hence, show that NA2  NF  ND .

[1]

(b) Given that N is the mid-point of AB, prove that NB 2 

1 AF  AC . 4

[3]

y 7. P y=x(x-3)2

x 0

Q

The diagram shows part of the curve y  x( x  3) 2 . The curve has a maximum point at P and a minimum point at Q. Find (a) the coordinates of P and Q.

[4]

(b) the area of the shaded region.

[5]

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8. In the diagram, OD = 3m, OC = 7m and DOC = DAO = CBO = 90o. It is given that DOA is a variable angle , where 0o <  < 90o. The point E is on the line BC such that ED is parallel to BA.

D

A 3  O

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C

E

B

(i) Show that AB  7 sin   3 cos .

[3]

(ii) Express AB in the form R sin(   ) where R is positive and α is acute.

[3]

(iii) Find the value of  for which AB = 6.5 m.

[3]

9. The function f is defined by f(x) = 3 sin 2x – 1. (i) State the amplitude of f.

[1]

(ii) State the period of f.

[1]

The equation of a curve is y = 3 sin 2x – 1 for 0o ≤ x ≤ 180o. (iii) Find the coordinates of the maximum and minimum points of the curve.

[2]

(iv) Sketch the graph of y = 3 sin 2x – 1 for 0o ≤ x ≤ 180o.

[2]

(v) Sketch, separately from (iv), the graph of y  3 sin 2 x  1 for 0o ≤ x ≤ 180o. State the range of y.

[3]

(vi) Using (v), state the number of solutions of the equation 3 sin 2 x  1  1 in the range 0o ≤ x ≤ 180o.

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[1]

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10.

dy 1  1  2 and P is a point on the curve. The x-coordinate dx 2x of P is positive and the equation of the tangent at P is y = 3x + 1.

A curve is such that

(i) Find the coordinates of P.

[3]

(ii) Find the equation of the curve.

[3]

(iii) Find the equation of the normal at P.

[2]

A point (x, y) moves along the curve in such a way that the y-coordinate increases at a constant rate of 0.15 units per second. (iv) Find the corresponding rate of change of the x-coordinate at point P.

[3]

y

11.

D

B C O

x

E A In the diagram, the points A (8, - 6) and D ( - 4, 10) are at the opposite ends of a diameter of a circle with centre C. The diameter BE is perpendicular to AD. (i) Find the equation of the circle.

[4]

(ii) Find the coordinates of the points B and E.

[8]

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