Tkss Prelim 2009 Am P2

1 Class

Reg Number

Candidate Name _____________________________________

TANJONG KATONG SECONDARY SCHOOL PRELIMINARY EXAMINATION 2009 SECONDARY FOUR

ADDITIONAL MATHEMATICS

4038/02

Paper 2 Friday 18 September

2 hours 30 minutes

Additional Materials: Writing Paper

READ THESE INSTRUCTIONS FIRST Write your name, class and index number in the spaces at the top of this page and on all separate writing paper used. Write in dark blue or black pen. You may use a soft pencil for any diagram or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all questions. Write your answers on the writing paper provided. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal in the case of angles in degree, 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 7 printed pages. Tanjong Katong Secondary School

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Prelim Add Mathematics P2

2 Mathematical Formulae

1. ALGEBRA Quadratic Equation For the equation ax2 + bx + c = 0, x=

 b  b 2  4ac . 2a

Binomial Theorem n (a + b)n = a n +   a n  1 b + 1

n n  2 2 n   a b + . . . +   an  r br + . . . + bn, 2 r

n n! n(n  1).......(n  r  1) 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  tan B tan (A ± B) = 1  tan A tan B sin 2A = 2 sin A cos A. 2 cos 2A = cos A  sin2 A = 2 cos2 A  1 = 1  2 sin2 A 2 tan A tan 2A = 1  tan 2 A sin A + sin B = 2 sin 12 (A + B) cos 12 (A  B) sin A  sin B = 2 cos

1 2

(A + B) sin

cos A + cos B = 2 cos

1 2

(A + B) cos

1 2

(A  B)

(A + B) sin

1 2

(A  B)

cos A  cos B = 2 sin

1 2

1 2

(A  B)

Formulae for ABC a b c   . sin A sin B sin C

a 2 = b 2 + c2  2bc cos A. =

Tanjong Katong Secondary School

1 bc sin A. 2

Prelim Add Mathematics P2

3

1.

The mass, M grammes, of an unknown substance decreases over time, t years. It is known that M = M0 e p t, where M0 and p are constants. The substance was first found at the end of 1970 and its mass, when found was 900.5 g. (i) State the value of M0. [1] (ii) At the end of 1995, the mass of the substance has decreased to 258 g. Find the value of p, giving your answer correct to 2 decimal places.

2.

(iii) Find the mass of the substance 10 years after it was found.

[1]

(iv) Find the average rate of decrease in the mass of the substance over the first 10 years.

[2]

The roots of the equation 2x2 + 6x  9 = 0 are α and β. (i)

Write down the value of α + β and αβ.

(ii) Find the quadratic equation whose roots are

3.

4.

[3]

[2] 1 1 and .  1  1

[5]

1  sin 2 x  cos 2 x ≡ 2 cos x. cos x  sin x (ii) Hence, solve the equation 3(cos x + sin x) = 4(1 + sin 2x + cos 2x) for 0  x  360.  (b) Sketch the graph of the function y = tan 2x for 0  x  , showing clearly the 2 asymptotes, if any. By adding a suitable straight line to your graph, state the number of solutions for  the equation cot 2x = . 2  4 x

(a) (i)

Prove the identity

[3] [3]

[2] [2]

(a) Solve the equation 2 lg 3 + lg 2x  lg (2x + 1) = 1.

[4]

(b) Solve the equation 3 y  5 y = 7 y + 1.

[3]

ab 1 (c) Given that lg   = lg a  lg b  , show that a = b.  2  2

[3]

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Prelim Add Mathematics P2

4

5.

A polynomial is given as f(x) = ax3  x2 + bx  8. (i) (ii)

Given that f(x) has remainders 31 and 9 when divided by (x  3) and (x  1) respectively, show that f(x) = 2x3  x2  2x  8.

[3]

Show that f(x) = 0 has only one real root.

[4]

(iii) Given that one of the turning points on y = f(x) has positive x-coordinate and the other turning point has negative x-coordinate, sketch the graph for y = f(x).

6.

[2]

The diagram shows a circle with points A, B, C and D on the circumference. CG is a tangent at C. F is a point on AC such that FD is parallel to CG and EF = FC. A

B E D F

G C

(a)

Explain why DE = DG.

[1]

(b)

Show that CG2 = BG  ED.

[2]

(c)

Show that CG2 = AE  EC + 2 ED2.

[2]

(d)

Write down a relationship connecting EC, EG and CG if AC is a diameter. If CG2 = 3ED2, show that EC = ED. Explain why E is the centre of the circle.

[4]

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Tanjong Katong Secondary School

Prelim Add Mathematics P2

5

7.

(a) Given that y =

2x2  3 dy , find an expression for . sin 3x dx

[3]

(b) The noise level from a siren, N decibels, depends on the distance from the siren. It is given that N = 10  2 ln (3x + 1), where x metres is the distance from the siren. Find (i)

dN , dx

(ii)

the rate of change in the noise level, if x is increasing at 2 m/s at the instant

[1]

when x = 7 m,

[2]

(iii) the value of x at which the noise level drops to zero. (c) Find the x-coordinate of the point on the curve y =

[2]

1 2x  1

at which the gradient of

1 the tangent is  . 8

8.

(a)

The shaded area in the diagram is bounded by the curve y = e2x, the y-axis, the x-axis and the line x = a.

[4]

y

y = e2x Find the value of a if the shaded area is 2.5 units2.

(b)

The diagram shows part of the graph of the equation x = (y  3)2(y + 2). The y-axis is a tangent to the graph at the point P.

0

x=a

x

a y

[5]

x = (y  3)2(y + 2)

P

x 2 (i)

Write down the coordinates of P.

[1]

(ii)

Find the area of the region that is bounded by the curve and the y-axis.

[4] [Turn over

Tanjong Katong Secondary School

Prelim Add Mathematics P2

6

9.

The diagram shows two circles C1 and C2, whose centres are A(4, 0) and B(0, 4) respectively. The y-axis is a tangent to circle C1 at the origin O. y C1

A(4, 0)

x

O

P B(0, 4)

C2

(i)

Write down the equation of circle C1.

[1]

(ii)

The circles touch at point P where APB is a straight line. Show that the coordinates of P is 2 2  4,  2 2 .

[5]





(iii) Find the equation of circle C2.

10.

[3]

A particle moves in a straight line, such that after t seconds, its velocity v m/s is given by 2 v = 2t  . Given that the particle is 3 m from a fixed point O at the start, find 2t  1 (i)

the time at which the particle is instantaneously at rest,

[2]

(ii)

an expression for s, the displacement of the particle from point O,

[3]

(iii) an expression for a, the acceleration of the particle and hence explain why the speed of the particle is always increasing.

[2]

(iv) the total distance travelled during the first two seconds.

[3]

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Tanjong Katong Secondary School

Prelim Add Mathematics P2

7

11.

The figure shows a triangle whose sides are of lengths 3 cm, x cm and (8  x) cm.

x cm 3 cm

(8  x) cm (i) (ii)

(iii)

Given that S = 12 perimeter of triangle  , write down the value of S. The area of the triangle, A cm2, is given by the expression A = S ( S  a )( S  b)( S  c ) , where a, b and c are the lengths of the sides of the triangle. Write down an expression for the area of the triangle in terms of x. dA Find and hence, find the value of x for which the area of the triangle is a dx maximum.

End of Paper

Tanjong Katong Secondary School

Prelim Add Mathematics P2

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