Chij Prelim Am 2 2009

Class

Register Number

Name

CHIJ SECONDARY (TOA PAYOH) PRELIMINARY EXAMINATION 2009

SECONDARY FOUR (SPECIAL /EXPRESS) & FIVE(NORMAL)

ADDITIONAL MATHEMATICS

4038/02

PAPER 2

23 September 2009 2 hours 30 minutes

Additional Materials: Writing Paper

READ THESE INSTRUCTIONS FIRST Write your name, register number and class 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 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 document consists of 6 printed pages including the cover page. [Turn over

chijsectp.4S/E/5(N).prelim.amath2.2009

2

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

ALGEBRA

 b  b 2  4 ac . 2a

Binomial expansion  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!  r  r!(n  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.

cos 2A = cos2 A – sin2 A = 2 cos2 A – 1 = 1 – 2 sin2 A 2 tan A . tan 2 A  1  tan 2 A 1 1 ( 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

sin A + sin B = 2 sin

Formulae for ΔABC a b c .   sin A sin B sin C a 2  b 2  c 2  2bc cos A . 1 Δ = bc sin A . 2

chijsectp.4S/E&5N.prelim.amath2.2009

3

Answer all the questions showing all your working clearly. 1

A liquid cools from its original temperature of T0 C to a temperature ToC in x minutes. Given that T  37  58e

2

4

5

x 12

, find

(i)

the value of T0 ,

[1]

(ii)

the value of T when x = 9,

[1]

(iii) the value of x when T = 50,

[3]

(iv) the rate at which T is decreasing when x = 24,

[2]

(v)

the value which T approaches when x becomes very large.

[1]

(i)

Given that  and  are the roots of the equation 3 x 2  4 x  5  0 , form an   equation whose roots are and . 5 5

[4]

The roots of the equation 2 x 2  5 x  9  0 are  and  and those of 2 x 2  7 x  5k  0 are (  3) and (   3) . Find the value of k.

[4]

(i)

Find the values of x between 0 and 4 for which 2 cot(3 x  2)  1.3 .

[3]

(ii)

Solve the equation 12 sin x  12  5 cos ec x for 0  x  360.

[4]

(ii)

3



(iii) Prove the identity

sin 2 x cos 2 x 2 sin 3 x .   sin x cos x sin 2 x

[3]

(i)

Solve the equation

log 5 (52 x 1  20 5 )  2 x .

[3]

(ii)

Solve the simultaneous equations. log 2 xy  5 log8 x  log 4 y

(i)

(ii)

[5]

The polynomial f ( x)  3x 4  px3  qx  43 gives a remainder of 76x +113 when divided by x 2  x  6 . Find the value of p and of q.

[4]

Solve the equation 3 x3  13 x  x 2  6 , giving your answers correct to 3 decimal places where necessary.

[5]

chijsectp.4S/E&5N.prelim.amath2.2009

4

6

The function f is defined by f ( x)  1  3 sin 2 x for 0  x  2 π . (i)

Find the amplitude and the period of f.

[2]

(ii)

State the minimum value of f (x) and the values of x when this occurs.

[2] [3]

(iii) Sketch the graph of the function y  f ( x)  2 .

2x can be solved by inserting a straight line on the π graph in (iii). State the equation of this line and insert this line on the graph. 2x Hence state the number of solutions of the equation 2  6 sin 2 x  3  in the π given range.

(iv) The equation 2  6 sin 2 x  3 

7

[4]

In the diagram, O is the centre of the small circle passing through the points A, B, C and D and A, O, C and T are on the big circle. Given that ABT and TCD are straight line, prove that (i)

ATC  2ADT  180 ,

[3]

(ii)

BC is parallel to AD,

[3]

(iii) TB  CD  TC  BA .

[3]

A

O D B

C T

chijsectp.4S/E&5N.prelim.amath2.2009

5 x

8

The diagram shows parts of the curve y  2e 2 (not drawn to scale) and the lines x  1 , x  k and x  4 .

y

y  2e

x 2

Area C

Area B 2

O

9

Area A 1

k

4

x

(i)

Find the total area of regions A, B and C giving your answer correct to 1 decimal place.

[4]

(ii)

Given that the area of region A is equal to region B, calculate the value of k giving your answer correct to 4 significant figures.

[4]

Express 12 cos x  5 sin x in the form R cos( x   ) where R is positive and  is acute. Show your working clearly. Hence find (i)

the acute angle x for which 12 cos x  5 sin x  7 ,

(ii)

the minimum value of which gives this value.

[4] [3]

1 and the angle(s) x between 0  and 360  2 (12 cos x  5 sin x)

[4]

chijsectp.4S/E&5N.prelim.amath2.2009

6

10

dy 1  and P (1, 0) is a point on the curve. The normal to dx x(5 x  1) the curve at P meets the y-axis at Q. A curve is such that

(i)

Find the coordinates of the mid-point of QP.

(ii)

Express

[3]

1 as partial fraction and hence find the equation of the curve. x(5 x  1)

[5]

A point (x, y) moves along the curve in such a way that the y-coordinate increases at a constant rate of 2 units per second.

(iii) Find the rate of change of the x-coordinate as the point passes through P.

11

[2]

y Circle C

Circle D

P

x

Y

Q

X

The diagram shows two circles C and D. Circle C is reflected in the y-axis to obtain Circle D. Given that P is (0, 2) and Y is ( 4,2) and that PX is the diameter of Circle C and PY is the diameter of Circle D, find

(i)

the equation of Circle C,

[3]

(ii)

the equation of Circle D.

[1]

The line y = 1 intersects the Circle C at points A and B.

Show that the x-coordinates of A and B are a  b 7 and a  b 7 respectively, where a and b are integers to be found.

[4]

chijsectp.4S/E&5N.prelim.amath2.2009

7 CHIJ Sec Toa Payoh Sec 4 Add Math PreliminaryP2 Exam 2009 Answers 1i)

95

4i)

1ii) 1iii)

64.4 17.9

4ii) 5i)

1iv) 1v)

-0.645 37

5ii) 6i)

2i)

75 x 2  14 x  3  0

6ii)

2ii)

12 5

6iv)

3i)

0.335, 1.38, 2.43, 3.48

8i)

3ii)

198.5, 341.5

8ii)

9)

13 cos (x + 22.62)

8 p = 5, q = 2

9i) 9ii)

34.8

2, 0.468, -2.135 3, 

10i) 10ii)

(0.5, 2)

3 7 , 4 4

10iii)

8

11i)

x 2  y 2  4x  4  0

434.2

11ii)

x 2  y 2  4x  4  0

3.017

11)

a = 2, b = 1

3 4

-2,

y  3.5 

x ,4 

1 , 157.4, 337.4 169

5 1  5x  1 x 5x  1 y  ln 4x

7i) AOC  2ADT ( at centre  2 at circumf )

ATC  AOC 180( in opp segment ) ATC  2ADT  180 ii) TBC  ABC  180( adj  on str line)

ADC  ABC  180( in opp segment ) But ATC  2ADC  180 Therefore, TCB  TDA and they form corresponding angles. So BC is parallel to AD. iii) Intercept thm

TB TC  AB CD

So TB x CD = TC x BA

chijsectp.4S/E&5N.prelim.amath2.2009