chij toa payoh 2011 am prelim p1

Class Register Number Name

CHIJ SECONDARY (TOA PAYOH)

PRELIMINARY EXAMINATION 2011 SECONDARY FOUR (EXPRESS) & FIVE (NORMAL)

ADDITIONAL MATHEMATICS 4038/01

PAPER 1 04 August 2011

2 hours

Additional Materials: Answer 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 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 number of marks for this paper is 80.

This document consists of 5 printed pages including the cover page.

[Turn over

chijtpss.4E/5N.prelim.a math1.2011

2

Mathematical Formulae

1. ALGEBRA Quadratic Equation

For the equation ax2 + bx + c = 0,

a

acbbx

2

42

Binomial expansion

,21

)( 221 nrrnnnnn bbar

nba

nba

naba

where n is a positive integer and

!

11

!)!(

!

r

rnnn

rrn

n

r

n

2. TRIGONOMETRY

Identities

sin2 A + cos

2 A = 1

sec 2 A = 1 + tan

2A

cosec 2 A = 1 + cot

2A

sin ( A B ) = sin A cos B cos A sin B

cos ( A B ) = cos A cos B sin A sin B

BA

BABA

tantan1

tantan)tan(

sin 2A = 2 sin A cos A

cos 2A = cos 2 A sin 2 A = 2 cos 2 A 1 = 1 2 sin 2 A

tan 2A = A

A2tan-1

tan2

sin A + sin B = 2 sin ) ()cos(2

1

2

1 BABA

sin A sin B = 2 cos ) ()sin(2

1

2

1 BABA

cos A + cos B = 2 cos ) ()cos(2

1

2

1 BABA

cos A cos B = 2 sin ) ()sin(2

1

2

1 BABA

Formulae for ABC

C

c

B

b

A

a

sinsinsin

a2 = b

2 +c

2 2bc cos A

Area of = ab2

1 sinC


3

Answer ALL the questions.

1 Find the value of a and of b for which 352 2 xx is a factor of .3)4(2 bxaxx [4]

2 Given that 22

)1ln( 2

x

xy for 1x , find the set of values of x for which y is an

increasing function of x. [5]

3 Given that the diagonal of a square is 244

251

, calculate 2P , where P is the perimeter

of this square. Give your answer in the form 2ba where a and b are real numbers. [5]

4 Solve the equation

(i) 2ln3)2ln()5ln( xx , [3]

(ii) 125loglog5 yy . [4]

5 A curve has the equation x

y1

5 and M is the point on the curve where 2x . Find

the angle, in degrees, that the normal to the curve at M makes with the y-axis. [6]

6 A quadratic equation is given by kxxxk 2)3( 2 , where k is a constant.

(i) Find the range of values of k such that the quadratic equation has real roots. [4]

(ii) Hence, or otherwise, explain if there exist values of k for which

kxxxk 2)3( 2 for all real values of x. [2]

7 The straight line 623 yx intersects the curve 28469 22 yxyx at the points A

and B. Find the exact value of the length of AB. [5]

8 (i) Show that 3cos3coscos2)5cos(cos2

1 2 . [3]

(ii) Hence or otherwise, for x0 radians, solve the equation 05cos3coscos . [4]


4

9

In the diagram, P )3,1( , Q )8,3( and R )1,1( are mid-points of the sides of triangle ABC.

Find

(i) the gradient of the line PR, [1]

(ii) the equation of the line BC, [2]

(iii) the equation of the line AB, [2]

(iv) the coordinates of B, [2]

(v) the area of the triangle PQR. [1]

10 (a) Given that

a

x

dx

1

4ln23

, find the value of .a [3]

(b) A curve has the equation xxy 62 .

(i) Find .dx

dy [3]

(ii) Hence, find the rate of change of x when x = 2, given that y is changing at a

constant rate of 4 units per second. [2]

A

C

B

P )3,1(

Q )8,3(

R )1,1(

x

y


5

11 The pairs of values of x and y in the table below satisfy approximately the equation by axe where a and b are constants.

x 1 2 3 5 7

y 1.10 2.28 2.85 3.83 4.41

(i) Plot y against ln x for the given data and draw a straight line graph. [3]

(ii) Use your graph to estimate the value of a and of b. [4]

(iii) Use your graph to estimate the value of x when 5xe y . [2]

12 The function f is defined, for 0x , by .22

cos3)(

xxf

(i) State the amplitude and period of f. [2]

(ii) Find the smallest value of x such that 0)( xf . [2]

(iii) Sketch on the same diagram, the graphs of 22

cos3

xy and

2sin2

xy for .7200 x [4]

(iv) Hence determine the value of a for which the equation

axx

2sin22

2cos3

has 3 solutions for .7200 x [2]

END OF PAPER


6

CHIJ Sec (TP) School Sec 4 AM Prel P1 2011 (Answers)

1 a =8, b =1 2 11 ex

3 a =56.5, b =36 4i 3

5 14.04 4ii 25, 1/5

6i 2

5

2 kk

7

3

13

6ii Not possible 8ii

3

2,

3,

6

5,

2,

6

9i 1 10a 6.5

9ii y = - x + 11 10bi

2

1

2

)6(2

524

x

xx

9iii 2y = 7x + 13 10bii

7

4

9iv B(1, 10) 11ii a = 3, b = 1.76

9v 9 11iii 1.38

12i Amplitude 3, period 720

12ii 96.4

12iv a = -5

chij toa payoh 2011 am prelim p1

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