St Gabriels Prelim 2009 Am P2

Name: …………………………………………(

)

Class: Sec ………………..

St Gabriel’s Secondary School

2009 ‘O’ Preliminary Examination Subject Paper Level/Stream Duration Date Setter

: : : : : :

Additional Mathematics 4038/02 4 Express / 5 Normal 2 hours 30 minutes 16 September 2009 Mrs Olsen / Mr Lincoln Chan

Additional materials: Answer paper (8 sheets) String

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 on both sides of the paper. You may use a 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 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 100.

This question paper consists of 6 printed pages including this cover page.

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2 Mathematical Formulae 1. ALGEBRA Quadratic Equation

For the equation ax 2  bx  c  0 , x

 b  b 2  4 ac 2a

Binomial expansion

n n n (a  b ) n  a n    a n 1b    a n  2b 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 2 A  2 sin A cos A

cos 2 A  cos 2 A  sin 2 A  2 cos 2 A  1  1  2sin 2 A tan 2 A 

sin A  sin B  2 sin sin A  sin B  2 cos

2 tan A 1  tan 2 A

1 2 1 2 1

 A  B  cos  A  B  sin

1 2 1 2 1

( A  B) ( A  B)

 A  B  cos ( A  B ) 2 2 1 1 cos A  cos B  2 sin  A  B  sin ( A  B ) 2 2 cos A  cos B  2 cos

Formulae for ABC

a sin A 2



2

b sin B



c sin C

2

a  b  c  2bc cos A

=

1 2

ab sin C

2009 AM 4E5N Prelim P2

3

1

In a private school with 1200 students, one student returned from vacation with a contagious flu virus. The spread of the virus through student body is given by

N

1200 , 1199e kt  1

where N is the total number of infected students after t days and k is a positive constant. 66 students are expected to be infected after 5 days. The school will cancel classes when 35% or more of the students are infected.

2

(i)

Find the value of k, giving your answer correct to 3 decimal places.

[2]

(ii)

How many students will be infected after 10 days?

[2]

(iii) After how many days will the school cancel classes?

[2]

The curve y = 2 + sin 3x is defined for 0 ≤ x ≤

2π . 3

(i)

State the amplitude and period of y.

[2]

(ii)

Sketch the curve y = 2 + sin 3x for this interval.

[2]

(iii) Write down the coordinates of the minimum turning point of the curve in this interval. [1] 2x (iv) Sketch, on the same diagram, the graph of y =  1 and hence state the number of π 2x solutions, in the interval, of the equation  1 = sin 3x. [2] π

1

x

3

Show that there is only one stationary point of the curve y = e 2  ln x, where x > 0 and determine the nature of the stationary point. [7]

4

(i)

Express

(ii)

Hence evaluate

5x  6 in partial fractions. ( x  2)( x 2  4)



5 3

5x  6 dx ( x  2)( x 2  4)

2009 AM 4E5N Prelim P2

[4]

[4]

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4 5

6

(i)

The equation 3x2 + px + 120 = 0, where p > 0, has roots  and  where    = 3. Evaluate  and  and hence, or otherwise, find the value of p. [4]

(ii)

Using the value of p obtained in (i), find the quadratic equation in x whose roots are the squares of the roots of the equation 3x2 + px + 120 = 0, where p > 0. [4]

(a)

Find the value of x and of y of the simultaneous equations 2 x + 2y = 9, 3 x1 = 9  3 y .

(b)

[4]

Find, in the form (a + b 3 ) cm, the height of a cone with volume whose radius is (1 + 3 ) cm.

7

2π (8  3 3 ) cm3 3 [4]

B C

D O

E

A

In the diagram, O is the centre of the circle which passes through A, B and C. A second circle passes through O, A, D and C. Given that DCB is a straight line, line OD and line AC meet at point E. Prove (i)

triangle AED and triangle CEO are similar,

[3]

(ii)

OC  ED = AD  EC,

[2]

(iii) ADC = 2  OAC.

[3]

2009 AM 4E5N Prelim P2

5

8

A particle moves in a straight line and after passing through a fixed point, O, its displacement, s m is given by s  2t (t  4) 2 . Find (i)

the time when the particle is again at O,

[2]

(ii)

the times when the particle is instantaneously at rest,

[4]

(iii) the total distance travelled in the first 5 seconds.

[4]

9 y y

x=3 y

1 (2 x  1) 3

O

3 1 P ,  2 8 x

Q

In the diagram above, the line PQ is normal to the curve y 

1 at the (2 x  1) 3

 3 1 point P  ,  .  2 8

(i)

Find the equation of the line PQ, giving your answer in the form ax  by  c  0 , where a, b and c are integers. [5]

(ii)

Find the length of OQ .

[2]

(iii) Show that the shaded area bounded by the line PQ, the curve y  x = 3 and the x-axis is approximately 0.0554 units 2 .

2009 AM 4E5N Prelim P2

1 , the line (2 x  1) 3

[5]

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6 P

10

S



15 cm 8 cm

Q

R

T

The diagram shows a trapezium PQRS. The point T lies on the side QR such that PT = 15 cm, ST = 8 cm, angle PTS = 90 and angle TPQ = . (i)

Show that QR  15 sin   8 cos and express it in the form R sin(   ) .

(ii) Find the value of  for which QR has a maximum length.

[5] [2]

(iii) Find the value of  for which QR = 12 cm and the corresponding perimeter of the trapezium. [6]

11

The diagram shows a trapezium WXYZ in which WZ is parallel to XY and WX is parallel to the y-axis. The coordinates of W, Z and Y are (3, 10), (7, 2) and (13, 0) respectively. The point S lies on XY such that angle XSW = 90o. y X

S W(3, 10)

Z(7, 2) O

Y(13, 0)

x

Find (i)

the gradient of XY,

[2]

(ii)

the equation of WS,

[2]

(iii) the coordinates of X and S,

[5]

(iv) the length of WS,

[2]

(v)

[2]

the area of triangle WXZ . END OF PAPER

2009 AM 4E5N Prelim P2

7

SAINT GABRIEL’S SECONDARY SCHOOL 2009 ‘O’ LEVEL ADDITIONAL MATHEMATICS PRELIMINARY EXAMINATION Secondary Four Express Paper 2

Answer Sheet 1(i)

k = 0.849 (3dp)

(ii)

963

(iii)

2(i)

amplitude = 1 ; Period =

(iii)

   ,1 2 

3

dy 1 2 x 1  e  ; it is a minimum point. dx 2 x

4(i)

5x  6 1 1 1    2 2 ( x  2)( x  4) ( x  2) ( x  2) ( x  2)

5(i)

p = 39

(ii)

x2  89x + 1600 = 0

6(a)

y = 0, x = 3

(b)

7  2 3 cm

7

Shown

8(i)

4 seconds

(ii)

t

9(i)

64 x  24 y  93  0

(ii)

10(i)

17 sin   28.1o 

(ii)

11(i)

2

8 days

2 3

(iv)

1 solution

1

(v)

(ii)

4 s or t  4s 3

2 y  x  17

(ii)

0.819

25 m 27

(iii)

47

93 units 64

(iii)

Shown

61.9o

(iii)

16.8o , 45.7cm

(iii)

X (3, 20) , S (7,12)

(iv)

4.47 units

20 square units

2009 AM 4E5N Prelim P2

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