Fairfield Am 2 Prelim 2009

FAIRFIELD METHODIST SCHOOL (SECONDARY) PRELIMINARY EXAMINATION 2009 SECONDARY 4 EXPRESS / 5 NORMAL ACADEMIC

ADDITIONAL MATHEMATICS

4038/02

Paper 2

Date: 19 Aug 2009

Additional Materials:

2 hours 30 minutes

Answer Paper Graph Paper

________________________________________________________ READ THESE INSTRUCTIONS FIRST Write your name, index number and class 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. 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 100.

For Examiner’s Use Paper 2

This question paper consists of 8 printed pages (including this cover page)

/100

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

 b  b 2  4ac 2a

Binomial expansion n  n1  n  n 2 2  n a b   a b  ...   a n r b r  ...  b n , 1  2 r

a  b n  a n  

n n! n(n  1)...(n  r  1) where n is a positive integer and     r!  r  r!(n  r )! 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  B) 

tan A  tan B 1  tan A tan B

sin 2 A  2 sin A cos A 2

cos 2 A  cos 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

a 2  b 2  c 2  2bc cos A 1   ab sin C 2 FMS(S) Sec 4 Express / 5 N(A) Preliminary Examination 2009 Additional Mathematics Paper 2

3

1.

An experiment was carried out to study the growth of a certain micro organism. The population, P, present at t hours after the initial observation, is given by the formula P  310  150e 0.2t . (i)

Calculate the population at the start of the experiment.

(ii)

Calculate the time taken for the population of the micro organism to reach 2500. Give your answer to the nearest hour.

(iii)

[1]

Find the population of the micro organism at the beginning of the 5th day of the experiment, leaving your answer in standard form.

(iv)

3.

4.

[1]

Calculate the average rate of increase of the population on the first 24 hours.

2.

[1]

[2]

(a)

Prove the identity 1  tan A  1  tan A  2 sec 2 A .

[2]

(b)

Hence, solve the equation 1  tan A  1  tan A   5 for 0  A  360 .

[3]

2

2

2

2

Find the coordinates of the stationary point of the curve y = e x cos x for

0  x   . Leave your answer in exact form.

[5]

A polynomial of degree 4, f (x) ,has a factor x 2  8 x  20 , where f ( x)  ax 4  7ax 3  bx 2  100 x  100 . Find (a) (i) the value of a and of b, (ii) the other quadratic factor of f(x).

[5] [2]

(b)

Show that f (x) has only 2 real roots.

FMS(S) Sec 4 Express / 5 N(A) Preliminary Examination 2009 Additional Mathematics Paper 2

[2]

4 5.

In the diagram below, A is a point outside the circle with centre O such that AY and AX are tangents to the circle. M is the midpoint of the chord XY and the line ACD cuts the circle at B.

D

O X M

C

Y B

A

6.

Prove that (i) ∆OMX is similar to ∆XMA,

[2]

(ii)

MX 2  OM  MA ,

[1]

(iii)

AC 2  AB  AD  CD  BC .

[5]

Solve the following equations. (a) (b) (c)

2 x 1  24 2 x  64  0 1  log 2  y 2  5   2 log 2  y  5  3 8  12 log 9 z  4  log z 9 .

FMS(S) Sec 4 Express / 5 N(A) Preliminary Examination 2009 Additional Mathematics Paper 2

[3] [3] [5]

5

7.

(a)

The diagram shows the graph of y  a tan bx  1 . Find the values of a and b. [2]

(b)

The equation of a curve is y  3 sin 2 x  2 for 0  x   . (i)

Find the coordinates of the minimum point.

[1]

(ii)

Sketch the graph of y  3 sin 2 x  2 for 0  x   .

[2]

(iii)

On the same diagram sketch the graph of y  2 cos x for [2] 0 x  .

(iv)

Hence, write down the number of solutions for the equation 3 sin 2 x . [2] cos x  1  2

FMS(S) Sec 4 Express / 5 N(A) Preliminary Examination 2009 Additional Mathematics Paper 2

6 8.

The diagram below shows part of the curve y  5 x 3  20 x 2 . The line AB cuts the curve at A, the maximum point of the curve, and B at x = 1. y A

y  5 x 3  20 x 2

B 0

(a)

(b)

x

Find (i)

the coordinates of the maximum point A,

[3]

(ii)

the area of the shaded region.

[4]

A point (x, y) such that x > 2 moves along the curve y  5 x 3  20 x 2 . Find the value of y when the rate of decrease of y is 15 times the rate of increase of x.

FMS(S) Sec 4 Express / 5 N(A) Preliminary Examination 2009 Additional Mathematics Paper 2

[4]

7 9.

The diagram below shows part of a garden which is under construction. It is given that AB = 5 m, BC = 2 m and DAB  BCD   °. Barricades were placed horizontally along DC and vertically along AD to prevent people from entering the garden premises.

A



5m

B 2m

 C

D

10.

(i)

Show that AD = 5 cos  2 sin  ,

[2]

(ii)

Express AD in the form of R cos    .

[4]

(iii)

State the maximum value of AD and the corresponding of  .

[2]

(iv)

Find the value of  when AD = 4.

[3]

(a)

Show that the roots of (a2 – bc)x2 + 2(b + c)x – 4 = 0 are real if a, b and c are real.

(b)

[3]

Find the range of values of p for which y   x 2 2( p  3) x  25 lies entirely below the x-axis.

(c)

[3]

The roots of the quadratic equation 3 x 2  2 x  7  0 are  and  . Find the quadratic equation whose roots are

FMS(S) Sec 4 Express / 5 N(A) Preliminary Examination 2009 Additional Mathematics Paper 2

  and .  1  1

[7]

8 Answer the whole of question 11(b) on a piece of graph paper. 11.

Given that the equation of a circle is x 2  y 2  10 x  8 y  25  0 ,

(a)

determine whether the point  2 , 3 is inside, outside or on the circle. [3] (b)

The table below shows experimental values two variables x and y. x

2

3

4

5

6

7.5

y

3.48

4.82

6.06

7.22

9.5

9.99

It is known that x and y are related by the equation y 2  ax b , where a and b are constants. (i)

Express the equation in a form suitable for drawing a straight line and state the variables whose values should be plotted. [2]

(ii)

Using these variables, plot on a graph paper the points corresponding to the values obtained from the table. Hence, state the value of y that has been recorded incorrectly. [3]

(iii) Ignoring the incorrect point, draw a suitable straight line to estimate the values of a and of b. [4] (iv) Use your graph to estimate the value of y to replace the incorrect value. [1]

THE END

FMS(S) Sec 4 Express / 5 N(A) Preliminary Examination 2009 Additional Mathematics Paper 2

9 Answers 1. (i) 460 2

(ii) 14 h (iii) 3.27  1010 (iv) 753

(a) LHS

1  2 tan A  tan 2 A  1  2 tan A  tan 2 A  2  2 tan 2 A  2(1  tan 2 A)  2 sec 2 A

A = 50.8,129.2,230.8,309.2

(b)  4

3 4

5

   e  ,   4 2   (a) (i) a = 3, b = 79. (ii) 3 x 2  3 x  5 (b) x = -2 or 10 , 3 x 2  3 x  5  0 has no real roots

(i) Let MOˆ X  x OMˆ X  90  x   MAˆ X OMˆ X  90 OMˆ X  AMˆ X  90





By AA OMX is similar to XMA

(ii) Since triangles OMX and XMA are similar, OM MX OX   XM MA OA  OM  MA  MX 2 AC 2  AM 2  MC 2 ( Pythagoras thm)





 AX 2  MX 2  MC 2 2

2

 AX  MX  MC 2   AD  AB   MX 2  MC 2 (tan  sec thm  AX 2  AD  AB) (iii)



  AD  AB   MX 2  MC 2



  AD  AB   MX  MC MX  MC    AD  AB   YC  XC    AD  AB   CD  BC  (Intersecting chord thm)

6

(a) (b) (c)

x=4,6 y = 1.5 z = 1.44

FMS(S) Sec 4 Express / 5 N(A) Preliminary Examination 2009 Additional Mathematics Paper 2

10 7.

(a) (b)

a=3,b=2  3  (i)  ,1  4  (ii/iii)

y  3 sin 2 x  2

y  2 cos x

(iv)

8.

(a)

(i) (ii)

(b) 9.

3 solutions

11   2  2 ,47  27   3 85 5 square units 108

45

(i)

A



5m

B 2m

 D AD = AX + XD FMS(S) Sec 4 Express / 5 N(A) Preliminary Examination 2009 Additional Mathematics Paper 2

C

11

AX 5 AX  5 cos 

XD 2 XD  2 sin 

cos  

sin  

AD = 5 cos   2 sin  (shown) (ii)

29 cos  21.8

(iii)

29 , 21.8

(iv) 10.

63.8

(a)



 

b 2  4ac  2b  2c   4 a 2  bc (4) 2



2

2

 4 4a  b  2bc  c



 4 4a 2  b  c 

2

 16a 2  4b  c 

2



2

For all real values of b and c, b  c   0  4b  c   0 and 16a 2  0 , 2



11.

2



(b)

for all real values of a. Hence, 4 4 a 2  b  c   0 for all real values of a, b and c. 2 p 8

(c)

12 x 2  16 x  7  0

(a)

centre of circle = ( 5, - 4 ) and radius = 4 point is outside the circle.

(b)

(i) (ii) (iii) (iv)

2

plot lg y against lg x plot points, y = 9.5 recorded incorrectly. a  3.98 b  0.8 y  8.32

FMS(S) Sec 4 Express / 5 N(A) Preliminary Examination 2009 Additional Mathematics Paper 2