Fairfield Am 1 Prelim 2009

NAME: ____________________________ (

)

CLASS: _______

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

ADDITIONAL MATHEMATICS

4038/01

Paper 1 Date: 26 August 2009 2 hours Additional Material:

Answer 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 80. For Examiner’s Use Paper 1

/ 80

Paper 2

/ 100

Total

%

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

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 Exp / 5 N(A) Preliminary Examination 2009 Additional Mathematics Paper 1

2

Name: _______________________________ (

1

)

Class: ______

Solve the simultaneous equations 2y + x = 5 y=|2–x|.

3x + 4 sin 2x = 0 for 0   x  180 2

[4]

2

Solve 4 sin x – 3 sin

[4]

3

 3 2 Given that A =   , find A 1 and hence solve the simultaneous equations by the matrix 4 7 method,

2x  3  3y  0 7x – 5 + 4y = 0. Explain why the method fails when applied to the equations 2 x  3 y  3 and 6 x 

5  9y . 2

[5]

4

A curve has a gradient function px2 + 4x where p is a constant. The tangent to the curve at the point (1, 7) is parallel to the straight line y + 2x – 6 = 0. Find

5

(i)

the value of p,

[2]

(ii)

the equation of the curve.

[3]

(i)

Express

(ii)

It is given that

9  2x  x2 in partial fractions. ( x  1)( x 2  9)

[3]

d2y 9  2x  x2 = . At x = 0, the gradient of the curve is parallel to dx 2 ( x  1)( x 2  9)

the line y = 5. Hence, or otherwise, find the gradient of the curve at the point where x = 3. FMS(S) Sec 4 Exp / 5 N(A) Preliminary Examination 2009 Additional Mathematics Paper 1

[2] 3

6

The equation of a curve is y =

x=

sin 2 x . Find the equation of normal at the point 2  cos 2 x

 , giving your answer in exact form. 3

[5]

12

7

8

The coefficient of x

15

1   in the binomial expansion of  ax  2  x  

is  1760 .

(i)

Find the value of a.

[4]

(ii)

Find the term independent of x.

[2]

The diagram shows a right angled triangle PQR, in which  TPQ = 60 , PT = x cm and QT : TR = 3 : 1. P 60 o x cm

R

T

Q

(i)

Find the length of QT, in terms of x, expressing your answer in surd form.

[2]

(ii)

4 3   60  Show that  RPT = tan  1   3  

[4]

FMS(S) Sec 4 Exp / 5 N(A) Preliminary Examination 2009 Additional Mathematics Paper 1

4

Name: _______________________________ (

)

Class: ______

9

A

B

D

C

The diagram shows a rectangle ABCD inscribed in a circle. Given AB = 4x cm and BC = 8 cm. (i)

Show that the area, P cm2, of the shaded region is given by P = 4( x 2  4  8 x ).

[3]

Given that x can vary,

10

11

(ii)

find the stationary value of P,

[3]

(iii)

determine whether this stationary value is maximum or minimum.

[1]

(i)

Prove that ( 2 cos   sin  ) 2 =

(ii)

Hence, 1

 4 (sin x  2 cos x)

5 3  2 sin 2  cos 2 . 2 2

2

(a)

find

(b)

solve 4 sin 2y – 3 cos 2y = 5 for 0   y  180  .

[3]

dx ,

[2] [3]

If y = e 2 x (1  x) , (i)

(ii)

show that y is an increasing function for all values of x > 

hence, evaluate



2 1

6e 2 x  4 xe 2 x dx.

FMS(S) Sec 4 Exp / 5 N(A) Preliminary Examination 2009 Additional Mathematics Paper 1

3 , 2

[3]

[3]

5

12

A particle, P, moves in a straight line and passes through a fixed point A with a velocity of 0.5 m/s. The acceleration, a m/s2, of the particle P, t seconds after passing through a point A is given by a = 1.4 – 0.6t.

13

(i)

Show that the particle comes instantaneously at rest at t = 5.

[3]

(ii)

Find the total distance travelled by the particle between t = 0 and t = 10.

[3]

(iii)

Sketch the velocity – time graph.

[2]

(a)

RSTU is a rhombus where the coordinates R and T are (– 6 ,– 3) and (2, 5) respectively.

(b)

(i)

Find the equation of the diagonal SU.

[2]

(ii)

If the gradient of RS is 7, find the coordinates of S and U.

[2]

(iii)

Hence calculate the area of the rhombus.

[2]

The diagram shows part of a straight line drawn to represent the equation y=

3x 2 . Find 2  x3

(i)

the gradient of the straight line,

(ii)

the value of  ,

(iii)

hence, find the values of h and k.

[5]

1 xy

(k, 1) 0



1 x3

(– 4, h)

THE END FMS(S) Sec 4 Exp / 5 N(A) Preliminary Examination 2009 Additional Mathematics Paper 1

6

Name: _______________________________ (

)

Class: ______

Paper 1 1. 2. 3.

4. 5. 6. 7. 8. 9. 10.

11.

12.

x = 3 & y = 1 or x = – 1 and y = 3 x = 0 ○, 120 ○ or 136.0 ○ 3 11 1  7  2  , x  or y  . A 1   13 13 13   4 3  Matrix is a singular matrix which implies that the inverse of the matrix does not exist. Therefore the method failed. (i) p=–6 (ii) y = – 2x3 + 2x2 + 7 1 2x (i) (ii) 0.693  2 x 1 x  9 3 25   y  x  10 3 2 3 (i) a=2 (ii) 126 720 3 (i) x 2 64 (ii) P = 29.9 or 16  (iii) P has a minimum value  5 1 3 (ii)(a) x  cos 2 x  sin 2 x  c , where c is an arbitrary constant 8 4 16 (ii)(b) 63.4 ○ dy 3 (i)  e 2 x 3  2 x  , for any value of x, e 2 x  0. When 3  2 x  0  x   dx 2 dy 3 Since  0 for all values of x   , therefore y is an increasing function. dx 2 (ii) 298 (ii) 40 m v (iii) 0.5 5

13.

(a)

14.

(i)

t

y = – x – 1; S(–5, 4), U (1, –2), 48 units2 2 (ii) 146.3 ○ (iii) 3

FMS(S) Sec 4 Exp / 5 N(A) Preliminary Examination 2009 Additional Mathematics Paper 1

h = –3, k = 2

7