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ANDSS 4E5NPrelim 2013 Additional Mathematics (4038/02) [Turn over

ANDERSONSECONDARY SCHOOL

2013 Preliminary Examination

Secondary Four Express / Five Normal

CANDIDATE NAME

CENTRE NUMBER

S INDEX NUMBER

ADDITIONAL MATHEMATICS 4038/02 Paper 2 3 September 2013 2 hours 30 minutes Candidates answer on writing papers. Additional materials: Writing paper (12 sheets)

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.

You may use a soft pencil for any diagrams or graphs.

Do not use staples, paper clips, highlighters, glue or correction fluid.

Answer all the questions.

If working is needed for any question it must be shown with the answer.

Omission of essential working will result in loss of marks.

Calculators should be used where appropriate.

If the degree of accuracy is not specified in the question, and if the answer is not exact, give the answer to three significant figures. Give answers in degrees to one decimal place.

For , use either your calculator value or 3.142, unless the question requires the

answer in terms of .

At the end of the examination, fasten all your work securely together.

Hand in the question paper and your answer scripts separately.

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.

2

ANDSS 4E5N Prelim 2013 Additional Mathematics (4038/02) [Turn Over

Mathematical Formulae

1. ALGEBRA

Quadratic Equation

For the equation 02 cbxax ,

a

acbbx

2

42

Binomial expansion

nrrnnnnn bbar

nba

nba

naba

......

21)( 221 ,

wheren is a positive integer and !

)1(...)1(

)!(!

!

r

rnnn

rnr

n

r

n

2. TRIGONOMETRY

Identities

1cossin 22 AA

AA 22 tan1sec

AA 22 cot1cosec

BABABA sincoscossin)sin(

BABABA sinsincoscos)cos(

BA

BABA

tantan1

tantan)tan(

AAA cossin22sin

AAAAA 2222 sin211cos2sincos2cos

A

AA

2tan1

tan22tan

)(cos)(sin2sinsin21

21 BABABA

)(sin)(cos2sinsin21

21 BABABA

)(cos)(cos2coscos21

21 BABABA

)(sin)(sin2coscos21

21 BABABA

Formulae for ABC

C

c

B

b

A

a

sinsinsin

Abccba cos2222

Cabsin2

1

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ANDSS 4E5N Prelim 2013 Additional Mathematics (4038/02) [Turn Over

1 (a) (i) Find dx

x

4

1cos

1

2

. [2]

(ii) Evaluate dxx2sin22

4

, leaving your answer in terms of . [3]

(b) Show that 12

312)1(

x

xxx

dx

d.

Hence evaluate

13

5 123

2dx

x

x. [5]

2 Solve the following equations.

(a) )5(log4log 22 xx [4]

(b) 112 12)3(4 yy [3]

(c) 07152 zz ee [2]

3 Express 582 2 xx in the form cbxa 2)( , where a, b and c are integers. [2]

(i) State the nature of the turning point and its coordinates. [2]

(ii) Sketch the graph of 582 2 xxy for 50 x , indicating on your

graph the coordinates of the stationary point and of the points where the

graph meets the coordinates axes. [2]

(iii) Given another line ky is drawn on the same grid, state

(a) the value of k such that kxx 582 2 has 3 solutions, [1]

(b) the range of values of k such that kxx 582 2 has 2 solutions. [2]

4 The diagram shows the graph of xy 23 for kx 0 .

(a) Find the coordinates of A and of B. [2]

(b) Calculate the value of k. [3]

(c) Find the range of values of x such that 3123 x for kx 0 . [2]

O

y

x B

A

(k, 4)

4

ANDSS 4E5N Prelim 2013 Additional Mathematics (4038/02) [Turn Over

5 The function xy 2cos31 is defined for 20 x .

(i) State the period and amplitude of xy 2cos31 . [2]

(ii) Sketch the graph of . [2]

(iii) On the same axes in (ii), sketch the graph of 12

xy

for 20 x . [1]

6

The diagram shows a scale drawing of a Mathematics Room ABCE. CDE and AFE

are straight lines and BDEF is a rectangle. AB = 2 m, BC = 4 m and

BCDBAF , where is an acute angle measured in degrees.

(a) Show that the area of the Mathematics Room,A m2 is

42cos42sin5 A . [4]

(b) Show that A can be expressed in the form 4)2sin( R , where R is

positive and is acute. [4]

Hence, find

(i) the value of for which 7A , [2]

(ii) the maximum value of A and the corresponding value of . [3]

xy 2cos31

A

B

C

D

F E

2 m

4 m

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ANDSS 4E5N Prelim 2013 Additional Mathematics (4038/02) [Turn Over

7 A round table has radius 2 m and height 1 m.

A small lamp O is placed h m vertically above the centre of the table, where

40 h . The lamp casts a shadow of the table on the ground.

(i) Show that the area of the shadow, A m2, is given by

2

214

h

hA

. [2]

(ii) The lamp is lowered vertically at a constant rate of 8

1m/s, find the rate of

change of A when h = 3. [4]

8 The equation of a curve is xxy 2ln 2 , where 0x .

(i) Show that y increases as x increases. [3]

(ii) Find the exact value of the coordinates of the point on the curve at which the

normal to the curve is parallel to the line 056 xy . [3]

9 The diagram shows part of the graph of qxpxy 2 , where p and q are constants.

(i) Find the value of p and of q. [2]

The line AB produced intersects the curve at A(2, 0), B and C.

Find

(ii) the equation of AB, [3]

(iii) the coordinates of C, [3]

(iv) the area of the shaded region. [3]

1 m

h m

y

x O

2

A

B

C

3

6

ANDSS 4E5N Prelim 2013 Additional Mathematics (4038/02) [End Of Paper

10 Solutions to this question by accurate drawing will not be accepted.

y

O

The diagram shows a quadrilateral ABCD where A is (2, 0), B is (5, 1) and C is (7, 5).

ADCBAD = 90. Find

(i) the equations of AD and of CD, [4]

(ii) the coordinates of D. [2]

A point E lies on the line CD such that ABED is a square.

(iii) Find the coordinates of E. [2]

A point P on the line BC is such that the area of BDP is 3

1of the area of BDC.

(vi) Find the coordinates of P. [2]

A point X on the x-axis is such that BXABAX . (v) Find the equation of BX. [2]

11 A particle moves in a straight line so that t seconds after leaving a fixed point O,

its velocity v m/s, is given by 54)12(6 2 tv . Find

(i) the initial acceleration, [2]

(ii) the minimum velocity, [4]

(iii) the value of t when the particle is at instantaneous rest, [2]

(iv) the total distance travelled by the particle during the first 5 seconds. [4]

A (2, 0)

B (5, 1)

C (7, 5)

D

x

7

ANDSS 4E5N Prelim 2013 Additional Mathematics (4038/02) Solutions [End of paper]

ANDERSONSECONDARY SCHOOL

2013 Preliminary Examination Secondary Four Express / Five Normal

ADDITIONAL MATHEMATICS 4038/02 Paper TWO

Answer Scheme

Qn Answer Qn Answer

1(a)(i) cx

4

1tan4

7(ii)

27

4cm

2/s

1(a)(ii)

8

8(ii)

4

5ln

2

1,

2

1

1(b)

9

511

9(i) 2p 3q

2(a) x = 20 9(ii) 126 xy

2(b) 3.17y 9(iii) C is (1, 18) 2(c)

2

3lnz

9(iv)

3

251 sq units

3 322 2 x 10(i) AD: 63 xy , CD: 3

8

3

1 xy

3(i) The min point is (2, 3). 10(ii) D is (1, 3) 3(iii)(a) 3k 10(iii) E is (4, 4)

3(iii)(b) 0k , 53 k 10(iv) P is

3

12,

3

25

4(a) A is (0, 3) and B is

0,

2

11

10(v)

3

8

3

1 xy

4(b)

2

13k

11(i) 24m/s2

4(c)

2

12

2

1 x

11(ii) 54m/s

5(i) Period = , Amplitude = 3 11(iii) t = 2 6(b) 6.382sin41 A 11(iv) 620 m 6(b)(i) 3.33

6(b)(ii) 3.64

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ANDSS 4E5N Prelim 2013 Additional Mathematics (4038/02) Solutions [End of paper]

3(ii)

5(ii)

(0, 5) (2, 3)

(5, 15)