1

TONGA GOVERNMENT

MINISTRY OF EDUCATION AND TRAINING

TONGA FORM SIX CERTIFICATE

2015

MATHEMATICS

Time allowed: 3 Hours

YOU MUST HAND THIS BOOKLET TO THE SUPERVISOR BEFORE YOU LEAVE

THE EXAMINATION ROOM.

MARKER CODE

Student Personal Identification Number (SPIN)

QUESTION AND ANSWER BOOKLET

INSTRUCTIONS

1. This examination paper consists of TWO sections. Both Sections are COMPULSORY.

SECTION A: (20 marks) Multiple Choice Suggested time - 30 minutes

SECTION B: (100 marks) Ten questions with 10 marks each. Suggested time - approximately 15 minutes per question.

2. An Answer Sheet for Section A is on the last page of this booklet.

In SECTION B, write the answers to the questions in the spaces provided in this

booklet.

3. Write your Student Personal Identification Number (SPIN) on the top right hand corner

of this page and on top of page 31. Write the Marker Code in the box at the top left hand

corner of this page.

4. If you use extra sheets of paper be sure to show clearly the question being answered.

Write your SPIN on the top right hand corner of each sheet, and tie it securely at the

appropriate place in this booklet.

NOTE: (i) There should be a Mathematics Formulae Sheet (No. 8/3) with this booklet.

(ii) Non-programmable calculators are allowed into the examination room.

(iii) Unless stated, diagrams are not drawn to scale. Check that this booklet contains pages 2-31 in the correct order. Page 30 has been

deliberately left blank.

120

0

TOTAL MARKS

2

SECTION A MULTIPLE CHOICE (20 Marks)

ANSWER ALL THE QUESTIONS IN THIS SECTION

On the Answer Sheet located at the last page of this booklet, write the letter which

corresponds to the answer you consider correct. An example is shown below. Check

question numbers carefully. Allow about 30 minutes to answer the questions in this

section.

Each question is worth only one mark.

Example:

If you consider B is correct, write it like this:

To change your answer from B to C, cross out B and write the new answer by the box, like this:

3

Question One

The constant term when expanding 12312 xxx is

A. 4 .

B. 3 .

C. 4 .

D. 5 .

Question Two

A common denominator for the algebraic fractions, 2)2(

2

xand

4

12 x

is

A. 42 x .

B. 2)2( x .

C. )2)(4( 2 xx .

D. )2)(4( 2 xx .

Question Three

Which of the following is a quadratic equation?

A. 04)2

sin( 2

x

B. 04)sin( 2 xx

C. 042

sin

x

D. 04sin xx

4

Question Four

The gradient of the line 022 kyx is .3 What is the value of k ?

A. 2

3

B. 3

2

C. 3

2

D. 2

3

Question Five

Which of the following lines will meet the line 12 xy at only one point?

A. 122 xy

B. 32 xy

C. xy 63

D. 242 xy

Question Six

What is the equation of the circle centred at the origin and passes through

the point )8,6( ?

A. 1022 yx

B. 3622 yx

C. 6422 yx

D. 10022 yx

5

Question Seven

Evaluating

2

0

cosn

n equals

A. 1

B. 0

C. 1

D. 2

Question Eight

The tenth term of the arithmetic sequence, ,32log,16log,8log,4log is

A. 512log

B. 1024log

C. 2048log

D. 4096log

Question Nine

Two equally likely events

A. have the same probability of occurrence.

B. can occur together.

C. have no effect on the occurrence of each other.

D. have effect on the occurrence of each other.

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Question Ten

The parameters of a Normal distribution are

A. mean and standard deviation.

B. mean, standard deviation and z-scores.

C. mean and z-scores.

D. standard deviation and z-scores.

Question Eleven

A bag contains red marbles and 15 blue marbles. The probability of red

marbles is 5

2. How many marbles in the bag?

A. 25

B. 35

C. 55

D. 75

Question Twelve

All continuous even functions intersect with the line

A. y = 0

B. x = 0

C. x = y

D. y = -x

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Question Thirteen

The equation of the asymptote to the curve 25.0 xy is

A. 0y

B. 5.0y

C. 2y

D. 2y

Question Fourteen

The gradient of the normal to the curve 32 xy at the point ),( yx is

A. x2

B. x2

C. 1)2( x

D. 1)2( x

Question Fifteen

If 25 xy then the rate of change of y with respect to x is

A. 5

B. 2

C. 2

D. 5

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Question Sixteen

The parabola 7162 xaxy has a turning point at 2x . What is the value

of a?

A. 7

B. 4

C. 2

D. 4

Question Seventeen

Let )(xg be a polynomial. Which of the following functions has the same

derivative as )(xg ?

A. )(2 xgy

B. )(2 xgy

C. )(2 xgy

D. 2

)(xgy

Question Eighteen

The equation of the curve obtained from translating the graph of ,sinxy

c units horizontally is

A. cxy sin

B. xcy sin

C. xcy sin

D. )sin( cxy

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Question Nineteen

What is the value of 90cot ?

A. 0

B. 1

C.

D. Undefined

Question Twenty

The diagram shows the second quadrant of the circle 422 xy and the

line 2 xy . What is the area of the shaded region?

A. 05.1

B. 1.14

C. 57.1

D. 14.3

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SECTION B: LONG ANSWER QUESTIONS (100 MARKS)

Question One

a) Solve 02

13

3

xx

b) Factorise 485 2 xx

2 Marks

2

1

0

NR

2 Marks

2

1

0

NR

11

c) Make b as the subject of the equation bb

c aa log21

log

d) Solve 015)3(232 xx

3 Marks

3

2

1

0

NR

3 Marks

3

2

1

0

NR

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Question Two

A function )(xf is defined by )9)(3()( 2 xxxf .

i. Solve 0)( xf

ii. Find the coordinates of the turning points of the graph of )(xf and determine

their nature using calculus.

2 Marks

2

1

0

NR

4 Marks

4

3

2

1

0

NR

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iii. Sketch the graph of )(xf , showing the turning points and the points

where the curve cuts the x axis.

iv. Use your graph in part iii. or otherwise, to solve the equation 0)( xf .

3 Marks

3

2

1

0

NR

1 Mark

1

0

NR

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Question Three.

a) A bag contains green, black, and red markers. If I choose one marker at

random from the bag, the probability that it is red is 3

1. Is this statement

true or false? Explain why, in no more than one sentence.

b) A box contains five lollies all of exactly same appearance. Three of the

lollies are hard and two are soft. Solomone eats two lollies chosen

randomly from the box.

i. Draw a probability tree diagram for the two lollies that Solomone eats. Label each branch with the appropriate probability.

2 Marks

2

1

0

NR

3 Marks

3

2

1

0

NR

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ii. Calculate the probability that Solomone eats one hard lolly.

c) The birth weights of baby girls at Vaiola hospital are normally distributed

with a mean weight of 6.2 kg and standard deviation of 3.0 kg.

i. Between what two measurements will a baby girl almost very probably

weigh?

ii. What percentage of baby girls weigh above 3kg?

2 Marks

2

1

0

NR

1 Mark

1

0

NR

2 Marks

2

1

0

NR

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Question Four

a) The first three terms of an arithmetic series are 941

i. Find the value of the tenth term of the series.

ii. Calculate the sum of the first 30 terms of the series.

2 Marks

2

1

0

NR

2 Marks

2

1

0

NR

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b) A tree grows from ground level to a height of 2.1 metres in one year. In each

subsequent year, it grows 10

9as much as it did in the previous years.

i. Find the increase in height during the tenth year.

ii. Calculate the height of the tree after 20 years.

iii. What will be the maximum height of the tree?

2 Marks

2

1

0

NR

2 Marks

2

1

0

NR

2 Marks

2

1

0

NR

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Question Five

a) The diagram below shows the first quadrant of the circle 122 yx . The point

A has coordinates

0,

2

1and AB is perpendicular to the x axis.

i. Calculate the COB in radian measure.

3 Marks

3

2

1

0

NR

19

ii. Find the area of the shaded region.

b) Find all solutions of the equation, ,01cos2 x in the interval 3600 x .

c) Prove the trigonometric identity; 1)tan1)(sin1( 22 xx .

2 Marks

2

1

0

NR

2 Marks

2

1

0

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3 Marks

3

2

1

0

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Question Six

a) Sketch the graph of xy 2sin2 for 3600 x . Label your graph clearly.

b) Find the derivative of the function 2

2

1)( xxxf using the First Principle.

3 Marks

3

2

1

0

NR

3 Marks

3

2

1

0

NR

21

c) The diagram below shows a triangle with sides,7 cm, 13cm and x cm,

and an angle of 60 as marked. Use the Cosine rule to show that ,12072 xx

and hence find the value of .x

4 Marks

4

3

2

1

0

NR

22

Question Seven

a) Evaluate the following.

i.

dx

x1

12

ii.

1

2

112 dxx

b) The graph of )(xfy passes through the point )4,2( and 72)( xxf . Find

).(xf

2 Marks

2

1

0

NR

2 Marks

2

1

0

NR

1 Mark

1

0

NR

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c)

The graphs of 28 xxy and xy 2 intersect at the origin and B

i. Find the coordinates of .B

ii. Find the shaded area bounded by 28 xxy and xy 2 .

2 Marks

2

1

0

NR

3 Marks

3

2

1

0

NR

24

Question Eight

a) Consider the function xxg 3log)( .

i. Write down the domain and range of )(xg .

ii. Write down the equation of the asymptotes if there are any.

iii. Sketch the graph of xxg 3log)( . Label all the intercepts and the

asymptotes.

2 Marks

2

1

0

NR

1 Mark

1

0

NR

3 Marks

3

2

1

0

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iv. Find the inverse of the function xxg 3log)( .

b) Describe the behavior of the function 21

)( x

xf for large values of x .

2 Marks

2

1

0

NR

2 Marks

2

1

0

NR

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Question Nine

a) Find the centre and radius of the circle 06)1(422 yxyx .

b) Simplify yx

yxxy

11

.

3 Marks

3

2

1

0

NR

3 Marks

3

2

1

0

NR

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c) One angle of a right angled triangle is 30 and the mean length of its

hypotenuse and opposite side is 45 cm. Calculate the length of the adjacent

side of the triangle to the nearest centimetres.

4 Marks

4

3

2

1

0

NR

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Question Ten

a) Find the value of 2

21lim

1

x

x

x.

b) Consider the following points on a number plane; O the origin and points

)1,2(A and )1,3( B .

i. Show that OA is perpendicular to AB .

ii. Show that OAand AB have the same lengths.

2 Marks

2

1

0

NR

2 Marks

2

1

0

NR

1 Mark

1

0

NR

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iii. Find the midpoint D of the interval OB .

iv. Find the coordinates of a point C such that D is the midpoint of AC .

v. What shape best describes the geometric figureOABC? Explain your

answer.

1 Mark

1

0

NR

1 Mark

1

0

NR

3 Marks

3

2

1

0

NR

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THIS PAGE HAS BEEN DELIBERATELY LEFT BLANK.

31

MATHEMATICS