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TONGA GOVERNMENT

MARKER CODE

Student Personal Identification Number (SPIN)

MINISTRY OF EDUCATION AND TRAINING

TONGA FORM SIX CERTIFICATE 2014

MATHEMATICS QUESTION AND ANSWER BOOKLET

Time allowed: 3 Hours

YOU MUST HAND THIS BOOKLET TO THE SUPERVISOR BEFORE YOU LEAVE THE EXAMINATION ROOM.

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

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

SECTION B: (100 marks) Ten questions with 10 mark 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 the last page of this booklet. 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 and that pages 28-30 are blank.

120TOTAL MARKS

2 SECTION A (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:

1. The value of b in the function 6 given that (x – 3) is a factor is

A. 2 B. 3 C. 5 D. 6

2. The equation of the line through (7, 2) parallel to the line 4x – y + 2 = 0 is

A. 4x – y – 26 = 0 B. 4x + y – 26 = 0 C. 4x – y + 26 = 0 D. – 4x – y – 26 = 0

3. If the first three terms of a Geometric progression are (2, x, 18, …), the value of x would be

A. 4 B. 6 C. 8 D. 12

4. Two ev , then ents A and B are complementary events. If

A.

C.

B. 1  

D. 1

3 5. 2  2

The am t e and period of f(x) is pli ud

A.

2, 2

C.   B. 2,

2,D. , 2 2

6. If ′ , then the antiderivative is

2 5

A.

5

B.

C.

5

4 10

D. 5

7. If 3 2 then lim5 is

A. 10 3

C.

B. 10 3

10 D. 10 3

8. A function is defined by 7 128  if and only if

A. 1 7

C.

28B. 128 2

2 7 D. 7 128

9. If  where 0 2  then the value(s) of  is

A.

,

B.

C. ,

D.

4 10. Shown is the graph of function f

The inverse of the function f shown above is best represented by A. B.

C. D.

11. The correct factorization of the expression 3 3 2 2 is

A. 3 2

C. B. 3 2

3 2D. 3 2

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12. The sketch below shows the graphs of 1  and √ 1.

The co- nates of the intersection of the two graphs, i.e. point , is

ordi

A.

2,2

C. B. 2,3

3,3D. 3,2

13. If the fifteenth (15th) term of an arithmetic series is 50 and the sixtieth (60th)

term is 185, its common difference is

A. 3 B. 4 C. 5 D. 8

14. Using the Cosine Rule, the value of is

A. 45° 0

C. 5° B. ° 12

13D. 90°

6 15. The gradient of the normal line to the curve 3  at the point 2, 2 is

A. 1 B. 3 C. – 1 D. – 3

16. The graph of 4 is represented by

A B

C D

17. If a normal distribution has a mean of 25 and a standard deviation of 5, then the z – value corresponding to 30 is

A. 0 B. – 1 C. 1 D. 5

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18. simplify in its simplest fraction to:

A. B.

C. D.

19. The sol set to the equation sinution √   for 0 2 is

A. 0,

B.

0,

C. ,

D. ,

20. The integral of is

A.

B.

C.

D.

8 SECTION B: LONG ANSWERS (100 MARKS)

QUESTION ONE

a) Find all the values of x for which 4 (3 Marks)

b) Write as a log of a single number: 2  9 3 (2 Marks)

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c) Simplify fully: (2 Marks)

d) Find the sum of the geometric series: 2 4 8 . . . . 2048 (3 Marks)

10 QUESTION TWO

a) A box contains 5 red marbles and 6 black marbles. All marbles are of equal size and weight. An experiment consists of randomly drawing a marble and recording its color without replacement. The experiment is repeated twice.

Draw a tree diagram for the experiment. (4 Marks)

a) What is the probability of obtaining a black marble and a red marble in that

order? (1 Mark)

11 b) A group is to visit ‘Eua National Reserved Rainforest by bus. The bus will hold

50 people. Adults are charged $8 each for the trip and children $2.50 each. The bus is full and $196.50 is collected in total from the passengers.

Find the number of children on the bus.

You must show any equations that you use in solving the problem. (5 Marks)

12 QUESTION THREE

a) Draw the following graphs, clearly showing any key features

i. 3 1 2 (3 Marks)

ii. 0.4 (3 Marks)

13 b) i metic sequences. Consider the following ar th

1, 5, 9, 13,………   

2, 5, 8, 11,………     

Find n if the sums of the above arithmetic sequences are equal. (4 Marks)

14 QUESTION FOUR

a) Considering triangle PQR.

i. Use geometry to work out the size of angle Q. (1 Mark)

ii. Use the sine rule to calculate the length of side PR. (2 Marks)

b) A plane applies full power to its engines and accelerates for

1500 m along a runway, taking off at an angle of 20° and climbing in a straight line for 2500 m.

Calculate the plane’s distance in a straight line from the point where it applied full power. (2 marks)

15 c) The line 5 3 7 is perpendicular to the line 4 9, where k is a

constant.

Find the value of k. (3 Marks)

d) Sketch the graph of   6 8 on the grid below. (2 Marks)

16 QUESTION FIVE

a) Differentiate the following functions

i. 5 3 2 (1 Mark)

ii. (2 Marks)

b) The gradient at any point on a curve is given by 4 3

Find the equation of the curve if it passes through the point (1, 4). You must show any integral or derived function used in solving this problem.

(3 Marks)

17 c) Shapes A and B below are made from rectangles. For what value of x are the

areas of shape A and shape B the same?

(4 Mark)

18 QUESTION SIX

a) Find 4   (2 Marks)

b) Evaluate 2 1   (3 Marks)

19 c) Sela has to write a 5 000-word assignment. On the first day she manages to

write 100 words. Each day she writes three times as many words as she wrote the previous day.

How many words will Sela have written in total at the end of the fourth day? (2 Marks)

d) ‘Ema types her assignment on a computer. She types 700 words on the day

she begins the assignment. Each day she types 10% fewer words than the day before.

How long will she take to type the 5 000-word assignment? (3 Marks)

20 QUESTION SEVEN

Light bulbs of a particular brand were found to have a normally distributed length of life with a mean of 2000 hours and a standard deviation of 300 hours.

a) A light bulb is selected at random. What is the probability that it will last between 2200 hours and 2500 hours? (4 Marks)

b) Find the range of hours within which the life of a bulb will almost certainly

lie. (3 Marks)

21 c) From a point 50m from the base of a church building, the angle of elevation to

the bottom of the cross is 35°, while the angle of elevation to the top is 53° as shown in the diagram. Find the height, H of the cross alone, to the nearest meter. (3 Marks)

22 QUESTION EIGHT

a) A sector has an arc length of 6 cm and an area of 9 .

Calculate the center angle. (2 Marks)

b) Calculate the shaded area. The hexagon is regular, and the circle has a radius

of 8 cm.Each edges of the hexagon touches the circumference of the circle (3 Marks)

23 c) Find the coordinates of the turning points of the graph of

2 12 and determine their nature. You must show any integral or derived function used in solving this problem. (5 Marks)

24 QUESTION NINE

a) Find the coordinates of the points of intersection of the graphs of 5 14 and 4 . (4 Marks)

b) Equation of a relation is given by 2, where ,   .

i) Give the equations for the vertical and horizontal asymptotes.(2 Marks)

Vertical Asymptote: ______________________________

Horizontal Asymptote: ____________________________

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ii. Sketch the graph of 2, clearly showing the asymptotes and

the intercepts. (4 Marks)

26 QUESTION TEN

a) A circle has the equation 2 4 4 0

Write this equation in the form . (3 Marks)

b) Give the coordinates of the centre and the radius of the circle 2 3 9. (2 Marks)

Coordinates of the centre: _____________________

Radius: ____________________________________

27 c) Use the grid below to sketch the graph of the circle 2 3 9 (2 Marks)

a) Prove this trigonometric identity, (3 Marks)

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MATHEMATICS