Pacific Senior Secondary Certificate

MATHEMATICS

2010

QUESTION and ANSWER BOOKLET

Time Allowed: 3 hours

Marker Code   Student Personal Identification Number (SPIN)

               

No. 8/1

INSTRUCTIONS

1. This Examination Paper consists of TWO sections. ANSWER ALL QUESTIONS.

SECTION A: (20 marks) contains 20 multiple choice questions worth one mark each.

SECTION B: (100 marks) contains 10 questions requiring detailed answers. Each question is worth 10 marks.

2. An answer sheet for Section A is found in the FOLD OUT FLAP on the last page.

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 at the top of the fold out flap. 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. (ii) Unless stated, diagrams are not drawn to scale. Check that this booklet contains pages 2-25 in the correct order and that none of these pages is blank. Page 26 has been left blank deliberately. YOU MUST HAND THIS BOOKLET TO THE SUPERVISOR AT THE END OF THE EXAMINATION.

120 TOTAL MARKS

2

ANSWER ALL THE QUESTIONS IN THIS SECTION.

On the Answer Sheet on the back Cover, 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:

SECTION A (20 marks)

1. The quadratic 4

12 +−= xxy has

A. two turning points, a maximum and a minimum. B. exactly one turning which is a maximum. C. exactly one turning which is a minimum. D. two unique roots.

2. The remainder when the function 543)( 34 −++= xxxxQ is divided by )2( +x is

A. 27. B. 32. C. 48. D. 53.

3. Two events A and B are complementary events. If P(A) = q, then

A. P(B) = q B. P(B) = q− C. P(B) = 1 + q D. P(B) = 1 q−

4. The value of ∑ −5

1

13x is

A. 16. B. 17. C. 40. D. 44.

B

B C

3

5. The thn term for the sequence ........,31,15,7,3 is given by

A. 12 −= nnt

B. 12 −= nnt

C. 12 1 −= +nnt

D. 12 += ntn

6. The solution set to the equation 12cos −=α   for  πα 20 ≤≤  is

A. ⎭⎬⎫

⎩⎨⎧

43,

4ππ

B. ⎭⎬⎫

⎩⎨⎧

32,

3ππ

C. ⎭⎬⎫

⎩⎨⎧

23,

2ππ

D. ⎭⎬⎫

⎩⎨⎧ ππ ,

6

7. The gradient of the tangent line to the curve )6( −−= xxy at the point (3,6) is

A. 6 B. 0 C. 3− D. 6−

Questions 8 and 9 refer to the diagram below.

An arc ACB subtends an angle 5

4π radians at the centre O of the circle of radius 10 cm.

8. The length of the arc ACB is

A. cm8 B. cmπ8 C. cmπ20 D. cmπ40

A B

O

5

4π

10 cm

C

4

9. The area of the sector BOAC is

A. 240 cm

B. 240 cmπ

C. 280 cm

D. 280 cmπ

10. Radar stations A and B, 4.7 km apart, are on an East-West line. A technician at station A detects a plane at point C on a bearing of o61 . Another technician at station B detects the same plane on a bearing of o331 . Refer to the sketch below.

What is the distance from A to C?

A. 1.80 km B. 2.28 km C. 2.61 km D. 4.11 km

11. The sketch below shows the graphs of 11 +=−= xyandxy

    b

O a

y ),( ba  

1+= xy

x

4.7 km

b

N

B

C

o61

A

5

The co-ordinates of the intersection of the two graphs, i.e. point ),( ba is A. ( )1,1 B. ( )2,2 C. ( )2,3 D. ( )3,3

12.

The diagram shows the graph of the function xxy 93 +−= . The total shaded area shown above from 30 −== xtox is

A. 4

113 sq. units

B. 10 sq. units C. 32 sq. units

D. 4

243 sq. units

13. β For the given line above, the measure of angleβ is

A. 0100 B. 0120 C. 0135 D. 0150

A(-2,-10)

-2

y

a

a

y

-10

0 -3

x

x

6

14. A die is rolled 400 times and the outcomes are recorded in the table below.

Face 1 2 3 4 5 6

Frequency 35 42 82 76 90 75

The probability of rolling the number 5 is

A. 65

B. 51

C. 409

D. 905

15. The solution set to the inequation 132

−+

xx

<  1 is

A. ⎭⎬⎫

⎩⎨⎧ −−

23,

31

B. { xx : > }Rxwhere ∈23

C. ⎩⎨⎧

31:x < x < }Rxwhere ∈

23

D. { xx : < xor31

> }Rxwhere ∈23

 

16. From the top of a cliff, the angle of depression to a river below is o32 . The river is 282 m from a point directly below the top of the cliff. The height of the cliff is approximately

A. 176 m B. 239 m C. 314 m D. 451 m

17. X is a normal random variable. If a normal distribution has a mean of 17 and a standard deviation of 5, then the area under the standard normal curve between the z-value corresponding to 37=X and the mean is

A. 0.5772 B. 0.5228 C. 0.5000 D. 0.4772

7

18. If b=θsin and a=θcos then θcot equals

A. ab

B. b

ba

C. ba

D. ab

19. The value of 2

1lim2 −→ xx

is

A. 1 B. 0 C. -1 D. does not exist

20. The value of dxxx 22

1

2 +∫ − is

A. 3

17

B. 6

17

C. 23

D. 0

8

Section B (100 marks)

ANSWER ALL TEN QUESTIONS IN THIS SECTION.

Write the answer to each question in the spaces provided.

It is in your best interest to show ALL your working, as some marks are allocated for appropriate methods and partially correct answers.

Check question numbers carefully. Each question is worth a total of 10 marks. Allow about 2½ hours to answer the questions in this section.

QUESTION 1: (a) By factorizing and realizing the difference of two squares, show that

bb11

3 − is equivalent to ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ −⎟⎠⎞

⎜⎝⎛ + 11111

bbb.

(3 marks) (b) Use any method you have studied to find the solution to the equation

65 −= xx .

(3 marks)

9

(c) From a production line, batteries were selected at random and tested to determine the life of each (that is, the number of hours each battery lasts). The numbers of hours were:

186, 189, 182, 191, 183, 188, 176, 188, 189, 186 188, 191, 187, 181, 189, 186, 188, 181, 185, 183 189, 182, 186, 177, 188, 187, 188, 188, 189, 184 186, 180, 188, 189, 185, 187, 185, 183, 183, 188 190, 185, 188, 189, 188, 185, 186, 176

(i) Letting x and f represent the number of hours and frequency respectively, construct a frequency table for the above data.

(2 marks)

(ii) Using your table, calculate the mean life of a battery.

(2 marks)

10

10

QUESTION 2: (a) Solve the following equations for k :

(i) )9(33 54 kk =−

(3 marks) (ii) )12(log2)54(log 22 +=+ kk

(3 marks)

11

(b) Ms Black inherits $25,000. She decided to invest her money in a bank. She deposits part

of the inheritance at 8% annual interest and part at 10% annual interest. Her total annual income from these investments is $2300. Let  x and y represent the two investment amounts. (i) Write the two equations in terms of x and y that represent the above information.

(2 marks)

(ii) Solve the equations obtained in (i) above to determine how much money is invested at each rate.

(2 marks)

10

12

QUESTION 3:

(a) (i) Find two possible values of x if the sequence .........,21,,

27 x is a geometric

sequence.

(3 marks)

(ii) Consider the following arithmetic sequences. termnth..........,13,9,5,1

termnth.....,13,15,17,19

Find n if the sums of the above arithmetic sequences are equal.

(4 marks)

13

(b) A slide of constant slope is to be built on a level piece of land. There are to be twenty equally spaced supports with the longest 20 meters in length and the shortest 3 meters in length. Find the total length of all the supports.

slide 20m

land

3m

(3 marks)

10

14

QUESTION 4:

(a) Consider the rational function xwherex

xxf−−

=1

34)( ≠ Rxand ∈1 .

(i) Complete the statements: _____,_____,

→−∞→→+∞→

fxAsfxAs

Hence, state the equation of the horizontal asymptote.

(1 mark)

(ii) State the equation of the vertical asymptote.

(1 mark) (iii) State the domain and range of f.

Domain: Range:

(2 marks)

(iv) Sketch the graph of f for the given domain.

(3 marks)

15

(b) Find the inverse function 1−f .

(3 marks)

10

16

QUESTION 5:

(a) Consider the function xxg21cos: → for π20 ≤≤ x

(i) Use the pair of axes given below to sketch the graph of g, clearly labeling on the

x-axis where there are x-intercepts and also any turning points.

(2 marks)

(ii) On the same pair of axes, rotate by π radians the graph of g about )0,(π .

Clearly indicate this graph by using a different colour or a bold sketch for π20 ≤≤ x and label it h. (2 marks)

(b) A student surveyor wishes to measure the distance across a river. A sketch of it is shown in the diagram below. He finds that angle C is 112˚, angle A is 31˚ and b is 115.67 m. Find the width of the river, that is, length BC. (The space for the answer is given on the next page).

g(x)`

x

-1

1

π π2  

17

(3 marks)

(c) Prove that cosecα  −αα

sincos2

= αsin

(3 marks)

10

18

A(1,3)

y

B(c,3)

x

3

M

c

QUESTION 6: (a) Write in sigma notation the series 1197531 −+−+−+−+−+− .

(2 marks)

(b) The end points of a line segment AB has co-ordinates A(1 , 3) and B(c , 3). M is the mid-point of the line segment.

(i) Write the co-ordinates of M in terms of c. (2 marks)

(ii) If the length of the line segment AB is 4 units, then use the distance formula to show that the value of c is 5.

(4 marks)

(iii) Hence, state the co-ordinates of the midpoint M. _________________ (2 marks)

10

1

19

QUESTION 7: (a) A tangent line ℓ to the curve 3)( 2 +−= xxf passes through the point )2,1(

(i) Find the gradient of the normal to the tangent line l at the point (1,2).

(3 marks) (ii) Find the equation of the normal. .

(3 marks)

3

l  

(1,2) 

f(x)

x

20

(b) The position of an object moving on the x-axis at time t is given by

1623

2

5

3

1++−= tttx

Find the location of the object when the acceleration is 1 m/s2.

(4 marks)

10

21

QUESTION 8: (a) Consider the cubic function xxy 253 −= .

(i) Sketch the graph of xxy 253 −= on the pair of axes given, indicating clearly the x and y intercepts.

(3 marks)

(ii) State the interval(s) in which the function is strictly increasing. ________________________________________________________ (3 marks)

(b) A wire 12 cm in length is cut in two. One part is bent into the shape of a circle, and the other is bent into the shape of a square. Let the radius of the circle be r and let the side of the square be x .

Show that the total area, A, of the two shapes shown above can be expressed in terms of

the radius r and is given by 2)3(2

12 rrA ππ −+= .

(4 marks)

10

y

x

               

                x

x

r

22

QUESTION 9: (a) Consider the trigonometric equation 0coscos2 2 =+ xx for π20 ≤≤ x .

Solve completely the equation, stating all solutions in terms of π radians.

(4 marks)

(b) The diagram below shows the graph of 2xy −= and 4−=y

(i) Find the co-ordinates of point B.

(2 marks)

2xy −=  

C

0

B

A

y

x

y = 4−

23

(ii) Find the area bounded by the curve 2xy −= , the line 4−=y and the line 0=x .

(4 marks)

10

24

QUESTION 10: (a) The life of a particular brand of light bulb was tested and found to be normally

distributed with mean life of 1800 hours and standard deviation of 200 hours.

(i) What is the probability that a light bulb selected at random will last over 1200 hours?

(3 marks)

(ii) Out of a batch of 6000 bulbs, how many would you expect to last between 1400

hours and 1900 hours?

(4 marks)

25

10

(iii) Between what range of hours would you expect the life of a bulb to “very probably

lie”?

(3 marks)

26

This page has been left blank deliberately.

27

Student Personal Identification Number

(SPIN)

FOR MARKER USE ONLY

QUESTION MARKS

M/C 20

Q1 10

Q2 10

Q3 10

Q4 10

Q5 10

Q6 10

Q7 10

Q8 10

Q9 10

Q10 10

TOTAL 120

MATHEMATICS

2010

For Candidate Use

Write number of extra sheets used in the box. OR Write NIL in the box if no extra sheets used.

Write clearly the letter of the correct answer in the box provided. Make sure your answer is put alongside the right question number.  EXAMPLE:

If you consider B is the correct answer, 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:

B

B C

SECTION A MULTIPLE CHOICE (20 marks)

1. 11.

2. 12.

3. 13.

4. 14.

5. 15.

Check Question Number Check Question Number

6. 16.

7. 17.

8. 18.

9. 19.

10 20.

SA

20