MATHEMATICS PAPER 2 ANSWER BOOK

VCAA 2016

Marks: 150 Time: 3 hours

INSTRUCTIONS

1. This question paper consists of 27 pages and an Information Sheet of 2 pages. Please check that your paper is complete.

2. Read the questions carefully.

3. Answer all questions on the question paper and hand this in at the end of the examination.

4. Do not work on loose sheets of paper.

5. Extra space is provided at the end of the paper, should this be necessary.

6. Diagrams are not necessarily drawn to scale.

7. You may use an approved non-programmable and non-graphical calculator, unless otherwise stated.

8. Round off to one decimal place, unless specified otherwise.

9. Ensure that your calculator is in DEGREE mode.

10. All necessary working details must be clearly shown.

QUESTION 1 2 3 4 5 6 7 8 9 10 11 12 13 14 TOTAL

MAXIMUM 8 13 12 6 7 10 6 14 18 15 15 10 7 9 150

MARK

VCAA 2017: Mathematics Paper 2 Page 1 of 29

y

x

B(8 ; 1)A

C(6 ; 2)

D

SECTION A

QUESTION 1

The given sketch represents ABD with vertices A, (8;1)B and D.

The equation of AD is 2 5x y .

C(6 ; 2) is the midpoint of AB. D is vertically above C.

(a) Determine the coordinates of A. (2)

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(b) Determine the coordinates of D. (3)

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(c) Write down the equation of DC. (1)

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(d) Show that 90B AD

. (2)

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[8]VCAA 2017: Mathematics Paper 2 Page 2 of 29

QUESTION 2

In the diagram, the circle with centre C and with equation 2 26 4 12x x y y

intersects the y-axis at A. BA is a tangent to the circle.

B lies on the x-axis and B AY

.

Determine:

(a) the coordinates of C. (4)

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(b) the coordinates of A. (3)

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VCAA 2017: Mathematics Paper 2 Page 3 of 29

(c) The equation of the tangent AB. (3)

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(d) The size of , rounded off to one decimal place. (3)

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[13]

VCAA 2017: Mathematics Paper 2 Page 4 of 29

QUESTION 3

(a) In the given diagram ( 3; )A p and (2; )B q are points in the Cartesian plane.

60BO X

and . 10 unitsOA .

(1) Determine the value of q, in the simplest surd form. (2)

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(2) Calculate the size of AOB

, correct to one decimal place. (3)

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VCAA 2017: Mathematics Paper 2 Page 5 of 29

(b) (1) Prove the following identity:

1 sin 1 sin 4 tan1 sin 1 sin cos

x x xx x x

. (4)

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(2) Hence, solve for x if

1 sin 1 sin 01 sin 1 sin

x xx x

. (3)

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[12]

VCAA 2017: Mathematics Paper 2 Page 6 of 29

QUESTION 4

The figure shows the curves of the functions f and g, [0 ;180 ]x , with equations

( ) cosf x a x and ( ) sing x dx .

Use the graphs to answer the following questions:

(a) Determine the values of a and d. (2)

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(b) Use the given graphs to determine the value(s) of x for which

( ) ( ) 2.g x f x (1)

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(c) Determine the value(s) of x for which '( ) 0f x and '( ) 0g x . (3)

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VCAA 2017: Mathematics Paper 2 Page 7 of 29

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[6]

VCAA 2017: Mathematics Paper 2 Page 8 of 29

B A

C

P

40 km60 km

20 km

35°

QUESTION 5

A cyclist at training is on his way from P to A. When he reaches point C, the road forks. The road to the right leads directly to A, which is 60 km from C. P and C are 20 km apart

The road to the left leads to A via B.

C and B are 40 km apart. The angle between the roads (BC and AC), is 35 .

The cyclist travels at a constant speed of 28 km/hour.

(a) Calculate the difference between the distances of the two routes from C to A , correct to the nearest kilometre. (4)

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(b) Calculate the time (in hours and minutes) it will it take the cyclist to reach A from P if he chooses the road to the right leading directly to A from P. (3)

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VCAA 2017: Mathematics Paper 2 Page 9 of 29

QUESTION 6

(a)

AB and BD are tangents to the small circle centre E.

B is the centre of the large circle with tangents AE and ED.

If A E D=x find, in terms of x, the size of C B D.

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(5)

VCAA 2017: Mathematics Paper 2 Page 10 of 29

12

2

12

1

EB

CD

A

x

(b) In the diagram below PQ and QR are tangents to the circle.

PQR=640

Determine the size of ^

RSP .

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(5)

[10]

VCAA 2017: Mathematics Paper 2 Page 11 of 29

Q

R

S

P

640

QUESTION 7

The graph below shows the Men’s 50m Freestyle World Record progression from Jonty Skinner (23,86s South Africa 1976) to Peter Williams (22,18s South Africa 1988) to the current record holder Cesar Cielo (20,91s Brazil 2009).

1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 202019.00

20.00

21.00

22.00

23.00

24.00

25.00

YEAR

TIM

E (s

econ

ds)

(a) Draw in a ‘Line of Best Fit’ and predict a value for the correlation coefficient.

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(b) The line of best fit is given by: y=−0,0673 x+156,23

Predict the new World Record at next year’s (2018) World Swimming.Championships. Give your answer correct to two decimal places.

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VCAA 2017: Mathematics Paper 2 Page 12 of 29

(c) Comment on the reliability of your prediction.

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(2)

[6]

VCAA 2017: Mathematics Paper 2 Page 13 of 29

QUESTION 8

An exam, marked out of 150 marks, is written by a large group of students. The cumulative frequency graph for the marks is given below.

30 40 50 60 70 80 90 100 110 120 130 140 1500

50100150200250300350400450500

Cumulative Raw Scores

Raw Score (marks)

Num

ber o

f Stu

dent

s

(a) How many students wrote the test?

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(b) Extract the necessary information from the graph and draw a box and whisker plot on the scale given below.

(5)

(c) Comment on the skewness of the data.

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VCAA 2017: Mathematics Paper 2 Page 14 of 29

(d) What is the probability that a student, chosen at random, will have a score of more than 60%?

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(e) During the moderation process it was decided to adjust the raw scores.

Two options were discussed.

Option one: add 7,5 marks to each student’s raw score.

Option two: add 10% of the marks they did not get. For example: A student who scored 100 marks gains 5 marks. A student who scored 80 marks gains 7.

(1) How will option one effect the raw mean and standard deviation?

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(2) How will option two affect the raw mean and standard deviation?

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[14]

VCAA 2017: Mathematics Paper 2 Page 15 of 29

O x

y

A..B( 3; 5)

SECTION B

QUESTION 9

The diagram represents circles A with equation2 2( 3) ( 3) 1x y and one possibility of

circle B with centre (3;5)B and radius R.

(a) Write down the coordinates of A. (2)

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(b) Determine whether A, B and the point ( 1;4)C are collinear. (3)

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VCAA 2017: Mathematics Paper 2 Page 16 of 29

(c) Calculate all possible lengths of R for which circles A and B will neither

intersect nor touch. Leave your answer rounded off to one decimal place. (6)

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(d) The line 8y x is a tangent to circle B at D(p; q) .

Calculate the values of p and q if the radius of circle B is 3 2 units. (7)

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[18]

VCAA 2017: Mathematics Paper 2 Page 17 of 29

QUESTION 10

(a) Determine the general solution for x, correct to one decimal place:

1 cos2sin1 sin

xxx

(6)

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(b) Simplify to a single trigonometric ratio:

cos(90 ).cos( ) cos(180 )sin( )cos(360 ).cos( 360 ) sin(540 ).cos(90 )

x y x yx y x y

(6)

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VCAA 2017: Mathematics Paper 2 Page 18 of 29

(c) If cos 2sin cos , 0 90x y y y , show that 2 90x y

. (3)

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VCAA 2017: Mathematics Paper 2 Page 19 of 29

QUESTION 11

(a) (1) In the diagram below, which of the two shapes Fig 1 or Fig 2 is similar

to ABCDE? Give a reason for your answer.

Figure 1 Figure 2

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VCAA 2017: Mathematics Paper 2 Page 20 of 29

E

B

C

D

A

8cm

7cm

9cm

6cm

8cm

Y

V

W

X

U

24cm

20cm

26cm

18cm

24cm J

G

H

I

F

12cm

10,5cm

13,5cm

9cm

12cm

(2) Find the value of a shown on the figure below if ABCDE /¿/PQRST .

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VCAA 2017: Mathematics Paper 2 Page 21 of 29

T

Q

R

S

P

acm

15,6cm

(b) In the diagram below, two right angled triangles ∆ EFG and ∆ FGH are given.

The longest side of the two triangles intersects at K.

KJ is perpendicular to GF.

FH=16 units and EG=10 units.

(1) Prove that 10KJ JF

GF

.

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(2) Prove that 16KJ GJ

GF

.

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VCAA 2017: Mathematics Paper 2 Page 22 of 29

FG

H

K

J

E

10 units

16 units

(3) Show that

8013

KJ .

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[15]

VCAA 2017: Mathematics Paper 2 Page 23 of 29

QUESTION 12

In the figure below, the points A ,B ,C ,D andE lie on the circumference of a circle.

F is a point outside the circle such that AB=AF.

(a) Prove that DE=EF.

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(4)

VCAA 2017: Mathematics Paper 2 Page 24 of 29

12 1

1

.C

F

D

B

E

A

(b) The diagram is reprinted for your convenience.

Point C moves on the circumference of the circle so that DE bisects^

CDF .

(1) Prove that BE bisects^

ABC .

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(2) Prove that ^ ^

C ABC .______________________________________________________________

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VCAA 2017: Mathematics Paper 2 Page 25 of 29

12 1

1

.C

F

D

B

E

A

[10]

QUESTION 13

In the diagram below A and B are the centres of the smaller and larger circles respectively.

FD is a tangent to the smaller circle.

Prove that: AB.CF=12CD. EF.

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VCAA 2017: Mathematics Paper 2 Page 26 of 29

12

2

1

21

F12

D

C

A

B

E

VCAA 2017: Mathematics Paper 2 Page 27 of 29

A

B C

D

.

QUESTION 14

In the figure, B, C and D are three vertices of the rectangular floor of a hall.

A light is in the middle of the ceiling at A such that B, A and D are in the same vertical plane. The light is 3,75 m above the floor.

Area ; 70ADC

.

a) Show that AD = , rounded off to the nearest integer. (5)

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VCAA 2017: Mathematics Paper 2 Page 28 of 29

(b) Calculate the length of CD, correct to one decimal place. (4)

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[9]

VCAA 2017: Mathematics Paper 2 Page 29 of 29