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.
  1. Read the questions carefully.
  1. Answer all questions on the question paper and hand this in at the end of the examination.
  1. Do not work on loose sheets of paper.
  1. Extra space is provided at the end of the paper, should this be necessary.
  1. Diagrams are not necessarily drawn to scale.
  1. You may use an approved non-programmable and non-graphical calculator, unless otherwise stated.
  1. Round off to one decimal place, unless specified otherwise.
  1. Ensure that your calculator is in DEGREE mode.
  1. 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

SECTION A

QUESTION 1

The given sketch represents with verticesA, and D.

The equation of AD is.

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.(2)

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QUESTION 2

In the diagram, the circle with centre C and with equation

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

B lies on the x-axis and .

Determine:

(a)the coordinates of C.(4)

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(b)the coordinates of A.(3)

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(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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QUESTION 3

(a)In the given diagram and are points in the Cartesian plane.

and ..

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

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(2)Calculate the size of , correct to one decimal place.(3)

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(b)(1)Prove the following identity:

.(4)

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(2)Hence, solve forx if .(3)

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QUESTION 4

The figure shows the curves of the functions f and g, , with equations

and .

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

(1)

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(c)Determine the value(s) of x for which and .(3)

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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 20km 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 .

The cyclist travels at a constant speed of 28km/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 Pif he chooses the road to the right leading directly to A from P.(3)

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______[7]

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 find, in terms of , the size of .

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(5)

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

Determine the size of .

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(5)

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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).

(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:

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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(c)Comment on the reliability of your prediction.

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(2)

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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.

(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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(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.

Twooptions 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 studentwho 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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SECTION B

QUESTION 9

The diagram represents circles A with equation and one possibility of circle B with centre and radiusR.

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

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(b)Determine whether A, B and the point are collinear.(3)

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(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 is a tangent to circle B at .

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

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QUESTION 10

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

(6)

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(b)Simplify to a single trigonometric ratio:

(6)

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(c)If , show that .(3)

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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 1Figure 2

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(2) Find the value of shown on the figure below if.

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

The longest side of the two triangles intersects at K.

is perpendicular to .

units and units.

(1)Prove that .

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(2)Prove that .

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(3)Show that . ______

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QUESTION 12

In the figure below, the points lie on the circumference of a circle.

is a pointoutside the circle such that .

(a)Prove that .

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(4)

(b)The diagram is reprinted for your convenience.

Point moves on the circumference of the circle so thatDEbisects.

(1)Prove that bisects.

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(3)

(2) Prove that.______

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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: .

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QUESTION 14

In the figure, B, C and D are three vertices of therectangular 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 3,75m above the floor.

Area; .

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

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

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VCAA 2017: Mathematics Paper 2Page 1 of 28