TIME:

3 hours DATE: 06 August 2018

Total marks:

150

Setter:

CF

GRADE 12 PRELIM EXAMINATION

MATHEMATICS: PAPER II

Moderator:

DAS

PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY

1. This question paper consists of 20 pages and an Information Sheet of 2 pages (i – ii).

Please ensure that your question paper is complete.

2. Read all the questions carefully.

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

examination. Remember to write your name on the paper.

4. Diagrams are not necessarily drawn to scale.

5. You may use an approved non-programmable and non-graphical calculator, unless

otherwise stated.

6. All necessary working details must be clearly shown

7. Round off your answers to one decimal digit where necessary, unless otherwise stated.

8. Ensure that your calculator is in DEGREE mode.

9. It is in your own interest to write legibly and to present your work neatly.

Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Total

17 8 17 24 8 12 18 10 14 10 6 6 150

Department of Mathematics

Name of student:

2

SECTION A

QUESTION 1

In the diagram below, ABC is an isosceles triangle with )1;2(A and )9;4(B .

BCAB and BC is parallel to the y-axis. Angle is indicated.

(a) Show that the length of AB is 10 units.

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(b) Hence determine the coordinate of C .

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(c) Calculate the coordinates of K , the midpoint of .AC

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(d) Determine the equation of AC in the form .cmxy

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(e) Calculate the size of .

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(f) Calculate the area of ABC .

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(g) Write down the coordinates of D if ABCD is a parallelogram.

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4

QUESTION 2

The table below shows the results from a survey on expenditure of monthly data usage for 100

learners from a school in Grahamstown.

Expenditure (in rand) Frequency Cumulative Frequency

0 ≤ 𝑥 < 50 10 10

50 ≤ 𝑥 < 100 14

100 ≤ x < 150 52

150 ≤ x < 200 14

200 ≤ x < 250 a

250 ≤ x < 300 4

(a) Determine the value of a. _______________________________________________ (1)

(b) Complete the cumulative frequency table. (2)

(c) Draw an ogive for the data on the set of axes below. Clearly label your axes.

(4)

(d) What is the modal class for the data?

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QUESTION 3

(a) Use the diagram to prove the theorem which states that the opposite angles of a cyclic

quadrilateral are supplementary.

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(b) In the diagram, the vertices of ABD lie on the circle with centre O . Diameter AC and

chord BD intersect at T . Point W lies on AB . OT BD . 1A = 30°.

Determine, giving reasons, the size of:

(1) C

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(2) 2A

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(3) 2B

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(4) If it is further given that WABW , prove that TBWO is a cyclic quadrilateral.

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QUESTION 4

This question must be answered without the use of a calculator.

(a) If 8tan and 360;180 determine, by using a sketch, the value of:

(1) cossin8 sketch

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(2) 2sin

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(b) Simplify and hence determine the numerical value of:

100cos).180sin(

170sin.210tan).90cos(

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(c) Prove that: xx

xxx

cos2sin

sin2cos1tan

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(d) Determine the general solution for: 0cos2sin 2 xx .

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QUESTION 5

In the diagram below, the graph of )30sin()( xxf is drawn for the interval

150;30x .

(a) On the same set of axes sketch the graph of xxg 3cos)( for 150;30x . (3)

(b) Write down the period of g .

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(c) Estimate the x -coordinate of the points of intersection of f and g and hence write down

the values of x for which )()( xgxf in the interval ]150;30[ x .

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(d) Given ,2)()( xgxh write down the range of h .

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SECTION B

QUESTION 6

A Science teacher wants to create a model by which he can predict a learner’s test result based

on a previous test written on the same content. Test marks are given below as percentages.

First test (x) 55 45 57 80 96 50 76 70 17 82 66 33

Second test (y) 57 50 64 80 92 50 80 81 23 80 75 42

(a) Determine the equation of the line of best fit in the form BxAy , giving A

and B correct to 3 decimal places.

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(b) Determine the correlation coefficient of the data correct to 3 decimal places.

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(c) Describe the correlation between the two tests.

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(d) Use your equation in (a) to predict the test mark for a learner who attained 46% in the

first test.

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(e) Determine x and y correct to 3 decimal places and hence show that the point );( yx

lies on the line of best fit.

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QUESTION 7

In the diagram, diameter AB of circle ACB with centre D is given. The coordinates of A

and B are )8;1( and (5;0) respectively. C is a point on the x-axis.

(a) Determine

(1) the equation of the circle ACB .

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(2) the coordinates of C .

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(3) the equation of the tangent to the circle at C .

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(b) Another circle with equation 02984 22 yyxx is given.

(1) Determine, showing all working, whether the two circles are concentric (HAVE THE

SAME CENTRE).

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(2) Determine, with reasons, whether this circle lies inside or outside circle .ACB

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QUESTION 8

In the figure below, ABC has D and E on BC . BD 10 cm and DC 15 cm.

TCAT : 1:2 and AD // TE .

(a) Write down the numerical value of ED

CE

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(b) Show that D is the midpoint of .BE

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(c) If 5,2FD cm, calculate TE giving a reason for your answer.

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(d) Calculate the value of: Area Δ𝐴𝐷𝐶

Area Δ𝐴𝐵𝐷

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(e) Caluculate the value of: Area Δ𝑇𝐸𝐶

Area Δ𝐴𝐵𝐶

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QUESTION 9

In the diagram below, ABCD is a cyclic quadrilateral with CDAD . Chords AC

and BD intersect at .H BA is extended to E such that .// ACED

Prove that:

(a) DE is a tangent to circle ABCD at D .

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(b) EBD ⦀ EDA

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(c) EAEBED .2

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(d) 22 .. EABDHDED

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QUESTION 10

In the diagram, P , T and R are

three points in the same horizontal

plane. SR is a vertical tower of

height h metres. The angle of

elevation of S from T is . In

addition, ,ˆ TRP 30ˆPTR

and PT = 6 m.

(a) Express h in terms of TR and .

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(b) Express RPT ˆ in terms of .

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(c) Show that )tan31(3 h .

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[10]

QUESTION 11

Most modern soccer balls are stiched together from 32 panels of waterproofed leather, using

12 regular pentagons and 20 regular hexagons. The surface area of each hexagonal panel is

52,68 cm2. The distance from the vertex of each pentagon to the centre of the pentagon is

3,8 cm.

Calculate the total amount of leather that is used to make a soccer ball.

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QUESTION 12

A cylinder with radius 4 units fits snugly into a right-angled triangular box, the cylinder just

touching all three sides of the triangular box.

If xKP , 90ˆLJK and the hypotenuse of the triangle is 24 units, determine, with

reasons, the value(s) of 𝑥.

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Total: 150