Grade 12 Preliminary ExaminationSt Catherines ConventPaper 2 - September 2015

ST CATHERINES COMVENT

GRADE 12

MATHEMATICS: PAPER 2

September 2015

TIME:3 hoursMARKS:150

EXAMINER:Mrs V GermishuysMODERATOR:Mrs A Rossouw

NAME: ______

PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY:

(1)This question paper consists of 13 questions and 25 pages including the cover sheet

and is divided intotwo sections, Section A and Section B.

(2)Read the questions carefully.

(3)You may use an approved non-programmable and non-graphical calculator, unless stated otherwise in any question.

(4)Unless stated otherwise, round all answers to two decimal places where necessary.

(5)Clearly show all details of your workingsand calculations.

(6)It is in your own interest to write legibly and to present your work neatly.

(7) This question paper is also your answer booklet. Write all your answers in this booklet in the spaces provided.

SECTION A

QUESTION 1

In the diagram, AB BC and A lies on the y-axis. M is the mid-point of AC.

1.1Find the equation of the line through B and C.(3)

1.2Show that the coordinates of A are (0; 2)(2)

1.3Calculate .(3)

1.4Find the distance MBin simplest surd form.(2)

1.51.5.1Determine the equation of the circle which circumscribes triangle ABC.(2)

1.5.2Hence write down a possible coordinate D such that ABDC is a

cyclic quadrilateral.(2)

[14]

QUESTION 2

The marks in a certain mathematics exam can be summarised in the following way:

Mark interval % / 1 – 20 / 21 – 40 / 41 – 60 / 61 – 80 / 81 – 100
Mid-point / 10,5
Frequency ( f) / 5 / 12 / 37 / 21
Cumulative marks / 89

2.1Complete the table above and construct a cumulative frequency curve

for the data on the grid provided.(6)

2.2Use the graph to estimate the median mark and the inter-quartile range.(1)

2.3What can be concluded about the skewness of the data?(1)

[8]

QUESTION 3

3.1Simplify and express in terms of a single trigonometric ratio:

(5)

3.2Solve for : where .(4)

3.3If and , determine WITHOUT the use of

a calculator:.(4)

3.4.1Prove(4)

3.4.2Hence, find the general solution of: (3)

3.5.1Show that (2)

3.5.2Hence, evaluate , without the use of a calculator, if it is further given that .(4)

[26]

[11]

QUESTION 4

Refer to the diagram.

RQ is a tangent to circle QTSUP with centre O. SOQ and PT are straight lines.

and

Find, with reasons, the size of the lower case letters marked (a) to (e).

Fill in your answers in the table below:

Angle / Answer / Reason
a
b
c
d
e

[10]

QUESTION 5

In each case below, you are given a statement and a reason that are true for the incomplete diagram. Complete the diagram, showing what was necessary so that the statement and the reason are true.

5.1Statement:

Reason: Opp L’s of cyclic quad

5.2Statement:

Reason: Line from centre to midpoint

of chord

(2)

5.3In the figure below, A, B, C and D are points on a circle with centre O.

It is further given that , reflex and .

Calculate the size ofy, stating all reasons: (5)

[7]

QUESTION 6

6.1Two circles intersect at A and B. AC is a tangent to circle ABD at A and AD is a

tangent to the circle ACB at A. Straight line CEFD intersects the circles at E and F. AE = AF.

6.1.1Prove: (3)

6.1.2Show: AC.DF = AD.AF(2)

6.2In the figure below, ABC has D and E on BC. BD = 6cm and DC = 9cm.

AT : TC = 2 : 1 and AD // TE.

6.2.1Write down the numerical value of (1)

6.2.2Show that D is the midpoint of BE.(2)

6.2.3If FD = 2cm, calculate the length of TE. (2)

[10]

END OF SECTION A

SECTION B

QUESTION 7

Two of the cogs that form part of the clock in the Big Ben in London are represented in the diagram below. We will represent the two cogs by a smaller circle with centre O, the origin, and a larger circle with centre M. The point of contact of the two circles is at point P(-3; 2). The radius of the larger circle is . /

7.1Determine the equation of the smaller circle centred at the origin.(2)

7.2Determine the equation of the line OM .(2)

7.3Determine the equation of the common tangent, t, to both circles.(4)

7.4If at point M, write b in terms of a.(1)

7.5Determine the equation of the larger circle.(8)

[17]

QUESTION 8

Three circles are sketched below, with centres A, B and C respectively.

The equation of the first, centred at A, is .

Note: The radius of the circle, centred at B, is 1 unit greater than the circle centred at A and the radius of the circle, centred at C (p; q), is 1 unit greater than the circle centred at B. Each circle centre is shifted 1 unit right and then 1 unit up to determine the next circle centre.

8.1Determine the radius and the coordinates of the centre of the circle

centred at A. (3)

8.1Determine the equation of the circle, centred at C, in the form:

(3)

[6]

QUESTION 9

Mr Dube is retired and he supplements his pension by mowing lawns for customers who live in his neighbourhood. As part of a review of his charges for this work, he measures the approximate areas (x) (in m2) of a random sample of 12 of his customers' lawns and notes the time (y) in minutes, that it takes him to mow these lawns.

His results are shown in the table.

Area (x) (m2) / 360 / 120 / 845 / 602 / 1 190 / 530 / 245 / 486 / 350 / 1 005 / 320 / 250
Time (y) (minutes) / 50 / 28 / 130 / 75 / 120 / 95 / 55 / 70 / 48 / 110 / 55 / 60

9.1Use your calculator to determine the equation of the least squares regression line. Give your answers correct to 4 decimal digits. (3)

9.2Calculate the value of r, the correlation coefficient for the data,

correct to 4 decimal places(2)

9.3Given that Mr Dube charges a flat call out fee of R150, as well as R50 per half hour (or part thereof), estimate the charge for mowing a customer's lawn that has an area of 560 m2. (For example: 100 minutes would be taken as 2 hours) (3)

9.4The local high school want Mr Ryan to mow their rugby field which is rectangular, 100 metres long by 70 metres wide. Should you use the regression equation found in (a) to calculate the time it would take to mow this area? Give a reason for your answer. (2)

[10]

QUESTION 10

10.1.1If the period of is halved, what is the new equation?(1)

10.1.2If the amplitude of is doubled, what is the new equation?(1)

10.1.3If the graph of is translated left by , what is the

new equation?(1)

10.2Given: and

10.2.1 Determine algebraically the values of x for which if

(7)

10.2.2The graph of is sketched below, for the interval.

Sketch the graph of on the same set of axes.(3)

10.2.3Use your graphs to write down the values of x for which

in the given domain.(2)

[15]

QUESTION 11

11.1PQ is a tangent to the circle at C. AEB and ADC are straight lines. PQ || AB.

Prove:

11.1.1(4)

11.1.2(3)

11.1.3(3)

11.4In the figure shown, and PQ is not parallel to ST.

, and

Calculate .(4)

[14]

QUESTION 12

, , and .

Determine the value of , given PQ // BC.(4)

[4]

QUESTION 13

In the given diagram, rectangle BDEG is the base of a pyramid with A as the vertex.

AB= AG = a metres; AD = AE = b metres; CD = 6 metres; DE = 3 metres

, , and

13.1Prove that BG = (4)

13.2Similarly if DE = , show that .(2)

13.3Determine the height h.(3)

[9]

END OF SECTION B

SPACE FOR ROUGH WORKING

ST CATHERINES CONVENT

GRADE 12

MATHEMATICS: PAPER 2

AUGUST 2015

MARK RECORD SHEET

FOR OFFICIAL USE ONLY

QUESTION / Trigonometry / Geometry / Statistics
1 / /14
2 / /8
3 / /26
4 / /10
5 / /7
6 / /10
7 / /17
8 / /6
9 / /10
10 / /15
11 / /14
12 / /4
13 / /9
TOTALS / /50 / /82 / /18
/150 / %

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