st. david’s marist inanda caps 2019 prelim papers/st davids/… · the smaller circle (centre n)...

ST. DAVID’S MARIST INANDA

MATHEMATICS

PAPER 2 PRELIMINARY EXAMINATION GRADE 12

20 SEPTEMBER 2019

EXAMINER: MR L VICENTE MARKS: 150 MODERATOR: MRS S RICHARD TIME: 3 hrs

NAME:_____________________________________________________________ PLEASE PUT A CROSS NEXT TO YOUR TEACHER’S NAME:

Mrs Kennedy Mrs Goemans Mrs Nagy Mr Vicente Mrs Richard Mrs Black INSTRUCTIONS:

This paper consists of 32 pages. Including blank pages, which may be used for working, on pages 13; 15; 17; 29; 31 & 32

A separate two-page information sheet with formulae is included. Please check that your paper is complete.

Please answer all questions on the Question Paper. You may use an approved non-programmable, non-graphics calculator unless otherwise

stated. Answers must be rounded off to two decimal places, unless otherwise stated. Answers only will NOT necessarily be awarded full marks. It is in your interest to show all your working details. Work neatly. Do NOT answer in pencil. Diagrams are not drawn to scale.

SECTION A

Q1 [12]

Q2 [9]

Q3 [19]

Q4 [19]

Q5 [7]

Q6 [12]

TOTAL [78]

LEARNER’S MARKS

SECTION B Q7 [11]

Q8 [9]

Q9 [16]

Q10 [14]

Q11 [8]

Q12 [9]

Q13 [5]

TOTAL [72]

LEARNER’S MARKS

Total: /150

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SECTION A QUESTION 1 [12 marks] The table below shows the speeds (in km/h) bowled by Liam Plunkett in the first 21 balls he bowled in the Cricket World Cup Final.

130 108 114 140 135 132 122

111 141 109 135 140 131 122

112 131 135 118 136 127 136

1.1. Calculate the mean bowling speed for Liam Plunkett. (1) 1.2. Write down the five-number summary for this set of data. (4) 1.3. Draw a box-and-whisker diagram for this set of data. (2)

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1.4. Comment on the skewness of data. (2) 1.5. Calculate the standard deviation of the speeds bowled. (1) 1.6. The technician measuring the bowling speeds discovers that the “speed-gun”

used to measure the ball speeds was not calibrated properly and that each of

the 21 balls was x km/h less than indicated. Write down, in terms of x (if

applicable), the:

a) actual mean (1)

b) actual standard deviation (1)

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QUESTION 2 [9 marks] A survey collects information about 12 families monthly income (measured in

thousands of rands) and how many “inferior goods” they purchase in their monthly

grocery shopping.

Monthly Income

5 52 40 18 20 60 67 191 71 80 110 90

No. of inferior goods

purchased

38 29 28 34 21 18 19 4 19 17 6 15

2.1. Calculate the correlation co-efficient between the two sets of data. (1) 2.2. Comment on the strength of relationship between the two variables. (1)

0

5

10

15

20

25

30

35

40

0 50 100 150 200 250

No

. of

Infe

rio

r G

oo

ds

Monthly Income

Scatter Plot

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2.3. Calculate an equation for the least squares regression line (line of best-fit). (2) 2.4. Describe what influence income has on how many inferior goods are

purchased. (1)

2.5. Give a reason why this trend can not continue indefinitely. (1) 2.6. a) If a family has a monthly income of R50 000, predict how many inferior

goods they will purchase in a month. (2)

b) Comment on the reliability of this prediction. (1)

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QUESTION 3 [19 marks] In the diagram below, M(3 ; 1), Q and N lie on the circumference of circle with centre P(-1 ; 4) and form MQN . NPM is a straight line.

3.1. Determine the equation of the circle. (4)

3.2. What is the size of ˆNQM (give a reason)? (1)

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3.3. Show that the co-ordinates of Q are 4;0 . (3)

3.4. Calculate the gradient of MN. (2) 3.5. Hence, calculate the size of . (5)

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3.6. Determine the equation of the tangent to the circle at M. (4)

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QUESTION 4 [19 marks]

4.1. Given that sin13 w , determine, in its simplest form, the value of each of the

following in terms of w. with the aid of a diagram and without using a

calculator.

a) tan13 (3)

b) cos 13 (3)

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4.2. Simplify the following expression to a single trigonometric function:

sin 450 tan 180 sin23 cos23

cos44 sin

x x

x

(6)

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4.3. Prove the following identity:

a) 1

tan sin coscos

(3)

b) Hence; determine the general solution to: 9

tan sin cossin

(4)

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The graph of cos 30f x x where 180 ;180x is sketched below.

5.1. Sketch the graph of sin3g x x , on the same set of axes. (2)

5.2. Using the graphs above, determine the value(s) of x for 180 ;180x for

which f x g x . (3)

5.3. Write down:

a) The period of h if 3h x g x . (1)

b) The equation of f x if f x is shifted 45 to the left. (1)

−180 −150 −120 −90 −60 −30 30 60 90 120 150 180

−3

−2

−1

1

2

3

x

y

f

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QUESTION 6 [12 marks] 6.1. Use the diagram below to prove that the angle subtended at the centre of a

circle is equal to twice the angle subtended at the circumference of the circle.

Given: D is the centre

of the circle.

A, B and C are points on the circle.

a) Construction: (1)

Required to Prove: ˆ ˆ2ADC B

b) Proof: (5)

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6.2. In the diagram below

O is the centre of the circle

A, B and C lie on the circumference of the circle.

Line DAE is a tangent to the circle at point A.

4ˆ 62A

2ˆ 25A

Calculate the size of angle 1C (6)

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SECTION B QUESTION 7 [11 marks] A, B and G are points on the base of a cut-out of a right, triangular based, pyramid.

TG is a vertical side of the shape. The angles of elevation of T from B and A are x

and y respectively. A, B and G lie in the same shaded horizontal plane.

ˆGBA w and ˆ 130BGA .

7.1. Prove that tan .sin 50

sintan

x ww

y

(6)

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7.2. Now find the volume of the cut out shape in picture labelled TBAG if

GT = 60cm ; 40x ; 60y , (5)

[Volume of Triangular Prism = 1

3Area of Base Vertical Height]

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QUESTION 8 [9 marks] 8.1. Determine, without the use of a calculator, the value of:

cos 65 cos 20 sin 245 cos 70x x x x (5)

[given: 0 25x ]

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8.2. Determine, without the use of a calculator, the following product:

tan1 tan2 tan3 ...tan89 (4)

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QUESTION 9 [16 marks] In the diagram below, two circles intersect at A and B (8;4) as shown. The centres of

the circles (N and M) lie on the line 6x . The tangents at B to the circles are

perpendicular to one another and pass through the centres of the circles.

9.1. If the equation of the smaller circle is given by 2 2 12 6 40x y x y ,

determine the co-ordinates of N. (5)

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9.2. Determine the equation of the larger circle with centre M. (5) 9.3. The smaller circle (centre N) is shifted down so that the x-axis is a tangent,

and the bigger circle (centre M) is moved so its new centre is the origin.

Determine if the two circles intersect at one point, two points or not at all. (6)

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QUESTION 10 [14 marks] In the diagram below, MATH is a cyclic quadrilateral with MH = TH.

Chords MT and AH intersect at Q. AM is extended to S such that HS TM .

Prove that: 10.1. HS is a tangent to circle MATH at H. (4)

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10.2. SAH ||| SHM (4)

10.3. 2SH SA SM (2)

10.4. 2 2. .SH QH AH SM (4)

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QUESTION 11 [8 marks] In the diagram below, two circles have a common tangent TAB. PT is a tangent to

the smaller circle. PAQ, QRT and NAR are straight lines.

Let Q x

11.1. Name, with reasons, three other angles equal to x. (3)

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11.2. Prove that APTR is a cyclic quadrilateral. (5)

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QUESTION 12 [9 marks] Two unequal circles intersect at A and B. They share a common tangent SMT, which

touches the circle at S and T. ABM is a straight line. Let ˆMSB x and ˆMTB y .

Prove that MS = MT [Hint: a construction is necessary]

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A quadrilateral ABCD with side lengths as shown in the sketch is rotated through 60

counter-clockwise around A. Determine the length of DD’ using a mathematical

argument to justify your answer (the dashed line in the sketch)?

Total: 150 Marks

st. david’s marist inanda caps 2019 prelim papers/st davids/… · the smaller circle (centre n)...

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