PRELIMINARY EXAMINATION MATHEMATICS PAPER 2 SEPTEMBER 2021

PRELIMINARY EXAMINATION

MATHEMATICS PAPER 2

SEPTEMBER 2021

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MARKS : 150 DURATION : 3 HOURS

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Name of Candidate

Name of Educator

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

MARK

MAXIMUM 10 10 20 25 10 20 14 11 10 6 7 7 150

INFORMATION and INSTRUCTIONS

1. This question paper consists of 30 PAGES and an Information Sheet of 2 pages.

2. Please check that your paper is complete.

3. Read the questions carefully.

4. Answer all questions on the question 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.

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.

11. Give reasons for statements, unless specified otherwise.

SECTION A

QUESTION 1

a) For a certain data set, the following box-and-whisker diagram was drawn:

1) What percentage of data lies between 9 and 15? (1)

2) What is the semi-interquartile range of the data? (1)

b) The table below shows the number of hours that a sales consultant spent with nine clients, in one year, and the value of the respective sales per client:

1) Write down the equation of the least squares regression line for the data to two decimal places. (2)

2) The sales consultant forgot to record the sales of one his clients. If the consultant spent 80 hours with this client, predict the value of the client’s sales. (2)

Number of Hours 30 50 100 120 150 190 220 240 260

Value of sales

(in thousands of rands) 270 275 500 420 602 150 800 850 820

3) Comment on the strength of the relationship between time spent with a client and the value of their sales. Justify your answer appropriately. (2)

4) What is the expected increase in sales for each additional hour spent with a client,

to the nearest rand? (2)

10 MARKS

QUESTION 2

The Grade 12 learners were interviewed about using a certain application to send SMS messages. The average number of SMS messages, 𝑚, sent by each learner per month, was summarised in the histogram below.

a) How many learners sent messages in total? (1)

b) Write down the modal class. (1)

This page left

BLANK

c) Use the grid below to draw an ogive to represent the data. (4)

OGIVE SHOWING NUMBER OF MESSAGES SENT

d) Use the ogive to identify the median for the data. (1)

e) Estimate the percentage of learners who sent more than 11 messages using this

application. (2)

f) Describe the skewness of the data. (1)

.

10 MARKS

QUESTION 3

a) Points A(–2; 3) and B(3; –2) lie in the Cartesian Plane.

1) Calculate the length of AB.

(2)

2) Determine the co-ordinates of M, the midpoint of AB. (2)

y

M P x O

B(3; –2)

A(–2; 3)

3) Determine the co-ordinates of the point P, the point where the straight line

AB cuts the x-axis. (5)

4) Determine the size of obtuse AO ̂ P.

(3)

5) Find the equation of the line which is parallel to OB, passing through A. (4)

b) A line is drawn through the points S(a; b) and T(c ; d).

1) Write down the gradient of the line ST, in terms of a, b, c and d. (2)

2) Write down the gradient of the line perpendicular to ST in terms of

a, b, c and d. (2)

20 MARKS

QUESTION 4

a) If sin 34 ^º = t , determine the following in terms of t:

1) tan 34 ^º (2)

2) cos 68 ^º (3)

3) sin 64 ^º (4)

b) 1) Simplify to a single trigonometric ratio:

^sin(180

º − x)

cos(−x) . tan (360 ^º + x) − sin ( 180 ^º + x ) cos (90 ^º + x) (7)

2) Without using a calculator, determine the value of: (4) sin ² 35 ^º − cos ² 35 ^º

4sin10 ^º . cos10 ^º

3) Prove the following identity:

2sin ² x

2tanx − sin 2x = _{tan x} ¹ (5)

25 MARKS

QUESTION 5

In the diagram, T, Q, V and S are points on the circle. QT and VS are produced to P and QS is produced to R such that 𝑃𝑅 ∥ 𝑇𝑉.

a)

b)

Give reasons for the following statements:

1) P ̂

2 =V ̂ ……….……… (1)

2) V ̂ = Q ̂ ………. (1)

Hence, prove that ΔPQR ||| ΔSPR. (3)

2

2

2 1 1

1

S T

V

R

Q

P

c) Now, complete the following:

1) ^PQ

PR = _... ^SP (1)

2) ^QR

... = ^PR _SR (1)

d) Calculate the length of PR if QR = 10,2 units and SR = 3,4 units. (3)

10 MARKS

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TOTAL SECTION A: 75 MARKS

SECTION B

QUESTION 6

The circle centred at M (-4;-5) touches the x-axis at D and cuts the y-axis at P and N. Tangents drawn to the circle at P and N meet at Q.

a) Explain why the co-ordinates of D are (-4;0). (2)

b) Show that the equation of the circle can be written as (x + 4) ² + (y + 5) ² = 25 (3)

c) Find the co-ordinates of P. (4)

D

P

N M(-4;-5)

Q

x

y

d) Say what kind of quadrilateral MNQP is, giving reasons. (3)

e) Find the co-ordinates of Q. (8)

20 MARKS QUESTION 7

In the diagram below, the graphs of f(x)= tan x and g(x)= sin 2x have been drawn for x ∈ [-180 ⁰ ; 90 ⁰ ].

a) Calculate the general solution for f(x) = g(x). (8)

f(x) f(x)

g(x)

b) Hence, or otherwise, determine the values of x if f(x) = g(x) and x ∈ [-180 ^º ;0 ^º ). (3)

(c) Use the solution(s) obtained in (b) to determine for which value(s) of x ∈ [-180 ^º ;90 ^º ) ^g

/ (x)

f(x) ≥ 0. (3)

14 MARKS

QUESTION 8

In the diagram, an isosceles triangle MLN with LM = LN is inscribed in a circle. The length of LM is fixed at 2 units and ML̂N = 𝜃.

a) Show that the area of ∆ LMN is 2 sin 𝜃 square units. (2)

b) If the radius of the circle in (a) is allowed to vary, calculate the following:

1) the value(s) of 𝜃 when the area of the inscribed ∆ LMN is one square unit. (3)

2) the radius of the circle for which the area of the inscribed ∆ LMN is a maximum. (Leave your answer in surd form if necessary). (6)

11 MARKS

QUESTION 9

a) Prove the theorem which states that the angle between a tangent and a chord is equal to the angle in the alternate segment.

Given: AC is a tangent to circle centre O at B.

E and D are points on the circle.

Required to prove: CBD = BED ˆ ˆ

Construction:. (1)

Proof: (4)

O D E

C

A B

b) In the figure, AB and AC are tangents to the circle through B, C and F. AD is drawn parallel to FC and meets CD and BF produced in D. BC is drawn and CD is

produced to E.

1) If CB ̂F = x, find two other angles each equal to x, giving reasons. (3)

Hence, prove that ABCD is a cyclic quadrilateral, giving reasons. (2)

10 MARKS

3

2

2 2 2 1

1

1 1

1

y x

F

E C D

B

A

2)

2 x

3

QUESTION 10

Multiple Choice: Circle the letter that corresponds to the CORRECT answer.

a) In the figure, BA ̂C = 90 ⁰ and AD ⊥ BC, then

(A) BD × CD = BC ² (B) BD × CD = AD ² (C) BA × CA = BC ² (D) AB × AC = AD ²

b) If ΔABC ||| ΔEDF and  ABC is not similar to DEF , then which of the following is NOT  TRUE?

(A) AB.EF = AC.DE (B) BC.EF = AC.FD (C) BC.DE = AB.EF (D) BC.DE = AB.FD

c) If it is given that BC 1 area ΔPRQ

ABC ||| PQR with then equal to

QR 3 area ΔBCA

  =

(A) 9 (B) 3 (C) ¹ ₃ (D) ¹ ₉

6 MARKS A

B

C

D

QUESTION 11

In the diagram below, DEFG is a parallelogram and AB ∥ DG.

Prove that HC || GF. Give reasons.

7 MARKS A

B C

H

D E

G

F

QUESTION 12

In the diagram two circles overlap at B and C. Point A is chosen on one circle. Lines AB and AC are produced to meet the other circle at D and E respectively. DE is drawn.

A second point P is chosen. Lines PC and PB are produced to meet the other circle at Q and R respectively. QR is drawn.

PROVE that DE = QR

B

C

D A

E P

R

Q

7 MARKS ____________________________________________________________________________

TOTAL SECTION B: 75 MARKS

EXTRA SPACE FOR WORKING

EXTRA SPACE FOR WORKING