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ADvTECH © Preliminary Examination 2021: Mathematics Paper 2 Page 1 of 30
PRELIMINARY EXAMINATION
MATHEMATICS PAPER 2
SEPTEMBER 2021 ________________________________________________________________________
MARKS : 150 DURATION : 3 HOURS
________________________________________________________________________
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
ADvTECH © Preliminary Examination 2021: Mathematics Paper 2 Page 2 of 30
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.
ADvTECH © Preliminary Examination 2021: Mathematics Paper 2 Page 3 of 30
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)
ADvTECH © Preliminary Examination 2021: Mathematics Paper 2 Page 4 of 30
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
ADvTECH © Preliminary Examination 2021: Mathematics Paper 2 Page 5 of 30
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
ADvTECH © Preliminary Examination 2021: Mathematics Paper 2 Page 6 of 30
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)
ADvTECH © Preliminary Examination 2021: Mathematics Paper 2 Page 7 of 30
This page left BLANK
ADvTECH © Preliminary Examination 2021: Mathematics Paper 2 Page 8 of 30
c) Use the grid below to draw an ogive to represent the data. (4)
OGIVE SHOWING NUMBER OF MESSAGES SENT
ADvTECH © Preliminary Examination 2021: Mathematics Paper 2 Page 9 of 30
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
ADvTECH © Preliminary Examination 2021: Mathematics Paper 2 Page 10 of 30
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
x P O
B(3; –2)
A(–2; 3)
ADvTECH © Preliminary Examination 2021: Mathematics Paper 2 Page 11 of 30
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 AOP.
(3)
5) Find the equation of the line which is parallel to OB, passing through A. (4)
ADvTECH © Preliminary Examination 2021: Mathematics Paper 2 Page 12 of 30
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
ADvTECH © Preliminary Examination 2021: Mathematics Paper 2 Page 13 of 30
QUESTION 4
a) If sin 34º= t , determine the following in terms of t:
1) tan 34º (2)
2) cos 68º (3)
3) sin 64º (4)
b) 1) Simplify to a single trigonometric ratio:
sin(180
º− x)
cos(−x) . tan (360º+ x)
− sin(180º+ x) cos (90º + x) (7)
ADvTECH © Preliminary Examination 2021: Mathematics Paper 2 Page 14 of 30
2) Without using a calculator, determine the value of: (4)
sin235
º − cos235
º
4sin10º . cos10
º
3) Prove the following identity:
2sin2x
2tanx − sin 2x=
1
tan x (5)
25 MARKS
ADvTECH © Preliminary Examination 2021: Mathematics Paper 2 Page 15 of 30
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) P2=V ………………………………………………………….…………………… (1)
2) V= Q ………………………………………………………………………………. (1) Hence, prove that ΔPQR ||| ΔSPR. (3)
2
2
2
1
1
1
S
T
V
R
Q
P
ADvTECH © Preliminary Examination 2021: Mathematics Paper 2 Page 16 of 30
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
______________________________________________________________________
TOTAL SECTION A: 75 MARKS
ADvTECH © Preliminary Examination 2021: Mathematics Paper 2 Page 17 of 30
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
ADvTECH © Preliminary Examination 2021: Mathematics Paper 2 Page 18 of 30
d) Say what kind of quadrilateral MNQP is, giving reasons. (3) e) Find the co-ordinates of Q. (8)
ADvTECH © Preliminary Examination 2021: Mathematics Paper 2 Page 19 of 30
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 ∈ [-1800; 90
0].
a) Calculate the general solution for f(x) = g(x). (8)
f(x) f(x)
g(x)
ADvTECH © Preliminary Examination 2021: Mathematics Paper 2 Page 20 of 30
b) Hence, or otherwise, determine the values of x if f(x) = g(x) and x ∈ [-180º;0
º). (3)
(c) Use the solution(s) obtained in (b) to determine for which value(s) of x ∈ [-180º;90
º)
g/(x)
f(x) ≥ 0. (3)
14 MARKS
ADvTECH © Preliminary Examination 2021: Mathematics Paper 2 Page 21 of 30
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 MLN = 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)
ADvTECH © Preliminary Examination 2021: Mathematics Paper 2 Page 22 of 30
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
ADvTECH © Preliminary Examination 2021: Mathematics Paper 2 Page 23 of 30
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: ˆ ˆCBD =BED
Construction:. (1) Proof: (4)
DO
E
CBA
ADvTECH © Preliminary Examination 2021: Mathematics Paper 2 Page 24 of 30
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 CBF = 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
22
21
1
1
1
1
y
x
F
EDC
B
A
2)
2
x
3
ADvTECH © Preliminary Examination 2021: Mathematics Paper 2 Page 25 of 30
QUESTION 10
Multiple Choice: Circle the letter that corresponds to the CORRECT answer.
a) In the figure, BAC = 900 and AD ⊥ BC, then
(A) BD × CD = BC2
(B) BD × CD = AD2
(C) BA × CA = BC2
(D) AB × AC = AD2
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
ADvTECH © Preliminary Examination 2021: Mathematics Paper 2 Page 26 of 30
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
DE
G
F
ADvTECH © Preliminary Examination 2021: Mathematics Paper 2 Page 27 of 30
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
ADvTECH © Preliminary Examination 2021: Mathematics Paper 2 Page 28 of 30
7 MARKS
____________________________________________________________________________
TOTAL SECTION B: 75 MARKS
ADvTECH © Preliminary Examination 2021: Mathematics Paper 2 Page 29 of 30
EXTRA SPACE FOR WORKING
ADvTECH © Preliminary Examination 2021: Mathematics Paper 2 Page 30 of 30
EXTRA SPACE FOR WORKING