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MATHEMATICS PAPER 2
Examiner: J Greenslade Date: 31 August 2020
Moderator: D Harrison Time: 3 Hours
S Barclay Marks: 150
C Henning
L Alexander
Name: Class:
Teacher:
Please read the following instructions carefully:
1. This examination consists of:
A question paper of 23 pages; and
Additional working space of 2 pages
Please make sure your paper is complete
2. Please write your name on the front page.
3. All the questions must be answered in the spaces provided.
4. Read the questions carefully.
5. An appropriate calculator (non-programmable, non-graphical) may be used unless otherwise stated.
6. Show all your workings in all calculations. Full marks will not be given for the answers only.
7. Round off to two decimal places where necessary unless otherwise stated.
8. It is in your own interest to write legibly and to set out your work neatly.
Don’t worry; be happy. GOOD LUCK
Grade 12 Mathematics Prelim Paper II Page 2 of 27
2
SECTION A [78 MARKS]
QUESTION 1 10 marks
The table and scatter plot below shows the monthly income (in rands) of 11 different
people and the amount (in rands) that each person spends on the monthly
repayment of a motor vehicle.
MONTHLY
INCOME (in
rands)
6500 9000 10500 13500 15000 16500 17000 20000 25000 30200
MONTHLY
REPAYMENT
(in rands)
1200 2000 3000 3000 3500 5200 5000 5800 6000 7000
a) Determine the equation of the least squares regression line for the data.
Round to 4 decimal places. (2)
b) Show one point on scatter plot which will definitely lie on the trend line within
the given Monthly Income values. (2)
0
1000
2000
3000
4000
5000
6000
7000
8000
0 5000 10000 15000 20000 25000 30000 35000
Mo
nth
ly r
ep
ay
me
nt
of
mo
tor
ve
hic
le
Monthly Income
Monthly Income vs Monthly Repayment of Motor Vehicle
Grade 12 Mathematics Prelim Paper II Page 3 of 27
3
c) If a person earns R18 500 per month, predict the monthly repayment that
person could make towards a motor vehicle. (2)
d) Determine the correlation coefficient between the monthly income and the
monthly repayment of a motor vehicle. (Correct to FOUR decimal places) (1)
e) If a person earning R22 000 per month was paying only R2000 per month for
his motor vehicle would this increase or decrease the gradient of the trend
line? (1)
f) A person who earns R 28 000 per month has to decide whether or not to
spend R10 000 as a monthly repayment of a motor vehicle. If the above
information given in the table is a true representation of the population data,
which of the following would the person most likely decide on: (Circle the
appropriate letter)
A Spend R 10 000 per month because there is a very strong positive
correlation between the amount earned and the monthly repayment.
B NOT to spend R10 000 per month because there is a very weak
positive correlation between the amount earned and the monthly
repayment.
C Spend R10 000 per month because the point (28 000; 10 000) lies very
near to the trend line.
D NOT spend R10 000 per month because the point (28 000; 10 000) lies
very far from the trend line. (2)
Grade 12 Mathematics Prelim Paper II Page 4 of 27
4
QUESTION 2 9 marks
https://www.desmos.com/calculator/nakxssx40s
The box-and-whisker plot below represents the heights, in metres, of several
giraffes.
a) Describe the skewness of this data. Provide an explanation for your
description. (2)
b) Draw in an estimated mean for the data on the box-and-whisker plot. (1)
c) The height range of 4,8 ≤ ℎ ≤ 5,0 𝑚 consisted entirely of 7 young male
giraffes. The mean of the heights of the 7 young males was calculated to be
4,89𝑚. However, it was noticed that the height of one of the young males was
incorrectly recorded as 4,81𝑚 and not 4,91𝑚. Calculate the new mean for the
7 young males. (3)
d) If the standard deviation of the data represented in the box-and-whisker plot is
0,385587796381. Determine how many giraffes’ heights were measured if:
∑(�̅� − 𝑥)2 = 5,947117949. Show all your working out. (3)
Grade 12 Mathematics Prelim Paper II Page 5 of 27
5
QUESTION 3 9 marks
Given trapezium 𝐴𝐵𝐶𝐷, with 𝐴𝐵 ∥ 𝐶𝐷. 𝐸 is the 𝑥 −intercept of line 𝐴𝐵 and the
coordinates of 𝐶 and 𝐷 are (2; −3) and (−2; −5) repectively.
Determine the following:
a) The equation of 𝐴𝐵. (3)
b) Prove that: 𝐴𝐵 ⊥ 𝐵𝐶. (3)
c) i) The midpoint of 𝐸𝐶. (1)
ii) Hence, if the points 𝐸, 𝐵 and 𝐶 lie on a circumference of a circle.
Determine the equation of the circle. (2)
𝐸(−2; 0)
𝐷(−2; −5)
𝐶(2; −3)
𝐵
𝐴
𝑂
Grade 12 Mathematics Prelim Paper II Page 6 of 27
6
QUESTION 4 13 marks
a) Without the use of a calculator, show that:
sin 105° = √2
4 (√3 + 1) (5)
b) Calculate, without the use of a calculator, the value of:
cos 325° sin 745° − cos(−205°) cos 55° (4)
Grade 12 Mathematics Prelim Paper II Page 7 of 27
7
c) Prove the identity: 2 sin2 𝑥
2 tan 𝑥−sin 2𝑥=
1
tan 𝑥 (4)
Grade 12 Mathematics Prelim Paper II Page 8 of 27
8
QUESTION 5 16 marks
The diagram below shows a circle having centre 𝑀 which intersects the 𝑥-axis at 𝐴
and 𝐵 and the 𝑦-axis at 𝐷 and 𝐶. 𝑃𝐶𝑄 is a tangent to the circle at 𝐶, the point of
contact on the 𝑦- axis. 𝑃 lies on the 𝑥-axis.
The equation of the circle is: 𝑥2 + 𝑦2 − 6𝑥 − 16 = 0
a) Determine the coordinates of 𝑀 and the radius of the circle (2)
b) Determine the coordinates of the following if the coordinates of 𝑀 are (3; 0)
and the radius is 5 𝑢𝑛𝑖𝑡𝑠.
i) 𝐵 (1)
ii) 𝐶 (2)
𝑄
𝑂 𝐴 𝑀
𝑃
𝐶
𝐵
𝐷
Grade 12 Mathematics Prelim Paper II Page 9 of 27
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c) Determine the equation of the tangent 𝑃𝐶𝑄. (2)
d) If the length of 𝑃𝑀 is 81
3 units, calculate the length of 𝑃𝐶. (3)
e) Calculate the angle subtended by the chord 𝐷𝐶 at 𝐵, i.e. find 𝐷�̂�𝐶. (4)
f) If the given circle is moved 2 units right and 1 unit up, determine the equation
of the tangent to the circle in its new position passing through point 𝐶′. (2)
Grade 12 Mathematics Prelim Paper II Page 10 of 27
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QUESTION 6 15 marks
In the diagram, 𝑃𝑄𝑅𝑆 is a cyclic quadrilateral
with 𝑃𝑆 ∥ 𝑄𝑅. 𝑈𝑄𝑇 is a tangent to the circle at
𝑄 and 𝑅𝑇 is a tangent at 𝑅. 𝑃𝑅 = 𝑅𝑄.
a) Prove that �̂�3 = �̂�2 (3)
b) If, �̂�1 = 𝑥 find, THREE other angles each equal to 𝑥. (4)
≫
≫ 2
1
𝑇
𝑈
𝑃 𝑆
𝑅 𝑄
1
1
3 2
2
4 3
Grade 12 Mathematics Prelim Paper II Page 11 of 27
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c) If 𝑃𝑆 = 𝑆𝑅:
1) Prove �̂�2 =1
2𝑥 (4)
2) Complete the following:
�̂�3 = 180° − 2𝑥 Reason: (1)
3) Hence, calculate the value of 𝑥. (3)
Grade 12 Mathematics Prelim Paper II Page 12 of 27
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QUESTION 7 6 marks
The diagram below shows a vertical netball pole 𝑃𝐵. Gugu (𝐺) is standing on the
base line of the court and the angle of elevation from Gugu to the top of the pole
𝑃 is 𝑦. Kylie is standing in the court at 𝐾 and the angle of elevation from Kylie to the
top of the pole 𝑃 is 𝑥. Points 𝐺, 𝐾 and 𝐵 are all in the same horizontal plane. Gugu
is 𝑚 metres from the pole. 𝐺�̂�𝐾 = 100°.
a) Show that 𝐵𝐾 = 𝑚.tan 𝑦
tan 𝑥 . (3)
b) Calculate the length of 𝐾𝐺 if 𝐵𝐾 = 4,73 metres and 𝑚 = 3 metres. (3)
𝑚 𝐵
𝑃
𝐾
𝐺 100° 𝑦
𝑥
Grade 12 Mathematics Prelim Paper II Page 13 of 27
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SECTION B [72 marks]
QUESTION 8 6 marks
Given: 𝐴(−2; 𝑦) and 𝑂𝐴 = √13
Without a calculator, determine the value of the following:
i) cos 2𝜃 + 1 (2)
ii) sin2 (𝜃
2) (4)
𝐴(−2; 𝑦)
√13
𝑂
𝑦
𝑥 𝜃
Grade 12 Mathematics Prelim Paper II Page 14 of 27
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QUESTION 9 7 marks
In the diagram below, 𝑃𝑇 = 6 units, 𝑇𝑉 = 4 units, 𝑋𝑊 = 7 units, 𝑇𝑊 = 2 units, 𝑉𝑋 =
4 units, 𝑊𝑃 = 5 units and 𝑃𝐿 ∥ 𝑉𝑋.
Prove that:
a) ∆𝑃𝑇𝑊 III ∆𝑃𝑋𝑉 (3)
b) 𝑃𝐿 is a tangent to the circle passing through the points 𝑃, 𝑊 and 𝑇. (4)
𝑇
>
𝐿
𝑊
𝑋
𝑉
𝑃
>
1
1 2
1
2
3 2
Grade 12 Mathematics Prelim Paper II Page 15 of 27
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QUESTION 10 15 marks
a) Using the diagram below, prove the theorem which states that if 𝑆𝑇 ∥ 𝑄𝑅 in
∆𝑃𝑄𝑅, then 𝑃𝑆
𝑆𝑄=
𝑃𝑇
𝑇𝑅. (6)
𝑆
𝑅 𝑄
𝑃
𝑇
Grade 12 Mathematics Prelim Paper II Page 16 of 27
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b) In the diagram below, ∆𝐴𝐵𝐶 is shown with 𝐵𝐷: 𝐵𝐶 = 3: 13. 𝐴𝐵 ∥ 𝐹𝐷.
1) If 𝑅𝑃
𝑃𝐺=
9
10 determine
𝐴𝐹
𝐺𝐶 (5)
𝐺
𝐶
𝐹
𝐴
𝐷
𝐵
𝑃
𝑅
Grade 12 Mathematics Prelim Paper II Page 17 of 27
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2) Determine : 𝐴𝑟𝑒𝑎 ∆𝐴𝐵𝐶
𝐴𝑟𝑒𝑎 ∆𝐶𝐹𝐷 (4)
Grade 12 Mathematics Prelim Paper II Page 18 of 27
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QUESTION 11 19 marks
The diagram below illustrates a ball running in a hollowed half pipe. The line 𝑃𝑅 with
equation 𝑦 − 2𝑥 − 1 = 0 (the one side of the pipe) is a tangent to the circle (ball)
with centre 𝑀(4; 4). 𝑃𝑇 is a diameter of the circle. Determine:
a) The equation of 𝑃𝑇. (3)
b) The coordinates of 𝑃, the point of tangency. (4)
𝐵
𝑅 𝑥
𝑦
𝑃
𝑇(𝑥; 𝑦) √10
𝑀(4; 4)
𝑉
𝑅
𝑆
Grade 12 Mathematics Prelim Paper II Page 19 of 27
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c) The equation of the circle centre 𝑀. (2)
d) The coordinates of 𝑇. (2)
e) Describe the transformation circle centre 𝑀 must undergo, in order to be
tangential to its original position, yet still running between the parallel
tangents, i.e. in the position of circle centre 𝑆. (4)
Grade 12 Mathematics Prelim Paper II Page 20 of 27
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f) If the original ball (circle centre 𝑀) jumps out of the half pipe and lands in a
new position, illustrated as a circle with equation (2𝑥 − 1)2 + (2𝑦 − 13)2 = 20,
would the circle be just touching the circle centre 𝑀 or not touching at all or
touching in two places? Show all relevant working out. (4)
QUESTION 12 11 marks
In the diagram below, 𝐴𝑂𝐵 is the diameter of semi-circle, centre 𝑂. 𝐶𝐵 is a tangent
at 𝐵. 𝑂𝐾 ⊥ 𝐷𝐵 and 𝑂𝐾 produced cuts 𝐶𝐵 at 𝑆.
𝐾
𝐶
𝐾
𝑂
𝐷 𝐷
𝐵 𝑂 𝐴
𝑆
𝐾
1 1
1
2
2
2
1 2
Grade 12 Mathematics Prelim Paper II Page 21 of 27
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Prove that:
a) ∆𝐴𝐵𝐶 III ∆𝐴𝐷𝐵 III ∆𝐵𝐷𝐶 (3)
b) 𝑂𝑆 ∥ 𝐴𝐶 (2)
c) 4𝐶𝑆2 − 𝐷𝐶2 = 𝐴𝐷. 𝐷𝐶 (6)
Grade 12 Mathematics Prelim Paper II Page 22 of 27
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QUESTION 13 14 marks
In the diagram, the graphs of 𝑓(𝑥) = − sin(2𝑥 − 60°) + 1 and
𝑔(𝑥) = cos(𝑥 − 30°) + 1 are drawn for the interval 𝑥 ∈ [−180°; 270°]. 𝐶 is one of the
intercept points of the two graphs and 𝐴 is an 𝑥 −intercept of 𝑔 and 𝐵 is the a
maximum turning point of 𝑓.
a) Determine the 𝑥 −coordinates of:
1) 𝐴 (2)
2) 𝐵 (2)
b) Determine the range of 𝑦 = 2cos(𝑥−30°)+1 (4)
𝑔
𝐴
𝑓 𝐵
𝐶
Grade 12 Mathematics Prelim Paper II Page 23 of 27
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c) Determine the general solution of − sin(2𝑥 − 60°) + 1 = cos(𝑥 − 30°) + 1
and hence the coordinates of 𝐶. (6)
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