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

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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

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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

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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

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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

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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

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c) Prove the identity: 2 sin2 𝑥

2 tan 𝑥−sin 2𝑥=

1

tan 𝑥 (4)

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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

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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)

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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

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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

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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

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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

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2) Determine : 𝐴𝑟𝑒𝑎 ∆𝐴𝐵𝐶

𝐴𝑟𝑒𝑎 ∆𝐶𝐹𝐷 (4)

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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

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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)

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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

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Prove that:

a) ∆𝐴𝐵𝐶 III ∆𝐴𝐷𝐵 III ∆𝐵𝐷𝐶 (3)

b) 𝑂𝑆 ∥ 𝐴𝐶 (2)

c) 4𝐶𝑆2 − 𝐷𝐶2 = 𝐴𝐷. 𝐷𝐶 (6)

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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

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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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