Grade 12 Mathematics Paper 2 September 2018 Page 1 of 26
Total (150) Percentage
Examiner: Miss Eastes Moderated: Mrs. Gunning, Mrs. Algie
NAME:
QUESTIONS.
This question paper consists of 15 questions, 26 pages, plus an information sheet.
Answer all questions on this question paper.
Read and answer all questions carefully. Write legibly and present your work neatly.
All necessary working which you have used in determining your answers must be clearly shown.
Approved non-programmable calculators may be used except where otherwise stated. Where
necessary give answers correct to 1 decimal place unless otherwise stated.
Ensure that your calculator is in DEGREE mode.
Diagrams have not necessarily been drawn to scale.
Give reasons for all Euclidean Geometry questions unless otherwise stated.
SECTION A (74) SECTION B (76)
Question 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Mark
Out of 12 8 5 16 8 10 15 7 11 6 5 11 12 17 7
Grade 12 Mathematics Paper 2 September 2018 Page 2 of 26
SECTION A QUESTION 1
Measurements were collected from 12 people to determine whether people with longer arms are
taller than people with shorter arms.
Candidate Arm Span in cm (x) Height in cm (y)
A 156 162
B 157 160
C 159 162
D 160 155
E 161 160
F 161 162
G 162 170
H 165 166
I 170 170
J 173 185
K 178 180
L 184 180
H ei
gh t
(c m
Height in cm (y)
Grade 12 Mathematics Paper 2 September 2018 Page 3 of 26
1.1 Use your calculator to determine the equation of the least squares regression line.
Give your answer correct to 4 decimal places. (4)
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Grade 12 Mathematics Paper 2 September 2018 Page 4 of 26
QUESTION 2
Below is the cumulative frequency graph of the number of spectators passing through the ticket
booths at a stadium over a 2 hour period, starting at 14h00 and ending at 16h00.
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Grade 12 Mathematics Paper 2 September 2018 Page 5 of 26
2.3 Draw a histogram of the time taken for spectators to enter the stadium
on the ogive above. Use the table below if needed. (4)
Time in
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Grade 12 Mathematics Paper 2 September 2018 Page 6 of 26
QUESTION 3 In the diagram O is the centre of the circle and A, B and C are points on the circle.
Use the diagram to prove that: [5]
Construction:
Proof:
Grade 12 Mathematics Paper 2 September 2018 Page 7 of 26
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Grade 12 Mathematics Paper 2 September 2018 Page 8 of 26
QUESTION 4
ABCD is a rectangle with A(-3; 4) and B(1; 7), with O the origin
4.1 Determine the gradient of AB. (2)
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Grade 12 Mathematics Paper 2 September 2018 Page 9 of 26
4.4 If the line BD is described by the equation , determine
the coordinates of point D. (3)
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Grade 12 Mathematics Paper 2 September 2018 Page 10 of 26
QUESTION 5
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Grade 12 Mathematics Paper 2 September 2018 Page 11 of 26
QUESTION 6
Standing at C, 150 m from the bottom, B, of a vertical cliff AB, Thabo observes that the angle of
elevation of A, the top of the cliff, is .
He then walks 300 m to the point D, up a slope inclined at to the horizontal.
A, B, C, D and F are in the same vertical plane. Give reasons where necessary.
Calculate:
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Grade 12 Mathematics Paper 2 September 2018 Page 12 of 26
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Grade 12 Mathematics Paper 2 September 2018 Page 13 of 26
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Grade 12 Mathematics Paper 2 September 2018 Page 14 of 26
SECTION B QUESTION 8 The diameter, , of a circle is produced to intersect the tangent from , a point on the circle, at . , , and is the radius of the circle.
8.1 Prove . (3) ______________________________________________________________________________
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A
B
C
T
Grade 12 Mathematics Paper 2 September 2018 Page 15 of 26
QUESTION 9
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Grade 12 Mathematics Paper 2 September 2018 Page 16 of 26
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Grade 12 Mathematics Paper 2 September 2018 Page 17 of 26
QUESTION 10
In the diagram below the circle touches the x-axis at P(-1; 0) and passes through the point
R(-4; -1). M is the centre of the circle which cuts the y-axis at T and N.
MR, MP and MN are radii of circle with centre M.
Find the equation of the circle.
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Grade 12 Mathematics Paper 2 September 2018 Page 18 of 26
QUESTION 11
The figure below shows the cross-section of a wheel bearing. The smaller circles represent ball-
bearings which roll between two larger concentric circles, A and B, which both have centre M.
The equation of circle A is .
P is the centre of one of the small ball-bearings, each of which has a diameter of 2 units. Find the
equation which best describes the path of P as it rolls between the two larger circles.
Write the answer in the form ( ) ( )
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Grade 12 Mathematics Paper 2 September 2018 Page 19 of 26
QUESTION 12
units ; units ;
Give reasons where necessary.
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Grade 12 Mathematics Paper 2 September 2018 Page 20 of 26
QUESTION 13
Refer to the diagram below, ST is a diameter of the circle.
OS || PN, TO bisects .
Give reasons where necessary.
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(4)
Grade 12 Mathematics Paper 2 September 2018 Page 21 of 26
13.2 SO is a tangent to circle KUST
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(3)
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Grade 12 Mathematics Paper 2 September 2018 Page 22 of 26
QUESTION 14
Messi and Ronaldo decided to have a competition to see who could kick a soccer ball into the
hands of the statue. Both players got to choose where they would kick the ball from. The diagram
below is an illustration of their positions.
The base of the statue, Messi ( ) and Ronaldo ( ) are all in the same horizontal plane.
The angle of elevation from Messi ( ) to the left hand ( ) of the statue is .
The angle of elevation from Ronaldo ( ) to the left hand ( ) of the statue is ( ).
The distance the ball will travel from Messi ( ) to the left hand is metres and from Ronaldo ( )
to the left hand is metres.
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Grade 12 Mathematics Paper 2 September 2018 Page 23 of 26
14.2 Show that the distance between Messi and Ronaldo is given by:
√ . (6)
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Grade 12 Mathematics Paper 2 September 2018 Page 24 of 26
14.4 The pedestal that the statue stands on is 8 metres high.
The pedestal is a cube of which the edges have been cut off to make a regular octagon. (The picture of the statue and pedestal on page 22 is not drawn to scale) Messi draws a “semi-accurate” sketch of the pedestal from a bird’s eye view.
Using Messi 's sketch and the information given above,
determine the volume of the pedestal. (6)
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Grade 12 Mathematics Paper 2 September 2018 Page 25 of 26
QUESTION 15
Refer to the diagram below. O is the centre of the circle ABDC. If and
Give reasons where necessary.
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Grade 12 Mathematics Paper 2 September 2018 Page 26 of 26
Use this page, if needed, for working