Western Cape Education Department

Examination Preparation
Learning Resource 2018

MATHEMATICS
Grade 12 Paper 2

Razzia Ebrahim
Senior Curriculum Planner for Mathematics

Website: http://www.wcedcurriculum.westerncape.gov.za/index.php/fet-futher-education-training/sciences/
mathematics-fet-home
Website: http://wcedeportal.co.za
 Index

Content Page
1. Statistics 3 – 16

2. Analytical Geometry 17 – 33

3. Trigonometry General 34 – 41

4. Trigonometry Graphs 42 – 49

5. Trigonometry (sine, area, cosine rules) 50 – 59

6. Geometry Grade 11 60 – 83

7. Geometry Grade 12 84 – 102

8. 2018 Feb - March Paper 2 103 – 118

9. 2017 November Paper 2 118 – 131

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 STATISTICS
REVISION

Index
Content Page
1. 2016 June Paper 2 4–6

2. 2016 Feb-March Paper 2 6–7

3. 2015 November Paper 2 8–9

4. 2015 June Paper 2 10

5. 2015 Feb-March Paper 2 11 – 12

6. 2014 November Paper 2 13 – 14

7. 2014 Exemplar Paper 2 15 – 16

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Mathematics/P2/STATISTICS DBE/2016
SCE

QUESTION 1

On a certain day a tour operator sent 11 tour buses to 11 different destinations.

The table below shows the number of passengers on each bus.

8 8 10 12 16 19 20 21 24 25 26

1.1 Calculate the mean number of passengers travelling in a tour bus. (2)

1.2 Write down the five-number summary of the data. (3)

1.3 Draw a box and whisker diagram for the data. Use the number line provided in the
ANSWER BOOK. (2)

1.4 Refer to the box and whisker diagram and comment on the skewness of the data set. (1)

1.5 Calculate the standard deviation for this data set. (2)

1.6 A tour is regarded as popular if the number of passengers on a tour bus is one
standard deviation above the mean. How many destinations were popular on this
particular day? (2)
[12]

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Mathematics/P2/STATISTICS DBE/2016
SCE

QUESTION 2

On the first school day of each month information is recorded about the temperature at midday
(in °C) and the number of 500 mℓ bottles of water that were sold at the tuck shop of a certain
school during the lunch break. The data is shown in the table below and represented on the
scatter plot. The least squares regression line for this data is drawn on the scatter plot.

Temperature at
18 21 19 26 32 35 36 40 38 30 25
midday (in °C)
Number of bottles of
12 15 13 31 46 51 57 70 63 53 23
water (500 mℓ)

Scatter plot
75

70

65

60
Number of bottles of water (500 mℓ)

55

50

45

40

35

30

25

20

15

10

5
15 20 25 30 35 40
Temperature at midday (in °C)

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Mathematics/P2/Statistics DBE/2016
SCE

2.1 Identify an outlier in the data. (1)

2.2 Determine the equation of the least squares regression line. (3)

2.3 Estimate the number of 500 mℓ bottles of water that will be sold if the temperature is
28 °C at midday. (2)

2.4 Refer to the scatter plot. Would you say that the relation between the temperature at
midday and the number of 500 mℓ bottles of water sold is weak or strong? Motivate
your answer. (2)

2.5 Give a reason why the observed trend for this data cannot continue indefinitely. (1)
[9]

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Copyright Reserved 7 Please turn over
 Mathematics/P2/Statistics DBE/November 2015
NSC

QUESTION 1

The table below shows the total fat (in grams, rounded off to the nearest whole number)
and energy (in kilojoules, rounded off to the nearest 100) of 10 items that are sold at a fast-food
restaurant.

Fat (in grams) 9 14 25 8 12 31 28 14 29 20

Energy
1 100 1 300 2 100 300 1 200 2 400 2 200 1 400 2 600 1 600
(in kilojoules)

1.1 Represent the information above in a scatter plot on the grid provided in the
ANSWER BOOK. (3)

1.2 The equation of the least squares regression line is ŷ = 154,60 + 77,13x.

1.2.1 An item at the restaurant contains 18 grams of fat. Calculate the number
of kilojoules of energy that this item will provide. Give your answer
rounded off to the nearest 100 kJ. (2)

1.2.2 Draw the least squares regression line on the scatter plot drawn
for QUESTION 1.1. (2)

1.3 Identify an outlier in the data set. (1)

1.4 Calculate the value of the correlation coefficient. (2)

1.5 Comment on the strength of the relationship between the fat content and the number
of kilojoules of energy. (1)
[11]

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Mathematics/P2/Statistics DBE/November 2015
NSC

QUESTION 2

A group of 30 learners each randomly rolled two dice once and the sum of the values on the
uppermost faces of the dice was recorded. The data is shown in the frequency table below.

Sum of the values

Frequency
on uppermost faces
2 0
3 3
4 2
5 4
6 4
7 8
8 3
9 2
10 2
11 1
12 1

2.1 Calculate the mean of the data. (2)

2.2 Determine the median of the data. (2)

2.3 Determine the standard deviation of the data. (2)

2.4 Determine the number of times that the sum of the recorded values of the dice is
within ONE standard deviation from the mean. Show your calculations. (3)
[9]

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Mathematics/P2/Statistics DBE/2015
SC£

QUESTION 1

The data below shows the ages (in years) of people who visited the library between 08:00 and
09:00 on a certain morning.

3 4 4 s 1 1 1 1 1 1 1 1 1 1 1 1
23 29 32 36 40 47 56 66 68 76 82

1.1 Determine:

1.1.1 The mean age of the visitors (2)

1.1.2 The median of the data (1)

1.1.3 The interquartile range of the data (3)

1.1.4 The standard deviation of the data (2)

1.2 Draw a box and whisker diagram for the data. (3)

1.3 By making reference to the box and whisker diagram, comment on the skewness of
the data set. (1)
[12]

,\ft11h�11w11cs Pl DBfJl0/5
. cr

QUESTIO 2

Fourteen learner attended a Gcomctf} training course spread over 12 'aturda) . Leamcrs
\\Tote u ,comctry test at the end of the course. One learne1 ,.,,as absent for this te�t. The number
of aturdays attended and the mark (as a O i>) each learner obtaiued rnr lht: te I are shov.11 in the
table below.

Number of Saturdays attended 12 11 10 10 9 9 7 6 5 4 12 11

Mark (asa %) 96 91 78 83 75 62 70 68 56 34 88 90 5()

2.1 Calculate the equation of the least squ.ut.-s rcgn:-..,;ion line. (3)

2.2 (\1lcululc the ,om.:lation co Hicicnt. (2)

2.3 Comment on the strcng1h of Lhe relat101t.lih1p b�1,,c�n the Hlriahlt.."'i. (I)

2.4 ·1 he learner who was ahscnt for the lest attended the course on 8 aturda}s.
Pn:�lict the mark 1hal thi · learner \\tlllld have scored for the test. (2)
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Mathematics/P2/Statistics DBE/Feb.-Mar. 2015
NSC

QUESTION 1

The table below shows the distances (in kilometres) travelled daily by a sales representative for
21 working days in a certain month.

131 132 140 140 141 144 146

147 149 150 151 159 167 169
169 172 174 175 178 187 189

1.1 Calculate the mean distance travelled by the sales representative. (2)

1.2 Write down the five-number summary for this set of data. (4)

1.3 Use the scaled line on DIAGRAM SHEET 1 to draw a box-and-whisker diagram for
this set of data. (2 )

1.4 Comment on the skewness of the data. (I)

1.5 Calculate the standard deviation of the distance travelled. (2)

1.6 The sales representative discovered that his odometer was faulty. The actual reading
on each of the 21 days was p km more than that which was indicated. Write down,
in tem1s of p (if applicable), the:

1.6.1 Actual mean (1)

1.6.2 Actual standard deviation (1)

[13]

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Mathematics/P2/Statistics DBE/Feb.-Mar.2015
NSC

QUESTION 2

An ice-cream shop recorded the sales of ice cream, in rand, and the maximum temperature,
in °C, for 12 days in a certain month. The data that they collected is represented in the table
and scatter plot below.

T emperature in °C 24,2 26,4 21,9 25,2 28,5 32,1 29,4 35,1 33,4 28,1 32,6 27,2
Sales of ice cream in rand 215 325 185 332 406 522 412 614 544 421 445 408

Scatter plot
650

600 •
550

500

� 450 -
.5 400 • •. •
� 350

-�
�
• •
300

•
� 250
�
200
t
150
20 22 24 26 28 30 32 34 36 38
Temperature in °C

2.1 Describe the influence of temperature on the sales of ice cream in the scatter plot. (1)

2.2 Give a reason why this trend cannot continue indefinitely. (1 )

2.3 Calculate an equation for the least squares regression line (line of best fit ). (4)

2.4 Calculate the correlation coefficient. (1)

2.5 Comment on the strength of the relationship between the variables. (1)
[8]

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 Mathematics/P2/Statistics DBE/November 2014
NSC

QUESTION 1

At a certain school, only 12 candidates take Mathematics and Accounting. The marks, as a
percentage, scored by these candidates in the preparatory examinations for Mathematics and
Accounting, are shown in the table and scatter plot below.

Mathematics 52 82 93 95 71 65 77 42 89 48 45 57
Accounting 60 62 88 90 72 67 75 48 83 57 52 62

Scatter plot
100
Percentage achieved in Accounting

90

80

70

60

50

40
40 50 60 70 80 90 100
Percentage achieved in Mathematics

1.1 Calculate the mean percentage of the Mathematics data. (2)

1.2 Calculate the standard deviation of the Mathematics data. (1)

1.3 Determine the number of candidates whose percentages in Mathematics lie within
ONE standard deviation of the mean. (3)

1.4 Calculate an equation for the least squares regression line (line of best fit) for the
data. (3)

1.5 If a candidate from this group scored 60% in the Mathematics examination but was
absent for the Accounting examination, predict the percentage that this candidate
would have scored in the Accounting examination, using your equation in
QUESTION 1.4. (Round off your answer to the NEAREST INTEGER.) (2)

1.6 Use the scatter plot and identify any outlier(s) in the data. (1)
[12]

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 Mathematics/P2/Statistics DBE/November 2014
NSC

QUESTION 2

The speeds of 55 cars passing through a certain section of a road are monitored for one hour.
The speed limit on this section of road is 60 km per hour. A histogram is drawn to represent
this data.

Histogram
18 17
16
14 13
12
Frequency

10 9
8 7
6 5
4
2
2 1 1
0
40 50 60 70
0-20 2020-30 3030-40 40-50 50-60 60-70 70-80 8080-90 9090-100100
100-110
Speed (in km per hour)

2.1 Identify the modal class of the data. (1)

2.2 Use the histogram to:

2.2.1 Complete the cumulative frequency column in the table on

DIAGRAM SHEET 1 (2)

2.2.2 Draw an ogive (cumulative frequency graph) of the above data on the grid
on DIAGRAM SHEET 1 (3)

2.3 The traffic department sends speeding fines to all motorists whose speed exceeds
66 km per hour. Estimate the number of motorists who will receive a speeding fine. (2)
[8]

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 Mathematics/P2/Statistics DBE/2014
NSC – Grade 12 Exemplar

QUESTION 1

Twelve athletes trained to run the 100 m sprint event at the local athletics club trials. Some of
them took their training more seriously than others. The following table and scatter plot shows
the number of days that an athlete trained and the time taken to run the event. The time taken, in
seconds, is rounded to one decimal place.

Number of days
50 70 10 60 60 20 50 90 100 60 30 30
of training
Time taken
12,9 13,1 17,0 11,3 18,1 16,5 14,3 11,7 10,2 12,7 17,2 14,3
(in seconds)

Scatter plot
20

18
Time taken (in seconds)

16

14

12

10

8
0 20 40 60 80 100 120

Number of days of training

1.1 Discuss the trend of the data collected. (1)

1.2 Identify any outlier(s) in the data. (1)

1.3 Calculate the equation of the least squares regression line. (4)

1.4 Predict the time taken to run the 100 m sprint for an athlete training for 45 days. (2)

1.5 Calculate the correlation coefficient. (2)

1.6 Comment on the strength of the relationship between the variables. (1)
[11]

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 Mathematics/P2/Statistics DBE/2014
NSC – Grade 12 Exemplar

QUESTION 2

The table below shows the amount of time (in hours) that learners aged between 14 and 18
spent watching television during 3 weeks of the holiday.

Time (hours) Cumulative frequency

0 ≤ t < 20 25
20 ≤ t < 40 69
40 ≤ t < 60 129
60 ≤ t < 80 157
80 ≤ t < 100 166
100 ≤ t < 120 172

2.1 Draw an ogive (cumulative frequency curve) on DIAGRAM SHEET 1 to represent

the above data. (3)

2.2 Write down the modal class of the data. (1)

2.3 Use the ogive (cumulative frequency curve) to estimate the number of learners who
watched television more than 80% of the time. (2)

2.4 Estimate the mean time (in hours) that learners spent watching television during
3 weeks of the holiday. (4)
[10]

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 ANALYTICAL
GEOMETRY

Index

Content Page
1. 2016 November Paper 2 18 – 19

2. 2016 June Paper 2 20 – 21

3. 2016 Feb-March Paper 2 22 – 23

4. 2015 November Paper 2 24 – 25

5. 2015 June Paper 2 26 – 27

6. 2015 Feb-March Paper 2 28 – 29

7. 2014 November Paper 2 30 – 31

8. 2014 Exemplar Paper 2 32 – 33

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Mathematics P2/Analytical Geometry

DBE/November 2016

QUESTION 3

In the diagram, A(–7 ; 2), B, C(6 ; 3) and D are the vertices of rectangle ABCD.
The equation of AD is y = 2x + 16. Line AB cuts the y-axis at G. The x-intercept of
line BC is F(p ; 0) and the angle of inclination of BC with the positive x-axis is  .
The diagonals of the rectangle intersect at M.

D y

y = 2x + 16

C(6 ; 3)
A( –7 ; 2) M

x
O F(p ; 0)
G

3.1 Calculate the coordinates of M. (2)

3.2 Write down the gradient of BC in terms of p. (1)

3.3 Hence, calculate the value of p. (3)

3.4 Calculate the length of DB. (3)

3.5 Calculate the size of . (2)

3.6 Calculate the size of OĜB. (3)

3.7 Determine the equation of the circle passing through points D, B and C in the form
( x  a ) 2  ( y  b) 2  r 2 . (3)

3.8 If AD is shifted so that ABCD becomes a square, will BC be a tangent to the

circle passing through points A, M and B, where M is now the intersection of the
diagonals of the square ABCD? Motivate your answer. (2)
[19]

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Mathematics P2/Analytical Geometry

DBE/November 2016
QUESTION 4

In the diagram, M is the centre of the circle passing through T(3 ; 7), R and S(5 ; 2). RT is
a diameter of the circle. K(a ; b) is a point in the 4th quadrant such that KTL is a tangent to
the circle at T.

L
y

T(3 ; 7)
M
R

S(5 ; 2)

x
O

K(a ; b)

4.1 Give a reason why T ŜR  90° . (1)

4.2 Calculate the gradient of TS. (2)

4.3 Determine the equation of the line SR in the form y = mx + c. (3)

2
 1 1
4.4 The equation of the circle above is ( x  9) 2   y  6   36 .
 2 4

4.4.1 Calculate the length of TR in surd form. (2)

4.4.2 Calculate the coordinates of R. (3)

4.4.3 Calculate sin R. (3)

4.4.4 Show that b = 12a – 29. (3)

4.4.5 If TK = TR, calculate the coordinates of K. (6)

[23]

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 Mathematics P2/Analytical Geometry

DBE/2016

QUESTION 3

In the diagram below (not drawn to scale) A(–8 ; 6), B(8 ; 10), C and D(–2 ; 0) are the
vertices of a trapezium having BC | | AD. T is the midpoint of DB. From B, the straight line
1
drawn parallel to the y-axis cuts DC in F  8 ; 3  and the x-axis in M.
 3

B(8 ; 10)

A(–8 ; 6)
T
C

D(–2 ; 0) O M x

3.1 Calculate the gradient of AD. (2)

3.2 Determine the equation of BC in the form y = mx + c. (3)

3.3 Prove that BD  AD. (3)

3.4 Calculate the size of BD̂M. (2)

3.5 If it is given that TC | | DM and points T and C are symmetrical about line BM,
calculate the coordinates of C. (3)

3.6 Calculate the area of BDF. (5)

[18]

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Mathematics P2/Analytical Geometry

DBE/2016
QUESTION 4

A circle having C(3 ; –1) as centre and a radius of 10 units is drawn. PTR is a tangent to this
circle at T. R(k ; 21), C and P are the vertices of a triangle. TR = 20 units.

y R(k ; 21)

x
O C(3 ; –1)

P S

4.1 Give a reason why TC  TR. (1)

4.2 Calculate the length of RC. Leave your answer in surd form. (2)

4.3 Calculate the value of k if R lies in the first quadrant. (4)

4.4 Determine the equation of the circle having centre C and passing through T.
Write your answer in the form ( x  a) 2  ( y  b) 2  r 2 (2)

4.5 PS, a tangent to the circle at S, is parallel to the x-axis. Determine the equation
of PS. (2)

4.6 The equation of PTR is 3y – 4x = 35

4.6.1 Calculate the coordinates of P. (2)

4.6.2 Calculate, giving a reason, the length of PT. (3)

4.7 Consider another circle with equation ( x  3) 2  ( y  16) 2  16 and having centre M.

4.7.1 Write down the coordinates of centre M. (1)

4.7.2 Write down the length of the radius of this circle. (1)

4.7.3 Prove that the circle with centre C and the circle with centre M do not
intersect or touch. (3)
[21]

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Mathematics P2/Analytical Geometry

DBE/Feb.-Mar. 2016

QUESTION 3

In the diagram below, P(l ; I), Q(O ; -2) and R are the vertices of a triangle and PRQ = 0.
The x-intercepts of PQ and PR are M and N respectively. The equations of the sides PR
and QR are y = -x + 2 and x + 3y + 6 = 0 respectively. T is a point on the x-axis, as shown.

y P(I ; 1)

0 T
X

y = -x + 2

Q(O; -2)

3.1 Determine the gradient of QP. (2)

3.2 Prove that PQR = 90 ° . (2)

3.3 Determine the coordinates of R. (3)

3.4 Calculate the length of PR. Leave your answer in surd form. (2)

3.5 Determine the equation of a circle passing through P, Q and R in the form
(x - a ) + (y - b ) = r 2 •
2 2
(6)

3.6 Determine the equation of a tangent to the circle passing through P, Q and R at
point P in the form y = mx + c. (3)

3.7 Calculate the size of 0. (5)

[23)

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Mathematics P2/Analytical Geometry

DBE/Feb.-Mar. 20 I 6
QUESTION 4

In the diagram below, the equation of the circle with centre O is x 2 + y 2 = 20. The tangent
1
PRS to the circle at R has the equation y = - x + k . PRS cuts the y-axis at T and the x-
axis at S.

4.1 Determine, giving reasons, the equation of OR in the form y = mx + c. (3)

4.2 Determine the coordinates of R. (4)

4.3 Determine the area of �OTS, given that R(2 ; -4). (6)

4.4 Calculate the length of VT. ( 4)

[17)

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Mathematics P2/Analytical Geometry

DBE/November 2015
QUESTION 3

In the diagram below, the line joining Q(–2 ; –3) and P(a ; b), a and b > 0, makes an angle
of 45° with the positive x-axis. QP = 7 2 units. N(7 ; 1) is the midpoint of PR and M is
the midpoint of QR.

y S
P(a ; b)

N(7 ; 1)
O T 45°
x

R
M
Q(–2 ; –3)

Determine:

3.1 The gradient of PQ (2)

3.2 The equation of MN in the form y = mx + c and give reasons (4)

3.3 The length of MN (2)

3.4 The length of RS (1)

3.5 The coordinates of S such that PQRS, in this order, is a parallelogram (3)

3.6 The coordinates of P (6)

[18]

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Mathematics P2/Analytical Geometry

DBE/November 2015

QUESTION 4

In the diagram below, Q(5 ; 2) is the centre of a circle that intersects the y-axis at P(0 ; 6)
and S. The tangent APB at P intersects the x-axis at B and makes the angle α with the
positive x-axis. R is a point on the circle and PR̂S = θ .

y A

P(0 ; 6)

θ R

Q(5 ; 2)

B α O
x

4.1 Determine the equation of the circle in the form ( x − a ) 2 + ( y − b) 2 = r 2 . (3)

4.2 Calculate the coordinates of S. (3)

4.3 Determine the equation of the tangent APB in the form y = mx + c. (4)

4.4 Calculate the size of α. (2)

4.5 Calculate, with reasons, the size of θ. (4)

4.6 Calculate the area of ∆PQS. (4)

[20]

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Mathematics P2/Analytical Geometry

DBE/2015
QUESTION3

In the diagram below P, Q, R(4 ; 2) and S(2 ; 1) are the vertices of a quadrilateral with
PS 11 QR. M(0 ; 4) and N are the y-intercepts of QR and PS respectively. PS produced
cuts the x-axis at T.

R(4; 2)

3.1 Calculate the gradient of RS. (2)

3.2 Given that the equation of PQ 1s 2y = x + 12, calculate, with reasons, the length
of PQ. (4)

3.3 Determine the equation of PT in the form y = mx + c. (4)

3.4 Hence, write down the coordinates of N. (1)

3.5 Calculate, with reasons, the size of RNS , rounded to ONE decimal place. (5)
[16)

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Mathematics P2/Analytical Geometry

DBE/2015
QUESTION 4

The diagram below shows a circle with centre O at the origin. AB is a diameter of the circle.

½)
The straight line ACD meets the tangent EBD to the circle at D.
The coordinates of B and E are (3 ; - 4) and (- 3; - 8 respectively.

4.1 Determine the coordinates of A. (2)

4.2 Determine the equation of the circle passing through A, B and C. (3)

4.3 Write down the length of AB. (2)

4.4 If it is given that AD is m5 units, calculate the length of BO. Give reasons. (3)

4.5 Calculate the area of .6.ABD. (3)

4.6 Another circle passes through A, B and E. Determine, with reasons, the equation
of this circle. Write the answer in the form (x-a) 2 +(y-b) 2 = r 2• (6)
[19}

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Mathematics P2/Analytical Geometry

DBE/Feb.-Mar. 2015

QUESTION 3

In the diagram below points P(5; 13), Q(-1; 5) and S(7,5; 8) are given. SR 11 PQ where R
is the y-intercept of SR. The x-intercept of SR is B. QR is joined.

S(7,5; 8)

Q(-1; 5)

a
X

3.1 Calculate the length of PQ. (3)

3.2 Calculate the gradient of PQ. (2)

..., . ...,
.) .) Determine the equation of line RS in the form ax + by + c = 0. (4)

3.4 Determine the x-coordinate of B. (2)

3.5 Calculate the size of ORB. (3)

3.6 Prove that QBSP is a parallelogram. (4)

[18)

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Mathematics P2/Analytical Geometry

DBE/Feb.-Mar. 2015
QUESTION 4

4.1 In the diagram below, the circle centred at M(2 ; 4) passes through C(-1 ; 2) and
cuts the y-axis at E. The diameter CMD is drawn and ACB is a tangent to
the circle.
y

4.1.1 Determine the equation of the circle in the form (x-a) 2 + (y-b) 2 = r 2 . (3)

4.1.2 Write down the coordinates of D. (2)

4.1.3 Determine the equation of AB in the form y = mx + c. (5)

4.1.4 Calculate the coordinates of E. (4)

4.1.5 Show that EM is parallel to AB. (2)

4.2 Determine whether or not the circles having equations (x + 2) 2 + (y- 4) 2 = 25 and
(x-5) 2 + (y + 1) 2 = 9 will intersect. Show ALL calculations. (6)
[22]

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Mathematics P2/Analytical Geometry

DBE/November 2014

QUESTION 3

In the diagram below, a circle with centre M(5 ; 4) touches the y-axis at N and intersects the
x-axis at A and B. PBL and SKL are tangents to the circle where SKL is parallel to the
x-axis and P and S are points on the y-axis. LM is drawn.

y
S K L

N M(5 ; 4)

x
O A B

3.1 Write down the length of the radius of the circle having centre M. (1)

3.2 Write down the equation of the circle having centre M, in the form
( x  a) 2  ( y  b) 2  r 2 . (1)

3.3 Calculate the coordinates of A. (3)

3.4 If the coordinates of B are (8 ; 0), calculate:

3.4.1 The gradient of MB (2)

3.4.2 The equation of the tangent PB in the form y = mx + c (3)

3.5 Write down the equation of tangent SKL. (2)

3.6 Show that L is the point (20 ; 9). (2)

3.7 Calculate the length of ML in surd form. (2)

3.8 Determine the equation of the circle passing through points K, L and M in the form
( x  p) 2  ( y  q) 2  c 2 (5)
[21]

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Mathematics P2/Analytical Geometry

DBE/November 2014

QUESTION 4

In the diagram below, E and F respectively are the x- and y-intercepts of the line having
equation y  3x  8 . The line through B(1 ; 5) making an angle of 45° with EF, as shown
below, has x- and y-intercepts A and M respectively.
y

B(1 ; 5)
D 45° M

A E O x

4.1 Determine the coordinates of E. (2)

4.2 Calculate the size of DÂE. (3)

4.3 Determine the equation of AB in the form y = mx + c. (4)

4.4 If AB has equation x  2 y  9  0, determine the coordinates of D. (4)

4.5 Calculate the area of quadrilateral DMOE. (6)

[19]

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Mathematics P2/Analytical Geometry

NSC – Grade 12 Exemplar DBE/2014

QUESTION 3

In the diagram below, M, T(–1 ; 5), N(x ; y) and P(7 ; 3) are vertices of trapezium MTNP
having TN | | MP. Q(1 ; 1) is the midpoint of MP. PK is a vertical line and SP̂K = θ.
The equation of NP is y = –2x + 17.

y
N(x ; y)

T(−1 ; 5)

P(7 ; 3)
θ

Q(1 ; 1)
x
S 0 K

3.1 Write down the coordinates of K. (1)

3.2 Determine the coordinates of M. (2)

3.3 Determine the gradient of PM. (2)

3.4 Calculate the size of θ. (3)

3.5 Hence, or otherwise, determine the length of PS. (3)

3.6 Determine the coordinates of N. (5)

3.7 If A(a ; 5) lies in the Cartesian plane:

3.7.1 Write down the equation of the straight line representing the possible
positions of A. (1)

3.7.2 Hence, or otherwise, calculate the value(s) of a for which TÂQ = 45°. (5)
[22]

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Mathematics P2/Analytical Geometry

DBE/2014
QUESTION 4

In the diagram below, the equation of the circle having centre M is (x + 1)2 + (y + 1)2 = 9.
R is a point on chord AB such that MR bisects AB. ABT is a tangent to the circle having
centre N(3 ; 2) at point T(4 ; 1).

N(3 ; 2)

T(4 ; 1)
x
M
B

4.1 Write down the coordinates of M. (1)

4.2 Determine the equation of AT in the form y = mx + c. (5)

10
4.3 If it is further given that MR = units, calculate the length of AB.
2
Leave your answer in simplest surd form. (4)

4.4 Calculate the length of MN. (2)

4.5 Another circle having centre N touches the circle having centre M at point K.
Determine the equation of the new circle. Write your answer in the form
x2 + y2 + Cx + Dy + E = 0. (3)
[15]

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 TRIGONOMETRY
General

Index
Content Page
1. 2016 November Paper 2 35

2. 2016 June Paper 2 36

3. 2016 Feb-March Paper 2 37

4. 2015 November Paper 2 38

5. 2015 June Paper 2 39

6. 2015 Feb-March Paper 2 40

7. 2014 November Paper 2 41

8. 2014 Exemplar Paper 2 41

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Mathematics P2/Trigonometry General

DBE/November 2016
QUESTION 5

5.1 Given: sin 16  p

Determine the following in terms of p, without using a calculator.

5.1.1 sin196 (2)

5.1.2 cos 16 (2)

5.2 Given: cos(A  B)  cosAcosB  sinAsinB

Use the formula for cos(A  B) to derive a formula for sin(A  B) (3)

1  cos 2 2A
5.3 Simplify completely, given that 0  A  90. (5)
cos( A). cos(90  A)

5.4 3
Given: cos 2B  and 0° ≤ B ≤ 90°
5

Determine, without using a calculator, the value of EACH of the following in its
simplest form:

5.4.1 cos B (3)

5.4.2 sin B (2)

5.4.3 cos (B + 45°) (4)

[21]

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Mathematics P2/Trigonometry General

DBE/2016
QUESTION 5

5.1 In the diagram PR  TS in obtuse triangle PTS.

PT = 5 ; TR = 2; PR = 1; PS = 10 and RS = 3

T 2 R 3 S

5.1.1 Write down the value of:

(a) sin T̂ (1)

(b) cos Ŝ (1)

5.1.2 Calculate, WITHOUT using a calculator, the value of cos(T̂  Ŝ) (5)

5.2 Determine the value of:

1
 tan 2 (180   )
cos(360   ). sin(90   ) (6)

3
5.3 If sin x  cos x  , calculate the value of sin 2 x WITHOUT using a calculator. (5)
4 [18]

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Mathematics P2/Trigonometry General

DBE/Feb.-Mar. 2016
QUESTION 5

5.1 P (-.fl; 3) and S(a ; b) are points on the Cartesian plane, as shown in the diagram
� �
below. POR = POS = 0 and OS = 6.

y
P(-..fi; 3

S(a; b)

Determine, WITHOUT using a calculator, the value of:

5.1.1 tan 0 (1)

5.1.2 sin(-0) (3)

5.1.3 a (4)

4sinxcosx
5.2 5.2.1 Simplify to a single trigonometric ratio. (3)
2sin 2 x-1

4sinl5° cosl5 °
5.2.2 Hence, calculate the value of ------ WITHOUT using a
2 sin 2 15 ° -1
calculator. (Leave your answer in simplest surd form.) (2)
[13]

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Mathematics P2/Trigonometry General

QUESTION 5 DBE/November 2015

5.1 Given that sin 23° = k , determine, in its simplest form, the value of each of the
following in terms of k, WITHOUT using a calculator:

5.1.1 sin 203° (2)

5.1.2 cos 23° (3)

5.1.3 tan(–23°) (2)

5.2 Simplify the following expression to a single trigonometric function:

4 cos(− x). cos(90° + x)

sin(30° − x). cos x + cos(30° − x). sin x (6)

5.3 Determine the general solution of cos 2 x − 7 cos x − 3 = 0 . (6)

1
5.4 Given that sin θ = , calculate the numerical value of sin 3θ , WITHOUT using
3
a calculator. (5)
[24]

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Mathematics P2/Trigonometry General

DBE/2015
QUESTION 5

5.1 Given that cos p = - Js, where 180 ° < p < 360 °.

Determine, with the aid of a sketch and without using a calculator, the value of sin p. (5)

5.2 Determine the value of the following expression:

tan(l 80° - x). sin(x - 90°)

4 sin(36 0° + x) (6)

5.3 If sin A = p and cos A = q:

5.3.1 Write tan A in terms of p and q (1 )

5.3.2 Simplify p 4 -q 4 to a single trigonometric ratio (4)

cos 0 cos 20
5.4 Consider the identity: -- - = tan 0
sin 0 sin 0. cos 0

5.4.1 Prove the identity. (5)

5.4.2 For which value(s) of 0 in the interval 0 ° < 0 < 180 ° will the identity
be undefined? (2)

5.5 Determine the general solution of 2 sin 2x + 3 sin x = 0 (6)

[29)

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Mathematics P2/Trigonometry General

DBE/Feb.-Mar. 2015
QUESTION 5

If x = 3 sin 0 and y = 3 cos 0, determine the value of x 2 + y 2

•
5.1 (3)

5.2 Simplify to a single term:

sin(54<J'- x).sin(-x)-cos(l 8<J'- x).sin(90° + x) (6)

5.3 In the diagram below, T(x ; p) is a point in the third quadrant and it is given that
. p
sma= .
�
-vl + p 2

T(x ;p)

5.3.1 Show that x = -1. (3)

5.3.2 Write cos (l 8<J' +a) in terms of p in its simplest form. (2)

1-p 2
5.3.3 Show that cos 2a can be written as . (3)
I+ p2

2 tanx-sin2x
5. 4 5.4.1 For which value(s) of x will be undefined in the
2 sin 2 x
interval 0 ° :S x :S 180 ° ? (3)

2 tanx-sin2x
5.4.2 Prove the identity: -----= tanx
2sin 2 x (6)
[26)

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Mathematics P2/Trigonometry General

DBE/November 2014
QUESTION 6

6.1 Prove the identity: cos2 (180  x) tan(x 180)sin(720  x) cos x  cos 2x (5)

6.2 Use cos(   )  cos  cos   sin  sin  to derive the formula for sin(   ). (3)

6.3 If sin 76° = x and cos 76° = y, show that x 2  y 2 = sin 62°. (4)
[12]

NSS – Graad 12 Model – Memorandum DBE/2014

QUESTION 5

4
5.1 Given that sin α = –
and 90° < α < 270°.
5
WITHOUT using a calculator, determine the value of each of the following in its
simplest form:

5.1.1 sin (–α) (2)

5.1.2 cos α (2)

5.1.3 sin (α – 45°) (3)

8 sin(180° − x) cos( x − 360°)

5.2 Consider the identity: = −4 tan 2 x
sin 2 x − sin 2 (90° + x)

5.2.1 Prove the identity. (6)

5.2.2 For which value(s) of x in the interval 0° < x < 180° will the identity be
undefined? (2)

5.3 Determine the general solution of cos 2θ + 4 sin 2 θ − 5 sin θ − 4 = 0 . (7)

[22]

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 TRIGONOMETRY
Graphs

Index
Content Page
1. 2016 November Paper 2 43

2. 2016 June Paper 2 44

3. 2016 Feb-March Paper 2 45

4. 2015 November Paper 2 46

5. 2015 June Paper 2 47

6. 2015 Feb-March Paper 2

7. 2014 November Paper 2 48

8. 2014 Exemplar Paper 2 49

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Mathematics P2/Trigonometry Graphs

DBE/November 2016
QUESTION 6

In the diagram the graph of f ( x )  2 sin 2 x is drawn for the interval x  [–180° ; 180°].

f
1

x
–180° –90° 0° 90° 180°

–1

–2

6.1 On the system of axes on which f is drawn in the ANSWER BOOK, draw the graph
of g ( x )   cos 2 x for x  [–180° ; 180°]. Clearly show all intercepts with the axes,
the coordinates of the turning points and end points of the graph. (3)

6.2 Write down the maximum value of f ( x)  3. (2)

6.3 Determine the general solution of f ( x )  g ( x ). (4)

6.4 Hence, determine the values of x for which f ( x )  g ( x ) in the interval

x  [–180° ; 0°]. (3)
[12]

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Mathematics P2/Trigonometry Graphs

DBE/2016
QUESTION 6

6.1 Determine the general solution of 4 sin x  2 cos 2 x  2 (6)

6.2 The graph of g ( x)   cos 2 x for x  [180  ; 180 ] is drawn below.

y
2

g x
–180 o
–90o
O 90o 180o

–1

–2

–3

6.2.1 Draw the graph of f ( x)  2 sin x  1 for x  [180  ; 180 ] on the set of
axes provided in the ANSWER BOOK. (3)

6.2.2 Write down the values of x for which g is strictly decreasing in the
interval x  [180  ; 0] (2)

6.2.3 Write down the value(s) of x for which f ( x  30 )  g ( x  30 )  0 for

x  [180  ; 180 ] (2)
[13]

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Mathematics P2/Trigonometry Graphs

DBE/Feb.-Mar. 2016
QUESTION6

Given the equation: sin(x + 60 °) + 2cos x = 0

6.1 Show that the equation can be rewritten as tan x = - 4 - J?,. (4)

6.2 Determine the solutions of the equation sin(x + 60°) + 2cos x = 0 in the interval
-180° :Sx :S 180 °. (3)

6.3 In the diagram below, the graph of f(x) =-2 cos x is drawn for -120 ° :Sx :S 240°.

-60° 60° 120 ° 240°

-2

6.3.1 Draw the graph of g(x) = sin(x + 60 °) for -120 ° :S x :S 240 ° on the grid
provided in the ANSWER BOOK. (3)

6.3.2 Determine the values of x in the interval -120° :S x :S 240° for which
sin(x + 60 °) + 2cos x > 0. (3)
[13)

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Mathematics P2/Trigonometry Graphs

DBE/November 2015

QUESTION 6

In the diagram below, the graphs of f ( x) = cos x + q and g ( x) = sin ( x + p ) are drawn on
 1
the same system of axes for –240° ≤ x ≤ 240°. The graphs intersect at  0° ;  , (–120° ; –1)
 2
and (240° ; –1).

1
2
g

x
–240° –180° –120° –60° 0° 60° 120° 180° 240°

–1

–1 12
f

6.1 Determine the values of p and q. (4)

6.2 Determine the values of x in the interval –240° ≤ x ≤ 240° for which f (x) > g(x). (2)

6.3 Describe a transformation that the graph of g has to undergo to form the graph of h,
where h( x) = − cos x . (2)
[8]

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Mathematics P2/Trigonometry Graphs

DBE/2015
QUESTION6

In the diagram below the graphs of f (x) = sin bx and g(x) = -cos x are drawn for
-90 ° ::;; x::;; 90 ° . Use the diagram to answer the following questions.

90I °

-1

6.1 Write down the period of .f (1)

6.2 Determine the value of b. (1)

6.3 The general solutions of the equation sin bx = - cos x are x = 67,5 ° + k.90 ° or
x = 135 ° +k.180 ° where kEZ.
Determine the x-values of the points of intersection of f and g for the given
domain. (3 )

6.4 Write down the values of x for which sin bx + cos x < 0 for the given domain. (4)
[9]

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Mathematics P2/Trigonometry Graphs

DBE/November 2014
QUESTION 7

In the diagram below, the graph of f (x) = sin x + 1 is drawn for –90°  x  270°.

y
2

x
-90 -45 0 45 90 135 180 225 270

-1

-2

7.1 Write down the range of f. (2)

7.2 Show that sin x  1  cos 2 x can be rewritten as (2 sin x  1) sin x  0. (2)

7.3 Hence, or otherwise, determine the general solution of sin x  1  cos 2 x . (4)

7.4 Use the grid on DIAGRAM SHEET 2 to draw the graph of g ( x)  cos 2 x for
–90°  x  270°. (3)

7.5 Determine the value(s) of x for which f(x + 30°) = g(x + 30°) in the interval
–90°  x  270°. (3)

7.6 Consider the following geometric series:

1 + 2 cos 2x + 4 cos2 2x + ...

Use the graph of g to determine the value(s) of x in the interval 0°  x  90° for
which this series will converge. (5)
[19]

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Mathematics P2/Trigonometry Graphs

QUESTION 6

In the diagram below, the graphs of f (x) = tan bx and g(x) = cos (x – 30°) are drawn on the
same system of axes for –180° ≤ x ≤ 180°. The point P(90° ; 1) lies on f. Use the diagram to
answer the following questions.

y
f

A
P(90° ; 1)

g
x
−180° 0 180°

6.1 Determine the value of b. (1)

6.2 Write down the coordinates of A, a turning point of g. (2)

6.3 Write down the equation of the asymptote(s) of y = tan b(x + 20°) for
x ∈ [–180°; 180°]. (1)

6.4 Determine the range of h if h(x) = 2g(x) + 1. (2)

[6]
 TRIGONOMETRY
(sine, area, cosine rules)

Index

Content Page
1. 2016 November Paper 2 51

2. 2016 June Paper 2 52

3. 2016 Feb-March Paper 2 53

4. 2015 November Paper 2 54

5. 2015 June Paper 2 55

6. 2015 Feb-March Paper 2 56 - 57

7. 2014 November Paper 2 58

8. 2014 Exemplar Paper 2 59

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Mathematics P2/Trigonometry Formulae

DBE/November 2016
QUESTION 7

E is the apex of a pyramid having a square base ABCD. O is the centre of the base.
EB̂A   , AB = 3 m and EO, the perpendicular height of the pyramid, is x.

x
D

A

3
B

7.1 Calculate the length of OB. (3)

3
7.2 Show that cos  
9
2 x2  (5)
2

7.3 If the volume of the pyramid is 15 m3, calculate the value of  . (4)
[12]

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Mathematics P2/Trigonometry Formulae

DBE/2016
QUESTION 7

From point A an observer spots two boats, B and C, at anchor. The angle of depression of
boat B from A is . D is a point directly below A and is on the same horizontal plane as B
and C. BD = 64 m, AB = 81 m and AC = 87 m.

81 87

D
64

B C

7.1 Calculate the size of  to the nearest degree. (3)

7.2 If it is given that BÂC  82,6  , calculate BC, the distance between the boats. (3)

7.3 If BD̂C 110  , calculate the size of DĈB . (3)

[9]

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Mathematics P2/Trigonometry Formulae

DBE/Feb.-Mar. 2016
QUESTION 7

7.1 In the diagram below, �PQR is drawn with PQ = 20 - 4x, RQ = x and Q = 60°.

20-4x
Q
60°

7.1.1 Show that the area of �PQR = s.fix - ,J3x 2 • (2)

7.1.2 Determine the value of x for which the area of �PQR will be
a maximum. (3)

7.1.3 Calculate the length of PR if the area of �PQR is a maximum. (3)

7.2 In the diagram below, BC is a pole anchored by two cables at A and D. A, D and
C are in the same horizontal plane. The height of the pole is h and the angle of
elevation from A to the top of the pole, B, is /J. ABO= 2/J and BA= BO.

A C

Determine the distance AD between the two anchors in terms of h. (7)

[15]

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Mathematics P2/Trigonometry Formulae

DBE/November 2015

QUESTION 7

A corner of a rectangular block of wood is cut off and shown in the diagram below.
The inclined plane, that is, ∆ACD, is an isosceles triangle having AD̂C = AĈD = θ .
1
Also AĈB = θ , AC = x + 3 and CD = 2x.
2

x+3

D
θ
B θ 2x
1
2
θ

7.1 Determine an expression for CÂD in terms of θ. (1)

x
7.2 Prove that cos θ = . (4)
x+3

7.3 If it is given that x = 2, calculate AB, the height of the piece of wood. (5)
[10]

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Mathematics P2/Trigonometry Formulae

DBE/2015
QUESTION 7

Triangle PQ S forms a certain area of a park. R is a point on P S and QR divides the area of
the park into two triangular parts, as shown below, for a festive event.
PQ = PR= x units, RS =
3x
2
units and RQ = J3
x units.

3x
�--- R 2 -----
----=-
P,_ --�x
S
--:;- -::-::;;;-

7.1 Calculate the size of P . (4)

7.2 Hence, calculate the area of triangle Q RS in terms of x in its simplest form. (5)
[9]

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Mathematics P2/Trigonometry Formulae

DBE/Feb.-Mar. 2015
QUESTION6

6.1 In the figure, points K, A and F lie in the same horizontal plane and TA represents
A A A

a vertical tower. ATK = x, KAF =90 ° + x and KFA = 2x where 0 ° < x < 30 ° .
TK = 2 units.

K F

6.1.1 Express AK in terms of sin x. (2)

6.1.2 Calculate the numerical value of KF. (5)

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Mathematics P2/Trigonometry Formulae

DBE/Feb.-Mar.2015
6.2 In the diagram below, a circle with centre O passes through A, B and C.
BC = AC = 15 units. BO and OC are joined. OB = 10 units and BOC= x .

Calculate:

6.2.1 The size of x (4)

6.2.2 The size of ACB (3)

6.2.3 The area of �ABC (2 )

[16]

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Mathematics P2/Trigonometry Formulae

DBE/November 2014
QUESTION 5

In the figure below, ACP and ADP are triangles with Ĉ = 90°, CP = 4 3 , AP = 8 and DP = 4.
PA bisects DP̂C . Let CÂP  x and DÂP  y.

4 3

x 8
A P
y

5.1 Show, by calculation, that x = 60°. (2)

5.2 Calculate the length of AD. (4)

5.3 Determine y. (3)

[9]

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Mathematics P2/Trigonometry Formulae

QUESTION 7

sinA sinB
7.1 Prove that in any acute-angled ∆ABC, = . (5)
a b

7.2 The framework for a construction consists of a cyclic quadrilateral PQRS in the
horizontal plane and a vertical post TP as shown in the figure. From Q the angle of
elevation of T is y°. PQ = PS = k units, TP = 3 units and SR̂Q = 2 x°.

y
P Q

2x
S R

7.2.1 Show, giving reasons, that PŜQ = x . (2)

7.2.2 Prove that SQ = 2k cos x . (4)

6 cos x
7.2.3 Hence, prove that SQ = . (2)
tan y
[13]
MATHEMATICS/P2/GR 11_12

GEOMETRY
Grade11 Theorems

INDEX
Content Page
1. 2016 November Paper 2 63 – 65

2. 2016 June Paper 2 66 – 68

3. 2016 Feb-March Paper 2 69 – 70

4. 2015 November Paper 2 71 – 73

5. 2015 June Paper 2 74 – 76

6. 2015 Feb-March Paper 2 77 – 79

7. 2014 November Paper 2 80 – 82

8. 2014 Exemplar Paper 2 83

MATHEMATICS/P2/Geometry GR 11_12

DBE/November 2016

Give reasons for ALL statements and calculations in QUESTIONS 8, 9 and 10.

QUESTION 8

8.1 In the diagram below PQRT is a cyclic quadrilateral having RT || QP. The tangent
at P meets RT produced at S. QP = QT and PT̂ S  70 .

S
P
1
2

1
T 2
3

1
2
R Q

8.1.1 Give a reason why P̂2  70 . (1)

8.1.2 Calculate, with reasons, the size of:

(a) Q̂ 1 (3)

(b) P̂1 (2)

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MATHEMATICS/P2/Geometry GR 11_12

DBE/November 2016
8.2 A, B and C are points on the circle having centre O. S and T are points on AC
and AB respectively such that OS  AC and OT  AB . AB = 40 and AC = 48.

T
B

8.2.1 Calculate AT. (1)

7
8.2.2 If OS  OT , calculate the radius OA of the circle. (5)
15
[12]

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MATHEMATICS/P2/Geometry GR 11_12

DBE/November 2016

QUESTION 9

ABC is a tangent to the circle BFE at B. From C a straight line is drawn parallel to BF to
meet FE produced at D. EC and BD are drawn. Ê1  Ê 2  x and Ĉ 2  y.

E
x 1 3
2 D
1
x 2

A
3
1 2
4
B

2
1

9.1 Give a reason why EACH of the following is TRUE:

9.1.1 B̂1  x (1)

9.1.2 BĈD  B̂1 (1)

9.2 Prove that BCDE is a cyclic quadrilateral. (2)

9.3 Which TWO other angles are each equal to x? (2)

9.4 Prove that B̂ 2  Ĉ1 . (3)

[9]

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MATHEMATICS/P2/Geometry GR 11_12

DBE/2016
Give reasons for ALL statements in QUESTIONS 8, 9, 10 and 11.

QUESTION 8

8.1 In the diagram below, P, M, T and R are points on a circle having centre O.
PR produced meets MS at S. Radii OM and OR and the chords MT and TR
are drawn. T̂1  148°, PM̂O  18 ° and Ŝ  43 

18°
M O
1

148°
1 1
T 2
3 R

43o

Calculate, with reasons, the size of:

8.1.1 P̂ (2)

8.1.2 Ô1 (2)

8.1.3 OM̂S (2)

8.1.4 R̂ 3 if it is given that TM̂S = 6° (2)

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MATHEMATICS/P2/Geometry GR 11_12

DBE/2016

8.2 In the diagram below, the circle passes through A, B and E. ABCD is a
parallelogram. BC is a tangent to the circle at B. AE = AB. Let Ĉ1  x

A 1
2 B
1 2
3

1 2
x
3 1
D C
E 2

8.2.1 Give a reason why B̂1  x (1)

8.2.2 Name, with reasons, THREE other angles equal in size to x. (6)

8.2.3 Prove that ABED is a cyclic quadrilateral. (3)

[18]

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MATHEMATICS/P2/Geometry GR 11_12

QUESTION 9 DBE/2016

9.1 Complete the statement so that it is TRUE:

The angle between the tangent to a circle and the chord drawn from the point of
contact is equal to the angle … (1)

9.2 In the diagram below, two unequal circles intersect at A and B. AB is produced to
C such that CD is a tangent to the circle ABD at D. F and G are points on the
smaller circle such that CGF and DBF are straight lines. AD and AG are drawn.

A
1
2
D 2
1

1B
4 2
3

1 1 2
2 G F
C

Prove that:

9.2.1 B̂ 4  D̂1  D̂ 2 (4)

9.2.2 AGCD is a cyclic quadrilateral (4)

9.2.3 DC = CF (4)
[13]

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MATHEMATICS/P2/Geometry GR 11_12

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MATHEMATICS/P2/Geometry GR 11_12

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MATHEMATICS/P2/Geometry GR 11_12
NSC
DBE/November 2015
Give reasons for ALL statements in QUESTIONS 8, 9, 10 and 11.

QUESTION 8

8.1 In the diagram below, cyclic quadrilateral ABCD is drawn in the circle with centre O.

2
O D
1

B
C

8.1.1 Complete the following statement:

The angle subtended by a chord at the centre of a circle is … the angle subtended
by the same chord at the circumference of the circle. (1)

8.1.2 Use QUESTION 8.1.1 to prove that  + Ĉ = 180° . (3)

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MATHEMATICS/P2/Geometry GR 11_12

NSC DBE/November 2015

8.2 In the diagram below, CD is a common chord of the two circles. Straight lines ADE
and BCF are drawn. Chords AB and EF are drawn.

C
1
2

A
F
2
1
D

Prove that EF | | AB. (5)

[9]

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MATHEMATICS/P2/Geometry GR 11_12

NSC DBE/November 2015

QUESTION 9

In the diagram below, ∆ABC is drawn in the circle. TA and TB are tangents to the circle.
The straight line THK is parallel to AC with H on BA and K on BC. AK is drawn.
Let  3 = x .

T B
1 2 2
1

H
3 K
2
1

x
2
3
1 •S
A

9.1 Prove that K̂ 3 = x . (4)

9.2 Prove that AKBT is a cyclic quadrilateral. (2)

9.3 Prove that TK bisects AK̂B. (4)

9.4 Prove that TA is a tangent to the circle passing through the points A, K and H. (2)

9.5 S is a point in the circle such that the points A, S, K and B are concyclic.
Explain why A, S, B and T are also concyclic. (2)
[14]

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MATHEMATICS/P2/Geometry GR 11_12 DBE/2015
SCE

Give reasons for ALL statements in QUESTIONS 8, 9, 10 and 11.

QUESTIONS

In the diagram �ACD is drawn with points A and D on the circumference of a circle.
CD cuts the circle at B. P is a point on AD with CP the bisector of ACD. CP cuts the
chord AB at T. AT= AP, ATP= 65 ° and PCD= 25 °.

8.1 Determine the size of each of the following:

8.1.1 P2 (2)

8.1.2 D (2)

8.1.3 Al (2)

8.2 Is CA a tangent to the circle ABD? Motivate your answer. (2)

[8]

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MATHEMATICS/P2/Geometry GR 11_12 DBE/2015
SC£

QUESTION9

In the diagram O is the centre of the circle and BO and OD are drawn. Chords CB and DE
are produced to meet in A. Chords BE and CD are drawn. BCD = x.

9.1 Give the reason for each of the statements in the table. Complete the table provided in
the ANSWER BOOK by writing down the reason for each statement. (2)

Reason

9.2 If it is given that BE 11 CD, prove that:

9.2.1 AC=AD (4)

9.2.2 ABOD is a cyclic quadrilateral (3)

[9]

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MATHEMATICS/P2/Geometry GR 11_12 DBE/2015
SCE

QUESTION 11

11.1 In the diagram O is the centre of the circle and PA is a tangent to the circle at A.
B and C are points on the circumference of the circle.

A A

Use the diagram to prove the theorem that states that BAP = ACB. (6)

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MATHEMATICS/P2/Geometry GR 11_12

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 DBE/Feb.-Mar. 2015
MATHEMATICS/P2/Geometry GR 11_12
NSC

QUESTIONS

In the diagram below, the circle with centre O passes through A, B, C and D.
AB I DC and BOC= 110 ° .
The chords AC and BO intersect at E.
EO, BO, CO and BC are joined.

8. I Calculate the size of the following angles, giving reasons for your answers:

8.1.1 D (2)

8.1.2 A (2)

8.1.3 E2 ( 4)

8.2 Prove that BEOC is a cyclic quadrilateral. (2)

[10]

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 DBE/Feb.-Mar. 2015
MATHEMATICS/P2/Geometry GR 11_12
NSC

QUESTION9

9.1 Complete the statement of the following theorem:

The exterior angle of a cyclic quadrilateral is equal to . . . (1)

9.2 In the diagram below the circle with centre O passes through points S, T and V.
PR is a tangent to the circle at T. VS, ST and VT are joined.
C
''

p T R

A A

Given below is the partially c.:ompleted proof of the theorem that states that VTR =S.
Using the above diagram, complete the proof of the theorem on
DIAGRAM SHEET 3.

Construction: Draw diameter TC and join CV.

Statement Reason
Let: VTR = T1 = x
A A

V1 + V2 = ......... .

T 2 = 90 ° - X

:. C = ......... . Sum of the angles of a triangle

:. S =x
A A

:. VTR=S
(5)

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MATHEMATICS/P2/Geometry GR 11_12

DBE/November 2014
GIVE REASONS FOR YOUR STATEMENTS IN QUESTIONS 8, 9 AND 10.

QUESTION 8

8.1 In the diagram, O is the centre of the circle passing through A, B and C.
CÂB = 48°, CÔB  x and Ĉ 2  y .
A

48°

1 x
C 2 y
1
2
B

Determine, with reasons, the size of:

8.1.1 x (2)

8.1.2 y (2)

8.2 In the diagram, O is the centre of the circle passing through A, B, C and D.
AOD is a straight line and F is the midpoint of chord CD. OD̂F  30 and OF are
joined.

B O

1 30°
C F D

Determine, with reasons, the size of:

8.2.1 F̂1 (2)

8.2.2 AB̂C (2)

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MATHEMATICS/P2/Geometry GR 11_12

DBE/November 2014
8.3 In the diagram, AB and AE are tangents to the circle at B and E respectively.
BC is a diameter of the circle. AC = 13, AE = x and BC = x + 7.

A
x
E

13

x+7

8.3.1 Give reasons for the statements below.

Complete the table on DIAGRAM SHEET 3.

Statement Reason

(a) AB̂C  90

(b) AB  x
(2)

8.3.2 Calculate the length of AB. (4)

[14]

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MATHEMATICS/P2/Geometry GR 11_12

DBE/November 2014
QUESTION 10

The two circles in the diagram have a common tangent XRY at R. W is any point on the
small circle. The straight line RWS meets the large circle at S. The chord STQ is a tangent
to the small circle, where T is the point of contact. Chord RTP is drawn.
Let R̂ 4  x and R̂ 2  y
P
1 2

S 1
2
1
2 T
3 4
X 1
W 2
1
x 2 Q
y 3
4
3
2
R 1

10.1 Give reasons for the statements below.

Complete the table on DIAGRAM SHEET 6.

Let R̂ 4  x and R̂ 2  y
Statement Reason
10.1.1 T̂3 = x
10.1.2 P̂1 = x

10.1.3 WT | | SP

10.1.4 Ŝ1 = y

10.1.5 T̂2 = y
(5)

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 NSC – Grade 12 Exemplar
Give reasons for your statements in QUESTIONS 8, 9 and 10. DBE/2014

QUESTION 8

8.1 Complete the following statement:

The angle between the tangent and the chord at the point of contact is equal to ... (1)

8.2 In the diagram, A, B, C, D and E are points on the circumference of the circle such
that AE | | BC. BE and CD produced meet in F. GBH is a tangent to the circle at B.
B̂1 = 68° and F̂ = 20°.

A F

20°
1 E
2 3
68°
1 2
B
3 1
4 2 D

Determine the size of each of the following:

8.2.1 Ê (2)
1

8.2.2 B̂3 (1)

8.2.3 D̂1 (2)

8.2.4 Ê 2 (1)

8.2.5 Ĉ (2)
[9]

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 GEOMETRY
Grade 12 Theorems

Index

Content Page
1. 2016 November Paper 2 85

2. 2016 June Paper 2 86 – 87

3. 2016 Feb-March Paper 2 88 – 90

4. 2015 November Paper 2 91 – 92

5. 2015 June Paper 2 93 – 94

6. 2015 Feb-March Paper 2 95 – 96

7. 2014 November Paper 2 97 – 100

8. 2014 Exemplar Paper 2 101 – 102

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MATHEMATICS/P2/Geometry GR 12

DBE/November 2016

10.2 In the diagram HLKF is a cyclic quadrilateral. The chords HL and FK are produced
to meet at M. The line through F parallel to KL meets MH produced at G.
MK = x, KF = 2x, ML = y and LH = HG.

M
y
L
1 2
x
1 H
1 2 2 3
K 3
G

2x

2 1

10.2.1 Give a reason why GF̂M  LK̂M . (1)

10.2.2 Prove that:

(a) =
GH y (3)

(b) MFH | | | MGF (5)

(c) GF 3 x

FH 2 y (2)

y 3
10.2.3 Show that 
x 2 (3)
[20]

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MATHEMATICS/P2/Geometry GR 12

QUESTION 10 DBE/2016

10.1 In the diagram below, MVT and AKF are drawn such that M̂  Â, V̂  K̂ and T̂  F̂

M A


K F


V T

Use the diagram in the ANSWER BOOK to prove the theorem which states that if
two triangles are equiangular, then the corresponding sides are in proportion,
MV MT
that is 
AK AF (7)

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DBE/2016
10.2 In the diagram below, cyclic quadrilateral EFGH is drawn. Chord EH produced and
chord FG produced meet at K. M is a point on EF such that MG | | EK.
Also KG = EF

M 2 1 H
1 2

F 1 2
3
G
K

10.2.1 Prove that:

(a) KGH | | | KEF (4)

(b) EF2 = KE . GH (2)

(c) KG2 = EM . KF (3)

10.2.2 If it is given that KE = 20 units, KF = 16 units and GH = 4 units, calculate

the length of EM. (3)
[19]

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 DBE/Feb.-Mar. 2016
MATHEMATICS/P2/Geometry GR 12 NSC

QUESTION9

In the diagram below, EO bisects side AC of LlACE. EDO is produced to B such that
BO = OD. AD and CD produced meet EC and EA at G and F respectively.

9.1 Give a reason why ABCD is a parallelogram. (1)

ED
9.2 Write down, with reasons, TWO ratios each equal to (4)
DB
� �
9.3 Prove that A 1 = F2• (5)

9.4 It is further given that ABCD is a rhombus. Prove that ACGF 1s a cyclic
quadrilateral. (3)
[13)

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 DBE/Feb.-Mar. 2016
MATHEMATICS/P2/Geometry GR 12 NSC

QUESTION 10

10.1 In the diagram below, �ABC and �PQR are given with A= P, 13 = Q and C = R.

A p

D ----------------------
Q R

B C

DE is drawn such that AD = PQ and AE = PR.

10.1.1 Prove that MDE = �PQR. (2)

10.1.2 Prove that DE 11 BC. (3)

AB AC
10.1.3 Hence, prove that - = - (2)
PQ PR

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 DBE/Feb.-Mar. 2016
MATHEMATICS/P2/Geometry GR 12 NSC

10.2 In the diagram below, VR is a diameter of a circle with centre 0. S is any point on
the circumference. P is the midpoint of RS. The circle with RS as diameter cuts
VR at T. ST, OP and SV are drawn.

I 0.2.1 Why is OP 1- PS? (1)

10.2.2 Prove that �ROP 111 �RVS. (4)

10.2.3 Prove that �RVS 111 �RST. (3 )

10.2.4 Prove that ST2 =VT . TR. (6)

[21]

TOTAL: 150

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MATHEMATICS/P2/Geometry GR 12

NSC DBE/November 2015

QUESTION 10

In the diagram below, BC = 17 units, where BC is a diameter of the circle. The length of
chord BD is 8 units. The tangent at B meets CD produced at A.

B
17

E
8
C
F

D
A

10.1 Calculate, with reasons, the length of DC. (3)

10.2 E is a point on BC such that BE : EC = 3 : 1. EF is parallel to BD with

F on DC.

10.2.1 Calculate, with reasons, the length of CF. (3)

10.2.2 Prove that ∆BAC | | | ∆FEC. (5)

10.2.3 Calculate the length of AC. (4)

10.2.4 Write down, giving reasons, the radius of the circle passing through points
A, B and C. (2)
[17]

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MATHEMATICS/P2/Geometry GR 12

NSC DBE/November 2015

QUESTION 11

11.1 Complete the following statement:

If the sides of two triangles are in the same proportion, then the triangles are ... (1)

11.2 In the diagram below, K, M and N respectively are points on sides PQ, PR and
QR of ∆PQR. KP = 1,5; PM = 2; KM = 2,5; MN = 1; MR = 1,25 and
NR = 0,75.

P
2
1,5
M
2,5 1,25
K
1
R
0,75
N

11.2.1 Prove that ∆KPM | | | ∆RNM. (3)

11.2.2 Determine the length of NQ. (6)

[10]

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 DBE/2015
MATHEMATICS/P2/Geometry GR 12 SCE

QUESTION 10

10.1 Complete the following statement of the theorem in the ANSWER BOOK:

If a line divides two sides of a triangle in the same proportion, then . . . ( 1)

10.2 In the diagram ABC is a triangle with F on AB and E on AC. BC II FE.

AD 3
D is on AF with - = - AE = 12 units and EC = 8 units.
AF 5

AL-------�12c--------�E=----.,8 - - C
� - �

10.2.1 Prove that DE 11 FC. (3)

10.2.2 If AB= 14 units, calculate the length of BF. (3)

[7]

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 DBE/2015
MATHEMATICS/P2/Geometry GR 12

11.2 In the diagram C is the centre of the circle DAP. BA is a tangent to the circle
at A. CD is produced to meet the tangent to the circle at B. DP and DA are
drawn. E is a point on BA such that EC bisects DCA . Let C 1 = x .

11.2.1 Prove that .-1BAD 111 .1BCE. (7)

11.2.2 If it is also given that AB= 8 units and AC= 6 units, calculate:

(a) The length of BD (5)

(b) The length of BE (3)

(c) The size of x (3)

[24)

TOTAL: 150

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 DBE/Feb.-Mar. 2015
MATHEMATICS/P2/Geometry GR 12 NSC

9.3 In the figure, TRSW is a cyclic quadrilateral with TW = WS. RT and RS are
produced to meet tangent VWZ at V and Z respectively. PRQ is a tangent to the
circle at R. RW is joined. R 2 = 30 ° and R 4 = 50 ° .

9.3.1 Give a reason why R. 3 = 30 °. (1)

9.3.2 State, with reasons, TWO other angles equal to 30 ° . (3)

9.3.3 Determine, with reasons, the size of:

(a) (3)

(b) V (4)

9.3.4 Prove that WR 2 = RV x RS. (5)

(22)

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 DBE/Feb.-Mar. 2015
MATHEMATICS/P2/Geometry GR 12 NSC

QUESTION 10

In b. TRM, M = 90 °. NP is drawn parallel to TR with N on TM and P on RM. It is

further given that RT= 3PN.

R p M

l 0.1 Give reasons for the statements below.

Use DIAGRAM SHEET S.

Statement Reason

In t-.PNM and t-.RTM

l 0.1.1 N I =T ········································································

Mis common

l 0.1.2 :. t-.PNM Ill t-.RTM ........................................................................

(2)

PM 1
10.2 Prove that - - - (2)
RM 3

10.3 Show that RN2 - PN2 = 2RP2 • (4)

[8]

TOTAL: 150

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MATHEMATICS/P2/Geometry GR 12

QUESTION 9 NSC DBE/November 2014

9.1 In the diagram, points D and E lie on sides AB and AC of ABC respectively
such that DE | | BC. DC and BE are joined.
A

k h
D 1
1
E

9.1.1 Explain why the areas of DEB and DEC are equal. (1)

9.1.2 Given below is the partially completed proof of the theorem that states
AD AE
that if in any ABC the line DE | | BC then  .
DB EC
Using the above diagram, complete the proof of the theorem on
DIAGRAM SHEET 4.

Construction: Construct the altitudes (heights) h and k in ADE .

area ΔADE
1
ADh
 2  ........
area DEB 1
BDh
2

area ΔADE AE
 ...................... 
area DEC EC

But areaDEB = .............................. (reason: .................................)

area ΔADE
  ...............................
area DEB
AD AE
 
DB EC
(5)

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MATHEMATICS/P2/Geometry GR 12

NSC DBE/November 2014

9.2 In the diagram, ABCD is a parallelogram. The diagonals of ABCD intersect in M.
F is a point on AD such that AF : FD = 4 : 3. E is a point on AM such that
EF | | BD. FC and MD intersect in G.

A F D

G
E

B C

Calculate, giving reasons, the ratio of:

EM
9.2.1 (3)
AM

CM
9.2.2 (3)
ME

area FDC
9.2.3
area BDC (4)
[16]

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MATHEMATICS/P2/Geometry GR 12

NSC DBE/November 2014

QUESTION 10

The two circles in the diagram have a common tangent XRY at R. W is any point on the
small circle. The straight line RWS meets the large circle at S. The chord STQ is a tangent
to the small circle, where T is the point of contact. Chord RTP is drawn.
Let R̂ 4  x and R̂ 2  y
P
1 2

S 1
2
1
2 T
3 4
X 1
W 2
1
x 2 Q
y 3
4
3
2
R 1

10.1 Give reasons for the statements below.

Complete the table on DIAGRAM SHEET 6.

Let R̂ 4  x and R̂ 2  y
Statement Reason
10.1.1 T̂3 = x
10.1.2 P̂1 = x

10.1.3 WT | | SP

10.1.4 Ŝ1 = y

10.1.5 T̂2 = y
(5)

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MATHEMATICS/P2/Geometry GR 12 NSC DBE/November 2014

WR.RP
10.2 Prove that RT  (2)
RS

10.3 Identify, with reasons, another TWO angles equal to y. (4)

10.4 Prove that Q̂ 3  Ŵ2 . (3)

10.5 Prove that RTS | | | RQP. (3)

WR RS 2
10.6 Hence, prove that  .
RQ RP 2 (3)
[20]

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MATHEMATICS/P2/Geometry GR 12
NSC – Grade 12 Exemplar DBE/2014
QUESTION 9

In the diagram, M is the centre of the circle and diameter AB is produced to C. ME is drawn
perpendicular to AC such that CDE is a tangent to the circle at D. ME and chord AD intersect
at F. MB = 2BC.

M B
A C
3 1 2 1
2
1
2 x
F 3 2 3 4

1
D

9.1 If D̂ = x, write down, with reasons, TWO other angles each equal to x. (3)
4

9.2 Prove that CM is a tangent at M to the circle passing through M, E and D. (4)

9.3 Prove that FMBD is a cyclic quadrilateral. (3)

9.4 Prove that DC2 = 5BC2. (3)

9.5 Prove that ∆DBC | | | ∆DFM. (4)

DM
9.6 Hence, determine the value of . (2)
FM [19]

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MATHEMATICS/P2/Geometry GR 12
NSC – Grade 12 Exemplar DBE/2014
QUESTION 10

10.1 In the diagram, points D and E lie on sides AB and AC respectively of ∆ABC such
that DE | | BC. Use Euclidean Geometry methods to prove the theorem which states that
AD AE
= .
DB EC

D E

B C (6)

10.2 In the diagram, ADE is a triangle having BC | | ED and AE | | GF. It is also given that
AB : BE = 1 : 3, AC = 3 units, EF = 6 units, FD = 3 units and CG = x units.

A
3

B C
x

E
6 F 3 D

Calculate, giving reasons:

10.2.1 The length of CD (3)

10.2.2 The value of x (4)

10.2.3 The length of BC (5)

area ΔABC
10.2.4 The value of
area ΔGFD (5)
[23]

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 NATIONAL
SENIOR CERTIFICATE

GRADE 12

MATH.2

MATHEMATICS P2

FEBRUARY/MARCH 2018

MARKS: 150

TIME: 3 hours

This question paper consists of 13 pages, 1 information sheet

and an answer book of 27 pages.

MORNING SESSION

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Mathematics/P2 DBE/Feb.–Mar. 2018
NSC

INSTRUCTIONS AND INFORMATION

Read the following instructions carefully before answering the questions.

1. This question paper consists of 10 questions.

2. Answer ALL the questions in the ANSWER BOOK provided.

3. Clearly show ALL calculations, diagrams, graphs, et cetera that you have used in
determining your answers.

4. Answers only will NOT necessarily be awarded full marks.

5. You may use an approved scientific calculator (non-programmable and

non-graphical), unless stated otherwise.

6. If necessary, round off answers to TWO decimal places, unless stated otherwise.

7. Diagrams are NOT necessarily drawn to scale.

8. An information sheet with formulae is included at the end of the question paper.

9. Write neatly and legibly.

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 NATIONAL
SENIOR CERTIFICATE

GRADE 12

MATH.2

MATHEMATICS P2

FEBRUARY/MARCH 2018

MARKS: 150

TIME: 3 hours

This question paper consists of 13 pages, 1 information sheet

and an answer book of 27 pages.

MORNING SESSION

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Mathematics/P2 DBE/Feb.–Mar. 2018
NSC

INSTRUCTIONS AND INFORMATION

Read the following instructions carefully before answering the questions.

1. This question paper consists of 10 questions.

2. Answer ALL the questions in the ANSWER BOOK provided.

3. Clearly show ALL calculations, diagrams, graphs, et cetera that you have used in
determining your answers.

4. Answers only will NOT necessarily be awarded full marks.

5. You may use an approved scientific calculator (non-programmable and

non-graphical), unless stated otherwise.

6. If necessary, round off answers to TWO decimal places, unless stated otherwise.

7. Diagrams are NOT necessarily drawn to scale.

8. An information sheet with formulae is included at the end of the question paper.

9. Write neatly and legibly.

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Mathematics/P2 DBE/Feb.–Mar. 2018
NSC

QUESTION 1

An organisation decided that it would set up blood donor clinics at various colleges. Students
would donate blood over a period of 10 days. The number of units of blood donated per day by
students of college X is shown in the table below.

DAYS 1 2 3 4 5 6 7 8 9 10
UNITS OF
45 59 65 73 79 82 91 99 101 106
BLOOD

1.1 Calculate:

1.1.1 The mean of the units of blood donated per day over the period of 10 days (2)

1.1.2 The standard deviation of the data (2)

1.1.3 How many days is the number of units of blood donated at college X
outside one standard deviation from the mean? (3)

1.2 The number of units of blood donated by the students of college X is represented in
the box and whisker diagram below.

A B

1.2.1 Describe the skewness of the data. (1)

1.2.2 Write down the values of A and B, the lower quartile and the upper
quartile respectively, of the data set. (2)

1.3 It was discovered that there was an error in counting the number of units of blood
donated by college X each day. The correct mean of the data is 95 units of blood.
How many units of blood were NOT counted over the ten days? (1)
[11]

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QUESTION 2

The table below shows the number of hours that a sales representative of a company spent with
each of his nine clients in one year and the value of the sales (in thousands of rands) for that
client.

NUMBER OF HOURS 30 50 80 100 120 150 190 220 260

VALUE OF SALES
270 275 376 100 420 602 684 800 820
(IN THOUSANDS OF RANDS)

SCATTER PLOT
900
Value of sales (in thousands of rands)

800

700

600

500

400

300

200

100

0
0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300

Number of hours

2.1 Identify an outlier in the data above. (1)

2.2 Calculate the equation of the least squares regression line of the data. (3)

2.3 The sales representative forgot to record the sales of one of his clients. Predict the
value of this client's sales (in thousands of rands) if he spent 240 hours with him
during the year. (2)

2.4 What is the expected increase in sales for EACH additional hour spent with a client? (2)
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QUESTION 3

In the diagram, P, Q(–7 ; –2), R and S(3 ; 6) are vertices of a quadrilateral. R is a point on
the x-axis. QR is produced to N such that QR = 2RN. SN is drawn. PT̂O  71,57 and
SR̂N   .

P y

S(3 ; 6)

N
T 71,57° x
O R

Q(–7 ; –2)

Determine:

3.1 The equation of SR (1)

3.2 The gradient of QP to the nearest integer (2)

3.3 The equation of QP in the form y = mx + c (2)

3.4 The length of QR. Leave your answer in surd form. (2)

3.5 tan(90   ) (3)

3.6 The area of RSN , without using a calculator (6)

[16]

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QUESTION 4

In the diagram, PKT is a common tangent to both circles at K(a ; b). The centres of both
1
circles lie on the line y = x. The equation of the circle centred at O is x 2  y 2  180 .
2
The radius of the circle is three times that of the circle centred at M.

x
O

K(a ; b)
M

4.1 Write down the length of OK in surd form. (1)

4.2 Show that K is the point (–12 ; –6). (4)

4.3 Determine:

4.3.1 The equation of the common tangent, PKT, in the form y = mx + c (3)

4.3.2 The coordinates of M (6)

4.3.3 The equation of the smaller circle in the form ( x  a) 2  ( y  b) 2  r 2 (2)

4.4 For which value(s) of r will another circle, with equation x 2  y 2  r 2 , intersect the
circle centred at M at two distinct points? (3)

4.5 Another circle, x 2  y 2  32 x  16 y  240  0 , is drawn. Prove by calculation that

this circle does NOT cut the circle with centre M(–16 ; –8). (5)
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QUESTION 5

5
5.1 If cos 2   , where 2  [180 ; 270] , calculate, without using a calculator,
6
the values in simplest form of:

5.1.1 sin 2 (4)

5.1.2 sin 2  (3)

5.2 Simplify sin(180   x ). cos(  x )  cos(90  x ). cos( x  180 ) to a single trigonometric

ratio. (6)

5.3 Determine the value of sin 3 x. cos y  cos 3 x. sin y if 3 x  y  270 . (2)

5.4 Given: 2 cos x  3 tan x

5.4.1 Show that the equation can be rewritten as 2 sin 2 x  3 sin x  2  0. (3)

5.4.2 Determine the general solution of x if 2 cos x  3 tan x. (5)

5.4.3 Hence, determine two values of y, 144° ≤ y ≤ 216°, that are solutions of
2 cos 5 y  3 tan 5 y. (4)

5.5 Consider: g ( x)  4 cos( x  30)

5.5.1 Write down the maximum value of g(x). (1)

5.5.2 Determine the range of g(x) + 1. (2)

5.5.3 The graph of g is shifted 60° to the left and then reflected about the
x-axis to form a new graph h. Determine the equation of h in its
simplest form. (3)
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QUESTION 6

PQ and AB are two vertical towers.

From a point R in the same horizontal plane as Q and B, the angles of elevation to P and A
are θ and 2θ respectively.
AQ̂R  90  θ , QÂR  θ and QR  x .


P

90° + 

x B

2

6.1 Determine in terms of x and  :

6.1.1 QP (2)

6.1.2 AR (2)

6.2 Show that AB  2 x cos 2  (4)

AB
6.3 Determine if   12. (2)
QP
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QUESTION 7

In the diagram, PQRT is a cyclic quadrilateral in a circle such that PT = TR. PT and QR are
produced to meet in S. TQ is drawn. SQ̂P  70

T 3
1 2

1
2
Q
1 2
R

7.1 Calculate, with reasons, the size of:

7.1.1 T̂1 (2)

7.1.2 Q̂1 (2)

7.2 If it is further given that PQ || TR:

7.2.1 Calculate, with reasons, the size of T̂2 (2)

TR RQ
7.2.2 Prove that 
TS RS (2)
[8]

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QUESTION 8

In the diagram, PR is a diameter of the circle with centre O. ST is a tangent to the circle
at T and meets RP produced at S. S P̂ T  x and Ŝ  y.

x
y
S P O R

Determine, with reasons, y in terms of x. [6]

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QUESTION 9

In the diagram, DEFG is a quadrilateral with DE = 45 and GF = 80. The diagonals GE and
DF meet in H. GD̂E  FÊG and DĜE  EF̂G .

45

G E
H

80

9.1 Give a reason why DEG ||| EGF. (1)

9.2 Calculate the length of GE. (3)

9.3 Prove that DEH ||| FGH. (3)

9.4 Hence, calculate the length of GH. (3)

[10]

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QUESTION 10

10.1 In the diagram, O is the centre of the circle with A, B and C drawn on the circle.

Prove the theorem which states that BÔC  2Â. (5)

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10.2 In the diagram, the circle with centre F is drawn. Points A, B, C and D lie on the
circle. Chords AC and BD intersect at E such that EC = ED. K is the midpoint
of chord BD. FK, AB, CD, AF, FE and FD are drawn. Let B̂  x .

K
F
3
1 2
C
1
2 E4
3
1
2
3
A 2 3
1
D

10.2.1 Determine, with reasons, the size of EACH of the following in

terms of x:

(a) F̂1 (2)

(b) Ĉ (2)

10.2.2 Prove, with reasons, that AFED is a cyclic quadrilateral. (4)

10.2.3 Prove, with reasons, that F̂3  x . (6)

AE
10.2.4 If area AEB = 6,25 × area DEC, calculate . (5)
ED
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TOTAL: 150

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INFORMATION SHEET: MATHEMATICS

 b  b 2  4ac
x
2a
A  P (1  ni ) A  P (1  ni ) A  P(1  i) n A  P(1  i) n
n
Tn  a  (n  1)d Sn  2a  (n  1)d 
2
Tn  ar n1 a r n  1 ; r  1 S 
a
; 1  r  1
Sn 
r 1 1 r

F

x 1  i   1
n
 P
x[1  (1  i )  n ]
i i
f ( x  h)  f ( x )
f ' ( x)  lim
h 0 h
 x  x y  y2 
d  ( x 2  x1 ) 2  ( y 2  y1 ) 2 M  1 2 ; 1 
 2 2 
y  y1
y  mx  c y  y1  m( x  x1 ) m 2 m  tan
x 2  x1
 x  a 2   y  b 2  r 2
a b c
InABC:  
sin A sin B sin C
a 2  b 2  c 2  2bc. cos A
1
area ΔABC  ab. sin C
2
sin     sin.cos  cos.sin  sin     sin.cos  cos.sin 

cos     cos. cos  sin.sin  cos     cos.cos  sin.sin 

cos 2   sin 2 

cos 2  1  2 sin 2  sin 2  2 sin . cos

2 cos   1
2

n 2

x  xi  x 
x 2  i 1
n n
n( A )
P(A)  P(A or B) = P(A) + P(B) – P(A and B)
nS

yˆ  a  bx b
 x  x ( y  y)
(x  x) 2

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