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WYNBERG BOYS’ HIGH SCHOOL
Department of Mathematics
End of year examination
MARKS: 150
TIME: 3 hours
EXAMINER: Hu
MODERATOR: Vz
This question paper consists of 16 printed pages, including this cover page.
MATHEMATICS P2
11 NOVEMBER 2019
MORNING SESSION
GRADE 11
11 Mathematics/P2 Page 2 of 16 WBHS/Nov 2019
INSTRUCTIONS AND INFORMATION
Read the following instructions carefully before answering the questions.
1. This question paper consists of 12 questions.
2. Answer ALL the questions in BLUE or BLACK ink only. Graphs must be drawn in
pencil when required.
3. Circle your teacher’s monogram on your answer book.
4. Clearly show ALL calculations, diagrams, graphs, etc. that you have used in determining
your answers. Use the ADDITONAL SPACES in your ANSWER BOOK if you need to.
5. In QUESTIONS 3 – 12, you are required to provide reasons for statements you make
arising from your use of any/all known geometry.
6. Answers only will not necessarily be awarded full marks.
7. You may use an approved scientific calculator (non-programmable and non-graphical) unless
stated otherwise.
8. If necessary, round off answers to TWO decimal places, unless stated otherwise.
9. Number the answers correctly according to the numbering system used in this question paper.
10. Write neatly and legibly.
11. The mark allocation by topic is as follows:
Data
Handling/
Statistics
Analytical
Geometry Trigonometry Measure Euclidean Geometry
Question 1 2 3 4 5 6 7 8 9 10 11 12
Maximum 7 15 19 11 29 9 16 5 9 14 10 6
11 Mathematics/P2 Page 3 of 16 WBHS/Nov 2019
QUESTION 1
A tyre manufacturer recorded how far a selection of minibus taxis travel before they needed new
tyres. The distances, in 1000’s of kilometres, covered by taxis that travelled the same route are
shown in the cumulative frequency graph (ogive) below.
1.1. How many taxis were sampled? (1)
1.2. Write down the modal class of the data. (1)
1.3. Estimate the median number of kilometres travelled before new tyres were required. (1)
1.4. Estimate the inter-quartile range. (3)
1.5. Provide a sensible reason why some taxis required a tyre change far sooner than others. (1)
[7]
Cumulative frequency curve showing the distances travelled by minibus taxis
before they needed new tyres
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
100
105
110
0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72
Distance travelled in 1000’s of kilometres
Cum
ula
tive
freq
uen
cy
11 Mathematics/P2 Page 4 of 16 WBHS/Nov 2019
QUESTION 2
2.1. The table below shows the distances, sorted by stage length in kilometres, of the 21
stages of the 106th Tour de France, held during July 2019 (https://www.letour.fr/en/).
Stage Dist. (km) Stage Dist. (km)
13 27,2 15 185
2 27,6 1 194,5
14 117,5 8 200,0
19 126,5 17 200
21 128 18 208
20 130 12 209,5
6 160,5 4 213,5
11 167,0 3 215,0
9 170,5 10 217,5
5 175,5 7 230,0
16 177
2.1.1. Calculate the mean stage distance. (2)
2.1.2. Calculate the standard deviation of the stage distances. (2)
2.1.3. The box-and-whisker for the data above is shown below. Write down the
values for each of A to E.
(4)
2.1.4. Explain why this data is negatively skewed. (1)
2.1.5. What percentage of the data lies between the values A and D? (1)
A B C D E
11 Mathematics/P2 Page 5 of 16 WBHS/Nov 2019
QUESTION 2 (continued)
2.2. The table and scatter plot below show the Life Sciences and English marks for 15
Grade 11 students in the June examinations. The least squares regression line has the
equation ˆ = +y a bx .
Life Sci % 32 39 47 48 48 57 62 67 74 75 77 82 84 90 94
English % 47 49 64 62 53 59 54 53 71 69 73 78 89 95 87
2.2.1. Determine the values of a and b. (2)
2.2.2. A student missed both examinations due to illness. If they were assigned a
Life Sciences mark of 54%, what would their predicted English mark be? (2)
2.2.3. What feature of the regression line indicates that increased performance in Life
Sciences is generally better than any accompanying increase in performance in
English. (1)
[15]
30
40
50
60
70
80
90
100
20 30 40 50 60 70 80 90 100
Life Sciences %
Engli
sh %
11 Mathematics/P2 Page 6 of 16 WBHS/Nov 2019
QUESTION 3
In the diagram below, ABCD is a parallelogram. E and H are the x- and y-intercepts respectively of
diagonal AC and AC DG . BE is perpendicular to the x-axis and AD 4 13= units.
3.1. Calculate the gradient of diagonal AC. (2)
3.2. Calculate the coordinates of J, the midpoint of AC. (2)
3.3. Show that AC has the equation 1
42
= − +y x . (3)
3.4. Write down the coordinates of E. (1)
3.5. Determine the equation of DG, in the form y mx c= + . (3)
3.6. Show that the perimeter of ABCD is ( )4 101 8 13+ units. (4)
3.7. Prove that BEGH is a parallelogram. (4)
[19]
y
x
A (−12; 10)
D (−4; −2)
G
E
C (16; −4)
B (8; 8)
H J
11 Mathematics/P2 Page 7 of 16 WBHS/Nov 2019
QUESTION 4
In the diagram below, S and P are the x- and y-intercepts respectively of the line passing through S, T
and P. SP has the equation 1
13
= +y x . ˆPSO = .
4.1. Determine the coordinates of S. (2)
4.2. Determine the equation of the line RT. (3)
4.3. Calculate the size of α . (2)
4.4. Hence calculate the size of β. (2)
4.5. Determine the radius of the circle passing through P, T and R. (2)
[11]
x
y
T
S
R (0; −4)
O
P
β
α
11 Mathematics/P2 Page 8 of 16 WBHS/Nov 2019
QUESTION 5
5.1. In the diagram below, OA = 4, OB = 8 and ˆXOB = θ. OA OB⊥ .
5.1.1. Calculate the value of x (express your answer in simplest surd form). (2)
5.1.2. Determine the value of θ. (2)
5.1.3. Determine the value of q (express your answer in simplest surd form). (2)
5.2. Given sin36 t = , express each of the following in terms of t by making use of a suitable
diagram.
5.2.1. sin144 (3)
5.2.2. cos234 (2)
5.3. Simplify the following to a single trig ratio of :
( ) ( )
( ) ( )2
sin 180 2cos 90 cos
2cos 360 cos
− − −
+ − − (8)
y
x
A (x; 2) 4
8
B (4; q)
θ O
11 Mathematics/P2 Page 9 of 16 WBHS/Nov 2019
QUESTION 5 (continued)
5.4. Given the following identity: cos sin
cos1 tan
x xx
x
−=
−
5.4.1. Prove the identity. (4)
5.4.2. For which values of x is the identity undefined? (2)
5.5. Determine the general solution to the equation cos sin 0x x+ = . (4)
[29]
11 Mathematics/P2 Page 10 of 16 WBHS/Nov 2019
QUESTION 6
The diagram below shows the graphs of ( ) cos=f x a x and ( ) tan=g x x , for 90 270− x .
6.1. Write down the value of a. (1)
6.2. Write down the range of g. (1)
6.3. Write down the period of f. (1)
6.4. Graph h is obtained by shifting graph f 90 to the right followed by a vertical shift
of 2 units upwards.
6.4.1. Write down the equation of h, in simplest form. (2)
6.4.2. Hence write down the coordinates of P, the image of R under this
transformation. (2)
6.5. Write down the values of x for which ( ) ( ) 0f x g x . (2)
[9]
x
y
− 90º 270º
2
R (180º; −2)
T
T
0º 180º 90º
11 Mathematics/P2 Page 11 of 16 WBHS/Nov 2019
QUESTION 7
7.1. Use the diagram below to prove that ˆ ˆsin K sin L
k l= .
(5)
7.2. In the diagram below, BC = CD, AB = t, ˆABC = and ˆBCD = 2 .
7.2.1. Write ˆBAC in terms of and α. (2)
7.2.2. Hence show that ( )sin 2
BDcos
t
−= . (4)
K
L
M
l
m
k
2θ
α
t
A
B
D
C
11 Mathematics/P2 Page 12 of 16 WBHS/Nov 2019
30cm
C
B
A
V
A
V
B C
F F
E
D
G G
H
K
26cm
36cm
B
S
V
L
40
12m
40m
θ
25,6m
16m
QUESTION 7 (continued)
7.3. Use the diagram below to answer the questions that follow.
7.3.1. Show that VB = 32m (to the nearest metre). (2)
7.3.2. Using VB = 32m, calculate the size of θ. (3)
[16]
QUESTION 8
A lampshade (shown below) has a square base HK of side 30cm and a slant height FC of 26cm.
The perpendicular height AB = 36cm with 1
AG AB3
= . G is a point on AB such that GF BC .
Calculate the amount of fabric used to make each lampshade. (5)
[5]
11 Mathematics/P2 Page 13 of 16 WBHS/Nov 2019
In Questions 9-12, you must provide an acceptable reason for each statement/answer you
provide.
QUESTION 9
In the diagram below, O is the centre of the circle and CV KT .
Determine the size of:
9.1. ˆCVK (2)
9.2. reflex ˆCOT (3)
9.3. ˆOKV (4)
[9]
O
C
K
T
V
35
11 Mathematics/P2 Page 14 of 16 WBHS/Nov 2019
Q
P
O
W
H
QUESTION 10
10.1. In the diagram below, O is the centre of the circle and QPW is a cyclic quadrilateral.
Prove the theorem which states that ˆ ˆQ W 180+ = .
(5)
10.2. In the diagram below, O is the centre of the circle. DZ bisects RE at W.
ˆ ˆEGZ ROZ x= = .
Determine the size of
10.2.1. ˆRWO . (2)
10.2.2. ˆERZ , in terms of x. (2)
10.2.3. x (5)
[14]
D
R
Z
G
E
O
W x
x
11 Mathematics/P2 Page 15 of 16 WBHS/Nov 2019
QUESTION 11
In the diagram below, LS is a tangent to the circle at L. BR LS and MD bisects ˆLMR .
ˆSLR x= .
11.1. Name, with reasons, three other angles equal to x. (6)
11.2. State why MBFR is a cyclic quadrilateral. (1)
11.3. Prove that BF = FR. (3)
[10]
M
L
B R
S D
F
x
1 2
1
2 3
1 2
3
11 Mathematics/P2 Page 16 of 16 WBHS/Nov 2019
QUESTION 12
In the diagram below, RT is a tangent to the smaller circle at M. ˆTMK x= and MD = MK.
VMK, RMT and SDK are straight lines.
Prove that RMKS is a parallelogram.
[6]
§
D
S
R
V
M
K
T
1
2 3
1 2
1 2
x
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