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