KING DAVID HIGH SCHOOL LINKSFIELD

GRADE 11

MATHEMATICS PAPER 2

NOVEMBER 2017

Writing Time: 2½ hours Total: 120 marks

INSTRUCTIONS AND INFORMATION Read the following instructions carefully before answering the questions:

1. This question paper consists of 20 pages (including the cover page). A separate information sheet is provided.

2. All the calculations must be clearly shown. You will NOT receive full marks if you merely write down the answer.

3. All the final answers must be rounded off to TWO decimal places, unless stated otherwise.

4. An approved calculator (non-programmable and non-graphical) may be used, unless stated otherwise.

5. Diagrams are not necessarily drawn to scale.

6. Write neatly and legibly.

7. Good luck.

1 [7]

2 [12]

3 [10]

4 [7]

5 [5]

6 [7]

7 [11]

8 [13]

9 [8]

10 [15]

11 [10]

12 [5]

13 [10]

Total [120]

NAME: ___________________________________________________

Grade 11 November 2017 Paper 2 Page 2

Q uestion 1 [7 marks]

(a) B(8;2) and C(3;a) are points on the Cartesian plane. Determine the value(s) of a ,

if BC 5 units. (3)

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(b) Prove that the points N( 1;1) , 2Q(p;p ) and R(1;2p 1) are collinear. (4)

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Grade 11 November 2017 Paper 2 Page 3

Question 2 [12 marks]

On a certain day, Jozi Tours sent 11 tour buses to 11 different destinations. The table below

shows the number of passengers on each bus.

Destination A B C D E F G H I J K

No. of

passengers 8 8 14 16 18 19 20 21 28 35 36

(a) Calculate the mean number of passengers travelling on a tour bus. Give the answer correct

to two decimal digits. (2)

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(b) Write down the five number summary. (3)

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(c) Draw a box and whisker diagram for this set of data. (2)

(d) Refer to the above diagram and comment on the skewness of the data, giving a reason for

your answer. (2)

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Grade 11 November 2017 Paper 2 Page 4

(e) Calculate the standard deviation for this data set. Give the answer correct to two decimal

digits. (2)

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(f) A tour is regarded as popular if the number of passengers on a tour bus, is more than 1

standard deviation above the mean. How many destinations were popular on this day? (1)

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Question 3 [10 marks]

The ages of teachers at a High School in Brisbane, are given in the table below:

(a) Complete the table: (2)

Age Frequency Cumulative

Frequency

25 < A ≤30 2

30 < A ≤35 8

35 < A ≤40 4

40 < A ≤45 5

45 < A ≤50 11

50 < A ≤55 19

55 < A ≤60 20

60 < A ≤65 6

Grade 11 November 2017 Paper 2 Page 5

(b) Draw an ogive on the set of axes provided, to represent the data in the table. (4)

Ogive of ages of teachers

(c) Use your ogive to find an estimate for the median age, clearly indicating this reading on

your graph. (2)

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(d) The school would like to give all teachers older than 57 a special present. Use your graph

to find an estimate for the percentage of teachers older than 57 years of age. (2)

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Grade 11 November 2017 Paper 2 Page 6

Question 4 [7 marks]

The graph of f(x) sin2x for x [ 180 ;90 ] is sketched on the axes below:

(a) State the period of f . (1)

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(b) Sketch g(x) cos(x 45 ) for x [ 180 ;90 ] on the same set of axes. Clearly indicate

all intercepts with the axes, as well as the end-points. (3)

(c) For what values of x 0 , will f(x).g(x) 0 ? (2)

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(d) Determine the value of f( 180 ) g( 180 ) . (1)

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Grade 11 November 2017 Paper 2 Page 7

Question 5 [5 marks]

Determine the general solution for x , in each of the following, rounding answers to one decimal

place:

(a) 3cosx 1 (2)

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(b) tan2x 1,86 (3)

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Grade 11 November 2017 Paper 2 Page 8

Question 6 [7 marks]

MNO is a tangent to the circle passing through the points N, P, Q, R and S. PQ//NS.

2P 30 and

1S 70

(a) Determine, with reasons, the size of each of the following angles:

(1) 1

N (2)

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(2) R (3)

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(b) Determine if PS is a diameter to the circle. Give a reason for your answer. (2)

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O

70 30

Grade 11 November 2017 Paper 2 Page 9

Question 7 [11 marks]

(a) Simplify the following to a single trigonometric function:

cos tan(180 ).cos(90 )

cos( )

(6)

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(b) Prove that:

1 sin 1 sin 4 tan

1 sin 1 sin cos

(5)

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Grade 11 November 2017 Paper 2 Page 10

y

P

R

T S(6;2)

Q(2;10)

x

_

M

O

Question 8 [13 marks]

In the diagram below PQRS is a quadrilateral, with Q(2;10)and S(6;2) . The diagonals of PQRS

bisect each other at M , at right angles. T and P are the intercepts of PR, with the respective axes.

(a) Determine the gradient of PR. (2)

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Grade 11 November 2017 Paper 2 Page 11

(b) Show that the equation of PR is given by 2y x 8 . (4)

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(c) Determine the co-ordinates of R. (4)

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(d) Calculate the size of ˆRTO , correct to one decimal place. (3)

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Grade 11 November 2017 Paper 2 Page 12

Question 9 [8 marks]

In the diagrams below, a leaning tower PQ is shown, which leans at an angle of , to the vertical.

The stays (steel ropes), PR and PS, help keep the tower stable.

From Q, the base of the tower, a vertical line QT is drawn.

QS = 30m, SR = 10m, 31R and 739QSP ,

(a) Give a reason as to why 78P1 ,

? (1)

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Grade 11 November 2017 Paper 2 Page 13

(b) Show that PS = 34,05m. (2)

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(c) Determine the size of , correct to the nearest degree. (5)

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Grade 11 November 2017 Paper 2 Page 14

Question 10 [15 marks]

QC is the diameter of the circle with centre M. The tangent to the circle at A meets QC produced

at P. E is the midpoint of AC and ME produced meets AP at D.

(a) Complete the statements below:

(1) 90E4ˆ _________________________________________ (1)

(2) 2 3ˆ ˆA A 90 ________________________________________ (1)

(3) _____ _____ 90 tan ⊥ radius (1)

(b) Hence, prove the following:

(1) MD // QA (2)

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Grade 11 November 2017 Paper 2 Page 15

(2) AMCD is a cyclic quadrilateral. (4)

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(3) 3

ˆD 2Q (3)

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(4) DC is a tangent to the circle. (3)

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Grade 11 November 2017 Paper 2 Page 16

Question 11 [10 marks]

(a) O is the centre of the circle. DE is a tangent to the circle at C. A and B are points on the

circumference of the circle. AB, AC and BC are joined.

Prove that ˆ ˆECB BAC . (5)

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Grade 11 November 2017 Paper 2 Page 17

(b) In the figure below, P, Q, R and S are points on the circle with centre O. PRT and USTV

are straight lines. USTV is a tangent to the circle at S. 50TSR ˆ and 150VTR ˆ .

Determine, with reasons, the size of RQP ˆ . (5)

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150 50

Grade 11 November 2017 Paper 2 Page 18

Question 12 [5 marks]

The Transamerica Pyramid is the tallest building in San Francisco. It is a square-based pyramid

with height 260 m and base width 53 m.

Decorative lights are installed over the festive season on each of the four the edges of the building

(labeled VA, VB, VC, VD in the figure). Will a 1km length of lighting be sufficient to cover the

length of the 4 edges? Show all calculations. (5)

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53 m

260 m

Grade 11 November 2017 Paper 2 Page 19

Question 13 [10 marks]

(a) If 1

sinx cos x2

, determine the value of sin(180 x).sin(90 x) , without using a

calculator. (5)

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(b) Determine the value of tan , if the distance between the points (cos ;sin ) and (3;4)

is 26 units. (5)

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Grade 11 November 2017 Paper 2 Page 20

Extra lines (if needed):

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