Mathematics Department

November 2018

Paper 2

Grade: 11 Time: 3 hours

Examiner: Mrs. T. Meintjies Marks: 150

Moderators: Mrs. S. Hickling, Mrs. F. Van der Merwe and Mrs. A. Heuer.

NAME: …………………………………………. TEACHER: ……………………………………………

Instructions:

1. The question paper consists of 25 pages, including the cover page, and formula sheet on the last page. Page 24 is for additional working. Please check that your paper is complete.

2. A non-programmable calculator may be used. 3. Please round off answers to two decimal places, where necessary. 4. Answer all the questions. 5. Please answer all questions in pen. 6. All calculations must be clearly shown. 7. Diagrams are NOT drawn to scale.

Q1 (10)

Q2 (22)

Q3 (22)

Q4 (9)

Q5 (10)

Q6 (9)

Q7 (16)

Q8 (6)

Q9 (11)

Q10 (9)

Q11 (14)

Q12 (7)

Q13 (5)

Total: /150

Grade 11 November Examination Page 2 of 25

SECTION A [73]

Question 1 [10]

In a Science Olympiad learners were expected to answer a multiple choice question paper. The time taken by the learners to the nearest minute to complete the paper, was recorded and data was obtained. The cumulative frequency graph representing the time taken to complete the paper is given below.

An incomplete frequency table for the data is given below.

Time taken to complete the paper in minutes

10 ≤ 𝑥 < 20 20 ≤ 𝑥 < 30 30 ≤ 𝑥 < 40 40 ≤ 𝑥 < 50 50 ≤ 𝑥 < 60

Frequency 𝑎 6 8 28 34

a) Determine the value of 𝑎 in the frequency table. (1) _____________________________________________________________________

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b) How many learners wrote the paper? (1) _____________________________________________________________________

Grade 11 November Examination Page 3 of 25

c) Identify the modal class of the data. (1) _____________________________________________________________________ d) Calculate:

1) The estimated mean time, in minutes, taken to complete the paper (2)

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2) The standard deviation (1)

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3) The 90th percentile. (2)

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4) The number of learners that took longer than 35 minutes to complete the paper (2) _____________________________________________________________________

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Grade 11 November Examination Page 4 of 25

Question 2 [22]

a) In the diagram below P(4 ; 5) , Q(2 ; 6) , R and S are the vertices of a parallelogram.

The equation of the line QR is 2𝑦 − 𝑥 − 16 = 0

1) Determine the equation of PS (3)

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2) Calculate the co-ordinates of S. (2)

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3) Calculate the mid-point of QS (2)

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4) Determine the co-ordinates of R. (2)

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5) Calculate the size of RSP. (4)

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Grade 11 November Examination Page 6 of 25

b) Given K(4 ; 3), J(0 ; −1) and L(p ; 1)

Determine the value of p for which:

1) K, J and L lie on the same straight line. (3)

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2) ∆KJL is right angled at J. (2)

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3) KL = JL (4)

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Grade 11 November Examination Page 7 of 25

Question 3 [22]

Simplify without the use of a calculator:

a) If −3𝑠𝑖𝑛𝛽 − 2 = 0 and 𝛽 ∈ (0°; 270°), draw an appropriate sketch and use it to determine

1 + 𝑡𝑎𝑛2𝛽 (5)

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b) If 𝑐𝑜𝑠 75° = 𝑚, express the following in terms of 𝑚, showing all your working:

1) 𝑐𝑜𝑠2 105° (2)

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2) 𝑠𝑖𝑛 15° (2)

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3) 𝑡𝑎𝑛 15° (2)

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c) Simplify: 𝑐𝑜𝑠(180°−𝑘)𝑠𝑖𝑛(𝑘−90°) − 1

𝑡𝑎𝑛2(540°+𝑘)𝑠𝑖𝑛(90°+𝑘)𝑐𝑜𝑠(−𝑘) (7)

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d) Determine the general solution of 2sin (𝑥 − 20°) = 1,464 (4)

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Grade 11 November Examination Page 9 of 25

Question 4 [9]

a) Given circle PVST with PV//ST and P = 75°. Calculate, with reasons, the size of R. (3)

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3

2

R

T

P

V S

1

1 2

R

1 2

P

Grade 11 November Examination Page 10 of 25

b) In the diagram below, points P, Q, R and T lie on the circumference of a circle. MW and

TW are tangents to the circle at P and T respectively. PT is produced to meet RU at U.

MPR = 75°, PQT = 29° and QTR = 34°

Calculate, with reasons, the values of 𝑎, 𝑏, 𝑐 and 𝑑. (6)

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R

Q

M

P

W

T U

1 1

2 3

1 1

29°

34° 𝑐

75°

𝑏

𝑎

𝑑

Grade 11 November Examination Page 11 of 25

Question 5 [10]

The graphs below represent the functions 𝑔(𝑥) = 𝑐𝑜𝑠2𝑥 and 𝑓(𝑥) = sin (𝑥 + 30°).

B has co-ordinates (140° ; 0,17)

a) Give the period of 𝑓 (1)

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b) Give the co-ordinates of A. (1)

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c) For which values of 𝑥 are 𝑓(𝑥) and 𝑔(𝑥) both increasing? (2)

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d) i) Using your calculator, find 𝑔(20°) (1)

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ii) Using your calculator, find 𝑓(20°) (1)

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e) For which values of 𝑥 is 𝑔(𝑥) < 𝑓(𝑥) for 𝑥 ∈ [0 ° ; 180° ] (2)

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f) If 𝑓 is translated 30° to the left to make ℎ, give the equation of ℎ, (2)

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−180 −150 −120 −90 −60 −30 30 60 90 120 150 180

−2

−1

1

2

x

y

B

A

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SECTION B [77]

Question 6 [9]

a) Learners at Phambili High School travel from three different neighbourhoods, A, B and C.

The table below shows the number of learners from each neighbourhood, and their mean

travelling times from home to school.

Neighbourhood A B C

Number of learners 135 225 200

Mean travelling time (in min) 24 32 𝑥

The mean travelling time for learners living in neighbourhood C is the same as the mean

travelling time for all 560 learners. Calculate the value of 𝑥. (4)

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b) Five numbers 𝑎, 𝑏, 𝑐, 𝑑 and 𝑒 are given, and:

𝑒 = 𝑐

𝑒 < 𝑎 < 𝑏

𝑑 is the maximum value

The modal number is 5

The difference between the fourth number and the second number is 4.

The range of the numbers is 9.

The average of the numbers is 8.

Determine the values of 𝑎, 𝑏, 𝑐, 𝑑 and 𝑒 (5)

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Grade 11 November Examination Page 13 of 25

Question 7 [16]

a) Determine, without a calculator, for which value(s) of 𝑘 is:

𝑘2 + 𝑠𝑖𝑛120° . 𝑐𝑜𝑠 30° = 𝑐𝑜𝑠 (𝑥+180°)

𝑐𝑜𝑠 (180°−𝑥) ? (6)

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b) Prove: 𝑠𝑖𝑛 𝑥 .𝑐𝑜𝑠 𝑥

1+ 𝑐𝑜𝑠2𝑥− 𝑠𝑖𝑛2𝑥 =

1

2𝑡𝑎𝑛 𝑥 (4)

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c) Solve for 𝑥: 𝑐𝑜𝑠2𝑥 + 𝑠𝑖𝑛𝑥 = 1 𝑥𝜖[−90°; 90°] (6)

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Grade 11 November Examination Page 15 of 25

Question 8 [6]

In the diagram, AC represents a vertical tower. Point A is the top of the tower and point D is a

point in the tower situated ℎ metres below A. The angle of depression from A and D to point B, in

the same horizontal plane as point C, are 𝑥 and 𝑦 respectively. The distance, in metres, between

points B and C is 𝑝.

a) Express ��1 in terms of 𝑥. (1)

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b) Express ��1 in terms of 𝑥 and 𝑦. (1)

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c) Hence, prove that 𝑝 =ℎ.𝑐𝑜𝑠𝑥.𝑐𝑜𝑠𝑦

sin (𝑥−𝑦) (4)

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A

C B

D

𝑥

𝑦

𝑝

ℎ

2

1

1

Grade 11 November Examination Page 16 of 25

Question 9 [11]

ABC is a semi-circle with centre O. AB is produced to K such that KCS is a tangent to the circle at C.

SO is perpendicular to AB. KCB = 25°

a) Prove that CTOB is a cyclic quadrilateral. (3)

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b) Determine the size of S (6)

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c) Is OK a tangent to the circle through S, O and C? Give a reason for your answer. (2)

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Grade 11 November Examination Page 18 of 25

Question 10 [9]

The base of the prism in the sketch below is an equilateral triangle with sides 𝑎 cm.

a) Show that the height of the base is ℎ =√3

2𝑎 (3)

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b) Find a formula for the area of the base in terms of 𝑎. (2)

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𝑎 𝑎

𝑎

ℎ

H

Grade 11 November Examination Page 19 of 25

c) Given 𝑎 = 5 𝑐𝑚 and 𝐻 = 15 𝑐𝑚, calculate:

1) the surface area of the prism (2)

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2) the volume of the prism (2)

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Grade 11 November Examination Page 20 of 25

Question 11 [14]

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) Prove the following:

1) MD//QA (4)

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Q

M

C

P A D

E

1

2 3

4

3

2 1

1 2

3

1

2 3

1 4

3 2

Grade 11 November Examination Page 21 of 25

2) AMCD is a cyclic quadrilateral. (4)

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3) D3 = 2Q (2)

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

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Grade 11 November Examination Page 22 of 25

Question 12 [7]

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

If OT = 𝑥 and OS =7

15OT, calculate the radius, OA, of the circle.

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Grade 11 November Examination Page 23 of 25

Question 13 [5]

The pilot of a plane coming in to land has to make sure that his plane is constantly equidistant

from the two outer landing lights L1 and L2. The line of landing lights are at a right angle to the

runway. The coordinates of L1 and L2, when plotted on a Cartesian plane,are (16 ; 30)and

(20 ; 25) respectively.

Find the equation of his flight path in the form 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0

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Grade 11 November Examination Page 24 of 25

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Grade 11 November Examination Page 25 of 25