total: 150 time: 3 hours instructions to candidates

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NOVEMBER 2018 MATHEMATICS

GRADE 11 PAPER 2

TOTAL: 150 TIME: 3 HOURS

INSTRUCTIONS TO CANDIDATES:

1. This paper consists of 13 questions. Answer ALL questions.

2. An approved calculator (non-programmable and/or non-graphical) may be used, unless

other instructions are given. If necessary, answers should be rounded off to two decimal

places unless other instructions are given.

3. An Answer Booklet is provided. All answers and working needs to be shown in the

appropriate place in this Booklet. If you use the Extra Space provided, please state the

question number of your answer.

4. It is in your own interest to write legibly and to present the work neatly.

5. All work must be shown. Answers only will not necessarily be awarded full marks.

6. A formula sheet is provided for your use. It is on the back of this question paper.

7. Geometry reasons need to be given in any question that uses geometrical reasoning.

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QUESTION 1:

The temperature in a small town was measured on 14 consecutive days by a Grade 11 learner. The results in °C were recorded at the same time every day and is shown in the table below.

20 38 40 30 35 22 40 24 25 36 12 14 18 20

1.1 Calculate the mean temperature for the data. (2)

1.2 Calculate the standard deviation for the data. (2)

1.3 How many of the temperatures are more than 1 standard deviation above

the mean? (2)

1.4 It was realised only afterwards that the actual temperatures were all 𝑥°C lower.

1.4.1 Describe what effect this will have on the mean. (2)

1.4.2 Describe what effect this will have on the standard deviation. (2)

[10]

QUESTION 2:

The frequency table below summarises the marks (out of 150) obtained by the Grade 11

learners of a particular school in their Mathematics exam.

Marks obtained Frequency Cumulative frequency 0 ≤ 𝑥 < 25 7 25 ≤ 𝑥 < 50 16

50 ≤ 𝑥 < 75 48 75 ≤ 𝑥 < 100 56

100 ≤ 𝑥 < 125 21 125 ≤ 𝑥 ≤ 150 12

2.1 Use the table in your answer book to complete the cumulative frequency column. (2)

2.2 Draw the ogive for the given data on the grid provided in your answer book. (3)

2.3 Use your ogive and draw a box and whisker plot in your answer book on the grid

provided. (4)

2.4 Approximately how many learners received a mark of above 75% in this exam? (2)

[11]

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QUESTION 3:

In the diagram below, A(−1; 5), B(2; 6), C and D are the vertices of parallelogram ABCD.

Vertex D lies on the 𝑥-axis. The equation of BC is 𝑥 + 2𝑦 = 14 .

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

3.2 Calculate the length of AD if D is (9;0) (leave your answer in simplified surd form). (2)

3.3 If the coordinates of F are (10; 2), show that DF is perpendicular to BC. (3)

3.4 Determine the area of parallelogram 𝐴𝐵𝐶𝐷. (3)

3.5 Calculate the size of angle 𝐴��𝐶. (5)

[16]

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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.

4.1.1 Determine the equation of the circle in the form

(𝑥 − 𝑎)2 + (𝑦 − 𝑏)2 = 𝑟2 . (3)

4.1.2 Write down the coordinates of D. (2)

4.1.3 Find the equation of the tangent AB in the form 𝑦 = 𝑚𝑥 + 𝑐. (4)

4.2 Calculate the maximum radius of the circle having equation

𝑥2 + 𝑦2 − 2𝑥 cos 𝜃 − 4𝑦 cos 𝜃 = −2 for any value of 𝜃 . (5)

[14]

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QUESTION 5:

5.1 Given: 25 cos 𝛼 = −7 where 180𝑜 < 𝛼 < 360𝑜 . Determine with the aid of a

sketch and WITHOUT using a calculator the value of 28𝑡𝑎𝑛𝛼 + 75𝑠𝑖𝑛𝛼. (4)

5.2 Simplify WITHOUT a calculator: sin 390𝑜.tan 225𝑜

tan 120𝑜.cos(−210𝑜) (6)

5.3 Simplify as far as possible: sin(90𝑜−𝛽).sin(180𝑜+ 𝛽).tan(360𝑜−𝛽)

sin(−𝛽) (6)

5.4 Prove the following identity: (1

sin 𝑥+

1

tan 𝑥)

2 =

1 + cos 𝑥

1 − cos 𝑥 (5)

5.5 Determine the general solution of 𝑥 if cos 𝑥 ; sin 𝑥 ; √3 sin 𝑥 are the first three

terms of a linear number pattern. (4)

[25]

QUESTION 6:

AB is a vertical tower of 𝑝 units high. D and C are in the same horizontal plane as B, the foot of the tower. The angle of elevation of A from D is

𝑥. 𝐵��𝐶 = 𝑦 𝑎𝑛𝑑 𝐷��𝐵 = 𝜃. The distance between D and C is 𝑘.

6.1 Prove that 𝑝 =𝑘 sin 𝜃 . tan 𝑥

sin(𝜃+𝑦) . (6)

6.2 Find BC if 𝑥 = 51,7𝑜 , 𝑦 = 62,5𝑜 , 𝑝 = 80𝑚 and 𝑘 = 95𝑚. (3)

[9]

𝑥

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QUESTION 7:

Given: 𝑓(𝑥) = cos 2𝑥 and 𝑔(𝑥) = sin(𝑥 + 60𝑜) for 𝑥 ∈ [−90𝑜; 180𝑜]

7.1 Use the axes provided to sketch the graphs of 𝑓 and 𝑔 for 𝑥 ∈ [−90𝑜; 180𝑜].

Show the 𝑥- and 𝑦-intercepts, turning points and endpoints. (6)

7.2 Write down the period of 𝑓 . (1)

7.3 Give the range of 𝑔 . (2)

7.4 Solve for 𝑥 if 𝑓(𝑥) = 𝑔(𝑥) where 𝑥 ∈ [−90𝑜; 180𝑜]. (6)

[15]

QUESTION 8:

Given the following formulae: 𝑆𝐴 = 4𝜋𝑟2 ; 𝑆𝐴 = 2𝜋𝑟2 + 2𝜋𝑟ℎ

The picture above shows a storage tank in which a farmer stores grain. The tank consists

of a right cylinder with a hemisphere on top. The perpendicular height of the tank to the

top is 75m and the radius of the tank is 10m. Calculate the total exterior surface area of this

tank. [5]

75m

10m

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QUESTION 9:

O is the centre of the circle in the diagram with chord CD parallel to diameter AB. AC is

produced to F and EG is a tangent to the circle. ABC = 35° and C2= 54° .

Calculate, WITH REASONS, the size of the following angles: 9.1 E1 (2) 9.2 C1 (2) 9.3 C3 (2) 9.4 AED (2)

9.5 E3 (3)

[11]

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QUESTION 10:

In the diagram O is the centre of the circle with P, A and K on the circumference.

Prove the theorem which states that: 𝐴��𝐾 = 2 × ��

[6]

QUESTION 11:

In the diagram below, M is the centre of circle ABCD. AM is parallel to DC and ��2 = 𝑥 .

Determine the following angles in terms of 𝑥. Supply all reasons.

11.1 ��2 (4)

11.2 �� (6)

[10]

𝑥

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QUESTION 12:

In the sketch below PA and PB are tangents to the circle.

𝐵𝐶//𝑇𝑃 and 𝐶𝐴 = 𝐶𝐵. Let 𝐶��𝐴 = 𝑥

12.1 Prove that CB bisects 𝐴��𝑃 . (4)

12.2 Prove that ABRP is a cyclic quadrilateral. (3)

12.3 Prove that TRP is a tangent to the circle through BCR. (8)

[15]

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QUESTION 13:

In the diagram below: • Points A, B, C and D lie on the circle. • A tangent is drawn at point D with a chord drawn from A to C.

Prove that ��2 + ��3 = ��. [3]

THE END

1

1

1

2

2

2 3

A

B C

D

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INFORMATION SHEET: MATHEMATICS

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c

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sinsinsin Abccba cos.2222

CabABCarea sin.2

1

sin.coscos.sin)sin( sin.coscos.sin)sin(

sin.sincos.cos)cos(

sin.sincos.cos)cos(

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n

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AnAP P(A or B) = P(A) + P(B) – P(A and B)

y a bx 2

( )( )

( )

x x y yb

x x

total: 150 time: 3 hours instructions to candidates

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