RONDEBOSCH BOYS’ HIGH SCHOOL 11 Papers/Rondebosch... · 2019-05-21 · Mathematics/P2 Gr11 - RBHS November 2018 3 QUESTION 1 The table below represents the marks achieved by a class

MARKS: 150 EXAMINER: S VERSTER

TIME: 3 HOURS MODERATOR: S CARLETTI

This question paper consists of 12 pages.

RONDEBOSCH BOYS’ HIGH SCHOOL

SENIOR CERTIFICATE

MATHEMATICS PAPER 2

12 NOVEMBER 2018

GRADE 11

SENIOR

CERTIFICATE

Mathematics/P2 Gr11 - RBHS November 2018

2

INSTRUCTIONS AND INFORMATION

Read the following instructions carefully before answering the questions.

1. This question paper consists of 14 questions.

2. Answer ALL the questions in the ANSWER BOOK.

3. Blue or black ink only may be used; pencils may be used for diagrams and sketches.

Write neatly and legibly.

4. Clearly show ALL calculations, diagrams, graphs, et cetera that you have used in

determining your answers.

5. Answers only will NOT necessarily be awarded full marks.

6. You may use an approved scientific calculator (non-programmable and non-graphical),

unless stated otherwise.

7. If necessary, round off answers to TWO decimal places, unless stated otherwise.

8. Diagrams are NOT necessarily drawn to scale.


3

QUESTION 1

The table below represents the marks achieved by a class of Grade 11 students.

Mark (𝒎) Frequency Cumulative frequency

10 ≤ 𝑚 < 20 4

20 ≤ 𝑚 < 30 11

30 ≤ 𝑚 < 40 15

40 ≤ 𝑚 < 50 30

1.1 Complete the missing information in the frequency table. (2)

1.2 Draw an ogive representing the above information on the axes provided. (3)

1.3 If 30% of the students failed the test, what was the pass mark? (2)

[7]

QUESTION 2

A student records his distance from school from 07h00 – 07h50 (i.e. before school starts).

The scatterplot below represents the data he collected.

2.1 Determine the equation of the least squares regression line. (3)

2.2 Use your equation to predict how far away from school he would be after

travelling for 30 minutes. (2)

2.3 What was the average speed travelled in 𝑘𝑚/ℎ? (2)

2.4 Explain what might have happened from 07h20 – 07h28. (2)

[9]

Dis

tan

ce f

rom

sch

ool

(km

)

Time (mins)


4

QUESTION 3

Seven numbers in ascending order are:

5; 𝑥; 12; 16; 21; 𝑦; 36

They have a mean of 18 and an interquartile range of 22. Determine the values of

𝑥 and 𝑦. [4]

QUESTION 4

In ∆𝐴𝐵𝐶: 𝐴(−2; 4), 𝐵(𝑎; 1) and 𝐶(2; −5). The angle between 𝐵𝐶 and the 𝑥 − axis is 63,43°.

4.1 Calculate the length of 𝐴𝐶. Leave your answer in surd form. (2)

4.2 Determine the gradient of 𝐵𝐶. (2)

4.3 Hence, or otherwise, determine the value of 𝑎. Show all working. (2)

4.4 Determine the magnitude of 𝐴�̂�𝐵. (4)

4.5 If 𝑎 = 5, calculate the area of ∆𝐴𝐵𝐶. (3)

[13]

°


5

QUESTION 5

𝑃(1; 2), 𝑄(3; 1), 𝑅(−3; 𝑘) and 𝑆(2; −3) are points on the Cartesian plane. Determine the

value(s) of 𝑘 if:

5.1 𝑃𝑄𝑅 are collinear. (3)

5.2 𝑅𝑆 = 5√2. (5)

[8]

QUESTION 6

The circle sketched below has diameter 𝐴𝐵 with 𝐴(0; 1) and 𝐵(2; −5).

6.1 Determine the coordinates of 𝑂, the centre of the circle. (2)

6.2 Determine the equation of the circle. (4)

6.3 If this circle was inscribed by a square, as shown in Figure 2, determine the area

of the shaded region. (4)

[10]

Figure 1 Figure 2

𝑥

𝑦


6

QUESTION 7

A dartboard consists of three concentric circles, with diameters dividing it into 12 equal sectors,

as shown in the diagram. The radii are 10 𝑐𝑚; 8 𝑐𝑚; and 6 𝑐𝑚 respectively and 𝑃(√3; 2) is the

centre of the circles. The diameter 𝐼𝐶 is parallel to the 𝑥 − axis.

Diagonal 𝐵𝐻 has equation 𝑦 =1

√3𝑥 + 1. Scores for a particular sector are shown on the outer

ring of the dartboard. An example of a dart is shown in the outer ring and in sector 𝐽𝑃𝐾.

7.1 If 𝐾𝐸 ⊥ 𝐻𝐵, determine the equation of 𝐾𝐸. (3)

7.2 In which ring (outer, middle or inner) is the point 𝑅(8; 6) located? Justify your

answer with a suitable calculation. (3)

7.3 In which sector is the point 𝑅(8; 6) located? Justify your answer with a suitable

calculation. (4)

In this game of darts the points are allocated as follows: if a dart lands in the inner ring,

the score of the sector remains the same. If the dart lands in the middle ring, the score is

doubled. If the dart lands in the outer ring, the score of the sector is halved. For example,

the score of the dart shown is 50.

7.4 Determine the score obtained by the dart landing at 𝑅. (1)

[11]


7

QUESTION 8

8.1 In the diagram below, reflex 𝑇�̂�𝑃 = 𝛼 and 𝑃 has coordinates (−5; −12). Determine the

values of the following, without the use of a calculator:

8.1.1 cos 𝛼 (2)

8.1.2 sin 30° + sin(180° − 𝛼) (4)

8.2 If sin 𝜃 =𝑘

2 , where 0 ≤ 𝜃 ≤ 90°, calculate the following in terms of 𝑘:

8.2.1 cos(90° − 𝜃) (2)

8.2.2 tan 𝜃 (2)

8.3 Simplify as far as possible:

1

cos(360°−𝜃) sin(90°−𝜃) − tan2(180° + 𝜃) (6)

8.4 Determine the general solution of: 4 sin2 𝑥 = 2 − 7 cos 𝑥 (6)

8.5 Given: sinn 𝜃−cosn 𝜃

tann 𝜃−1 = cosn 𝜃

8.5.1 Prove the above identity, ignoring any restrictions. (3)

8.5.2 Hence determine, without a calculator:

sin2018 45°−cos2018 45°

tan2018 45°−1 ×

tan2012 45°−1

sin2012 45°−cos2012 45° (3)

[28]

Ox

y

P(-5;-12)

T

α


8

QUESTION 9

The graph of 𝑔(𝑥) = − cos 2𝑥 for 𝑥 ∈ [−180°; 180°] is drawn below:

9.1 Draw the graph of 𝑓(𝑥) = 2 sin 𝑥 − 1 for 𝑥 ∈ [−180°; 180°] on the same set

of axes provided in the ANSWER BOOK. (4)

9.2 Determine the range of 𝑓. (2)

9.3 Write down the value(s) of 𝑥 for which 𝑓(𝑥 + 30°) − 𝑔(𝑥 + 30°) = 0

for 𝑥 ∈ [0°; 180°]. (2)

[8]

QUESTION 10

In ∆𝑃𝑄𝑅, 𝑄𝑅 = 3 𝑢𝑛𝑖𝑡𝑠, 𝑃𝑅 = 𝑥 𝑢𝑛𝑖𝑡𝑠, 𝑃𝑄 = 2𝑥 𝑢𝑛𝑖𝑡𝑠 and �̂� = 𝜃.

10.1 Show that cos 𝜃 = 𝑥2+3

4𝑥 (3)

10.2 Hence, or otherwise, calculate the values of 𝑥 for which

the triangle exists. (4)

[7]

𝑔

3

2xx

θ

P

R

Q


9

QUESTION 11

11.1 In the figure, 𝑂 is the centre of the circle with 𝐴, 𝐵, 𝐶 and 𝐷 on the circumference.

Prove the theorem that states �̂� + �̂� = 180°.

(5)

11.2 In the diagram below, circle 𝑃𝑄𝑅𝑆𝑇 has centre 𝑂. 𝑇𝑄 is a diameter.

𝑄�̂�𝑁 = 63°; �̂�1 = 35° and 𝑃𝑇 = 𝑄𝑅.

Calculate the size of the following angles, with reasons:

11.2.1 𝑇�̂�𝑄 (2)

11.2.2 �̂�1 (2)

11.2.3 �̂�1 (2)

11.2.4 �̂�3 (2)

11.2.5 �̂�2 (2)

11.2.6 �̂�2 (2)

[17]

O

D

C

B

A


10

QUESTION 12

In the diagram 𝑂 is the centre of the circle 𝐻𝐸𝐴𝑇𝑅. 𝐴𝑂𝐹 is parallel to 𝐸𝐻.

�̂�2 = 𝑥 and �̂�1 = 𝑦.

Calculate with reasons, in simplest form, the size of the following in terms of 𝑥 and/or 𝑦.

12.1 �̂�1 (2)

12.2 �̂�1 (2)

12.3 �̂� (2)

12.4 �̂�2 (3)

[9]

𝑥

𝑦


11

QUESTION 13

In the diagram, 𝐸𝐴 is a tangent to circle 𝐴𝐵𝐶𝐷 at 𝐴. 𝐴𝐶 is a tangent to circle 𝐶𝐷𝐹𝐺 at 𝐶.

𝐶𝐸 and 𝐴𝐺 intersect at 𝐷. The two circles intersect each other at 𝐶 and 𝐷.

�̂�1 = 𝑥 and �̂�1 = 𝑦.

Prove the following, with reasons:

13.1 𝐵𝐺 ∥ 𝐴𝐸 (5)

13.2 𝐴𝐸 is a tangent to circle 𝐹𝐸𝐷 (5)

13.3 𝐴𝐵 = 𝐴𝐶 (4)

[14]

𝑥

𝑦


12

QUESTION 14

The “power of a point” theorem reflects the relative distance of a given point from a given

circle. In the circle below 𝑃𝑇 is a tangent to the circle at 𝑇 and chord 𝑁𝑀 extended meets the

tangent at 𝑃. It is further given that 𝑃𝑇2 = 𝑃𝑀 × 𝑃𝑁. You may use this theorem for the

question below without referencing a reason.

Consider a triangle 𝐴𝐵𝐶, where �̂� = 60°. The inscribed circle of triangle 𝐴𝐵𝐶 is shown where

𝐴𝐵, 𝐵𝐶 and 𝐴𝐶 are tangents to the circle at points 𝐷, 𝐸 and 𝐹 respectively. 𝐺 is the point where

𝐴𝐸 intersects the circle.

Without the use of a calculator, determine the value of:

𝐴𝑟𝑒𝑎 ∆𝐴𝐷𝐹

𝐴𝐺 × 𝐴𝐸

[5]

Total: 150 marks

Documents

RONDEBOSCH BOYS’ HIGH SCHOOL 11 Papers/Rondebosch... · 2019-05-21 · Mathematics/P2 Gr11 - RBHS November 2018 3 QUESTION 1 The table below represents the marks achieved by a class