SUBJECT Mathematics PAPER 2 GRADE 11 DATE 21 NOV 2017 EXAMINER Mrs Sillman MARKS 150 NAME MODERATOR Gr 11 Teachers TEACHER DURATION 3 hours

QUESTION NO

DESCRIPTION MAXIMUM MARK

ACTUAL MARK

1 - 3 Analytical Geometry 27

4 Euclidean Geometry 10

5 Data Handling 7

6 - 7 Trigonometry 33

8 2D Trigonometry 6

9 - 10 Data Handling 18

11 Analytical Geometry 7

12 Trig Functions 12

13 Measurement 8

14 - 15 Euclidean Geometry 22

TOTAL 150

Grade 11 2 of 25 Mathematics Paper 2

INSTRUCTIONS:

1. This paper consists of 15 questions and 25 pages (which includes the diagram sheet and

formula sheet)

2. Read the questions carefully.

3. Answer all questions.

4. Number your answers clearly and use the same numbering as in the question paper.

5. You may use an approved non-programmable and non-graphical calculator, unless

otherwise stated.

6. Round off your answers to two decimal digits where necessary.

7. All necessary working details must be shown. Answers only, without the relevant

calculations will not necessarily be given marks. Equations may not be solved solely with a

calculator.

8. It is in your interest to write legibly and present your work neatly.

9. You may detach the Information Sheet and Diagram sheet at the back of the question

paper.

Grade 11 3 of 25 Mathematics Paper 2

QUESTION 1 9 MARKS

Given quadrilateral ABCD, with corners 𝐴(2; −2), 𝐵(4; 3), 𝐶(7; 4), 𝐷(5; −1). Diagram is NOT

drawn to scale.

a) Calculate the gradient of BC (2)

b) Calculate the length of CD (2)

c) Calculate the midpoint of diagonal AC (2)

d) Prove that ABCD is a parallelogram (3)

Grade 11 4 of 25 Mathematics Paper 2

QUESTION 2 8 MARKS

a) A straight line has the equation 𝑦 = −2𝑥 − 3.

i) Give the inclination of the line with the 𝑥 – axis. (2)

ii) Give the equation of a line that is perpendicular to 𝑦 = −2𝑥 − 3 and passes

through the point (6; 2). (3)

b) A line has an angle of inclination of 60 . What is the gradient of the line? (1)

c) Determine the equation of a line perpendicular to the 𝑥 −axis passing through the point

(4 ; 1) (2)

Grade 11 5 of 25 Mathematics Paper 2

QUESTION 3 10 MARKS

K (-3 ; 5), L (t ; 2) and M (1 ; -5) are points in the Cartesian plane. Calculate t if:

a) K, L and M are collinear (5)

b) KL ⊥ML (5)

Grade 11 6 of 25 Mathematics Paper 2

QUESTION 4 10 MARKS

In the diagram, O is the centre of the circle. 𝑌𝐼𝐸 = 50 . AB is a tangent at Y.

Determine, with reasons the size of the following angles:

a) 𝑌

b) 𝐸

c) 𝑌

d) 𝑌

e) 𝑅

(10)

Grade 11 7 of 25 Mathematics Paper 2

QUESTION 5 7 MARKS

The AP Maths marks of 23 boys are expressed in percentages in the following stem-and-leaf diagram:

3 8 4 2 3 8 5 0 5 6 9 6 1 3 6 8 9 7 2 2 2 7 8 8 3 5 6 9 1 3

Determine the:

a) median mark (1)

b) modal mark (1)

c) inter-quartile range of the marks (3)

d) standard deviation of the marks to one decimal place (2)

QUESTION 6 26 MARKS

a) Solve for 𝜃: 3𝑡𝑎𝑛𝜃 = √3 if 𝑥𝜖[0 ; 360 ] (3)

Grade 11 8 of 25 Mathematics Paper 2

b) Simplify without the use of a calculator:

1) ( ). ( ). ( )

( ). ( ) (7)

2) 𝑡𝑎𝑛120 . 𝑐𝑜𝑠210 + (7)

Grade 11 9 of 25 Mathematics Paper 2

c) Simplify to a single trigonometric ratio for 𝜃𝜖[0 ; 90 ]

(1 − 𝑠𝑖𝑛𝛼)(1 + 𝑠𝑖𝑛𝛼) (3)

d) Prove that:

.

.= 𝑡𝑎𝑛 𝑥 (6)

Grade 11 10 of 25 Mathematics Paper 2

QUESTION 7 7 MARKS

In the diagram below, 𝑃(𝑥; 24) is a point such that OP = 25 and 𝑅𝑂𝑃 = 𝛽, where 𝛽 is an obtuse angle.

a) Calculate the value of 𝑥 (2)

b) Determine the value of each of the following WITHOUT the use of a calculator:

1) 𝑠𝑖𝑛𝛽 (1)

2) cos (180 − 𝛽) (2)

3) tan (−𝛽) (2)

Grade 11 11 of 25 Mathematics Paper 2

QUESTION 8 6 MARKS

The diagram represents a small crane lifting a light pole.

AC = 1,5m AB = 2,2m BC = 2,9m

Determine 𝐴 and 𝐶 correct to two decimal places.

Grade 11 12 of 25 Mathematics Paper 2

QUESTION 9 10 MARKS

Snow skiers pass a point on a ski slope at a speed in km/h

The speeds are in the table:

Speed (km/h) Frequency (f) Cumulative frequency 0 ≤ 𝑥 < 10 10

10 ≤ 𝑥 < 20 20 20 ≤ 𝑥 < 30 45 30 ≤ 𝑥 < 40 71 40 ≤ 𝑥 < 50 21

a) Complete the table (2)

b) Draw a cumulative frequency curve (ogive) representing the information on the next page (4)

c) Indicate clearly on the cumulative frequency curve the lower quartile (𝑄 ) and the

upper quartile (𝑄 ). Mark where these points would be read off with and A and B

respectively. (2)

d) Use your graph to estimate the number of skiers who passed the point with speeds greater than 35km/h (2)

Grade 11 13 of 25 Mathematics Paper 2

Grade 11 14 of 25 Mathematics Paper 2

QUESTION 10 8 MARKS

The following table shows the Term 2 marks for English and Maths for a random sample of 10 Grade 11 boys

English (x)

54 54 64 62 70 70 72 80 78 83

Maths (y)

43 46 50 71 72 76 77 82 87 91

a) Calculate the correlation coefficient to 3 decimal places. Does this represent a strong or weak correlation? (2)

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

c) Use your line to predict the Maths mark, to the nearest whole number, of a boy who scores 30% in English (1)

d) Comment on the reliability of your prediction in (c) and give one reason for your answer (2)

Grade 11 15 of 25 Mathematics Paper 2

QUESTION 11 7 MARKS

Determine the size of the acute angle between the lines 5𝑦 = 𝑥 − 5 and 𝑦 = 5𝑥 − 1.

Show that the gradient of the bisector of this angle is 1. From this, determine the equation of

the bisector. (Hint: a diagram may be of help)

Grade 11 16 of 25 Mathematics Paper 2

QUESTION 12 12 MARKS

The sketch below shows the graphs of:

𝑓(𝑥) = sin 𝑎𝑥 and 𝑔(𝑥) = cos(𝑥 + 𝜃) for 𝑥𝜖[−180 ; 180 ]

a) Determine the period of 𝑓(𝑥) = 𝑠𝑖𝑛 𝑎𝑥 (1)

b) Write down the range of 𝑔(𝑥) (1)

c) Write down the values of 𝜃 and 𝑎 (2)

d) Determine the values of 𝑥, if 𝒙𝝐[−𝟏𝟖𝟎𝒐; 𝟎𝒐] for which:

1) 𝑓(𝑥) − 𝑔(𝑥) > 0 (2)

2) 𝑔(𝑥). 𝑓(𝑥) ≥ 0 (3)

e) Determine the minimum value of: 3 ( ) (3)

Grade 11 17 of 25 Mathematics Paper 2

QUESTION 13 8 MARKS

A cylindrical aerosol can has a lid in the shape of a hemisphere that fits exactly on top of the

can. The height of the can is 16cm and the radius of the base of the can is 2,9cm.

a) Calculate the surface area of the can with the lid in place, as shown in Figure 1 (5)

b) If the lid is 80% filled with liquid, as shown on Figure 2, calculate the volume of the

liquid in the lid. (3)

Grade 11 18 of 25 Mathematics Paper 2

QUESTION 14 8 MARKS

On the diagram below, AB is a tangent to the circle (centre O) at X.

W, X, Y and Z are points on the circle. 𝑂𝑌𝑊 = 35

a) Prove (with reasons) that 𝑋𝑍𝑌 = 35 (3)

Grade 11 19 of 25 Mathematics Paper 2

b) If 𝑋 = 30 , calculate (with reasons) the value of 𝑌 (3)

c) Hence, calculate (with reasons) the value of 𝑋 (2)

Grade 11 20 of 25 Mathematics Paper 2

QUESTION 15 14 MARKS

O is the centre of the bigger circle. O lies on the circumference of the smaller circle.

OWVU are points on the circumference of the smaller circle.

TSVU are points on the circumference of the bigger circle

a) Determine 𝑂 and 𝑆 in terms of 𝑥 (2)

Grade 11 21 of 25 Mathematics Paper 2

b) Name THREE angles equal to 𝑥, with reasons. (4)

c) Determine the size of 𝑊 in terms of 𝑥, with reasons (3)

d) Prove that WS = WV (5)

Grade 11 22 of 25 Mathematics Paper 2

EXTRA PAPER FOR WORKING OR WHERE MORE SPACE WAS REQUIRED

IF ANYTHING ON THIS PAGE NEEDS TO BE MARKED, LABEL IT CLEARLY WITH THE QUESTION NUMBER AND ALSO STATE THAT YOU HAVE WORKED HERE ON THE QUESTION ITSELF

Grade 11 23 of 25 Mathematics Paper 2

DIAGRAM SHEET

QUESTION 12

QUESTION 14

Grade 11 24 of 25 Mathematics Paper 2

QUESTION 15

Grade 11 25 of 25 Mathematics Paper 2

FORMULA SHEET