**PRELIMINARY EXAMINATION **

**MATHEMATICS PAPER 2 **

**SEPTEMBER 2021 **

**________________________________________________________________________ **

**MARKS : 150 ** ** ** ** ** ** ** ** ** ** ** ** ** ** DURATION : 3 HOURS **

**________________________________________________________________________ **

**Name of Candidate ** ** **

**Name of Educator ** ** **

**QUESTION ** **1 ** **2 ** **3 ** **4 ** **5 ** **6 ** **7 ** **8 ** **9 ** **10 ** **11 ** **12 ** **TOTAL **

**MARK **

**MAXIMUM ** **10 ** **10 ** **20 ** **25 ** **10 ** **20 ** **14 ** **11 ** **10 ** **6 ** **7 ** **7 ** **150 **

**INFORMATION and INSTRUCTIONS **

### 1. **This question paper consists of 30 PAGES and an Information Sheet of 2 pages. **

### 2. Please check that your paper is complete.

### 3. **Read the questions carefully. **

### 4. Answer all questions on the question paper.

### 5. Extra space is provided at the end of the paper, should this be necessary.

### 6. Diagrams are not necessarily drawn to scale.

### 7. **You may use an approved non-programmable and non-graphical calculator. **

### 8. **Round off to one decimal place, unless specified otherwise. **

### 9. **Ensure that your calculator is in DEGREE mode. **

### 10. **All necessary working details must be clearly shown. **

### 11. Give reasons for statements, unless specified otherwise.

**SECTION A**

**QUESTION 1 **

### a) For a certain data set, the following box-and-whisker diagram was drawn:

### 1) What percentage of data lies between 9 and 15? (1)

### 2) What is the semi-interquartile range of the data? (1)

### b) The table below shows the number of hours that a sales consultant spent with nine clients, in one year, and the value of the respective sales per client:

### 1) Write down the equation of the least squares regression line for the data to two decimal places. (2)

### 2) The sales consultant forgot to record the sales of one his clients. If the consultant spent 80 hours with this client, predict the value of the client’s sales. (2)

### Number of Hours 30 50 100 120 150 190 220 240 260

### Value of sales

### (in thousands of rands) 270 275 500 420 602 150 800 850 820

### 3) Comment on the strength of the relationship between time spent with a client and the value of their sales. Justify your answer appropriately. (2)

### 4) What is the expected increase in sales for each additional hour spent with a client,

### to the nearest rand? (2)

**10 MARKS **

**QUESTION 2 **

### The Grade 12 learners were interviewed about using a certain application to send SMS messages. The average number of SMS messages, 𝑚, sent by each learner per month, was summarised in the histogram below.

### a) How many learners sent messages in total? (1)

### b) Write down the modal class. (1)

**This page left **

**BLANK **

### c) Use the grid below to draw an ogive to represent the data. (4)

**OGIVE SHOWING NUMBER OF MESSAGES SENT **

### d) Use the ogive to identify the median for the data. (1)

### e) Estimate the percentage of learners who sent more than 11 messages using this

### application. (2)

### f) Describe the skewness of the data. (1)

### .

**10 MARKS **

**QUESTION 3 **

### a) Points A(–2; 3) and B(3; –2) lie in the Cartesian Plane.

### 1) Calculate the length of AB.

### (2)

### 2) Determine the co-ordinates of M, the midpoint of AB. (2)

**y **

**y**

*M * *P * **x ** *O *

**x**

### B(3; –2)

### A(–2; 3)

### 3) Determine the co-ordinates of the point P, the point where the straight line

### AB cuts the x-axis. (5)

### 4) Determine the size of obtuse AO ̂ P.

### (3)

### 5) Find the equation of the line which is parallel to OB, passing through A. (4)

### b) A line is drawn through the points S(a; b) and T(c ; d).

### 1) Write down the gradient of the line ST, in terms of a, b, c and d. (2)

### 2) Write down the gradient of the line perpendicular to ST in terms of

### a, b, c and d. (2)

**20 MARKS **

**QUESTION 4 **

### a) If sin 34 ^{º} *= t , determine the following in terms of t: *

### 1) tan 34 ^{º} (2)

### 2) cos 68 ^{º} (3)

### 3) sin 64 ^{º} (4)

### b) 1) Simplify to a single trigonometric ratio:

### ^{sin(180}

### º − x)

### cos(−x) . tan (360 ^{º} + x) − sin ( 180 ^{º} + x ) cos (90 ^{º} + x) (7)

*2) Without using a calculator, determine the value of: (4) * sin ^{2} 35 ^{º} − cos ^{2} 35 ^{º}

### 4sin10 ^{º} . cos10 ^{º}

### 3) Prove the following identity:

### 2sin ^{2} x

### 2tanx − sin 2x = _{tan x} ^{1} ^{ } (5)

**25 MARKS **

**QUESTION 5 **

### In the diagram, T, Q, V and S are points on the circle. QT and VS are produced to P and QS is produced to R such that 𝑃𝑅 ∥ 𝑇𝑉.

### a)

### b)

### Give reasons for the following statements:

### 1) P ̂

### 2 =V ̂ ……….……… (1)

### 2) V ̂ = Q ̂ ………. (1)

### Hence, prove that ΔPQR ||| ΔSPR. (3)

### 2

### 2

### 2 1 1

### 1

### S T

### V

### R

### Q

### P

### c) Now, complete the following:

### 1) ^{PQ}

### PR = _{...} ^{SP} (1)

### 2) ^{QR}

### ... = ^{PR} _{SR} (1)

### d) Calculate the length of PR if QR = 10,2 units and SR = 3,4 units. (3)

**10 MARKS **

**______________________________________________________________________ **

**TOTAL SECTION A: 75 MARKS **

**SECTION B **

**QUESTION 6 **

### The circle centred at M (-4;-5) touches the x-axis at D and cuts the y-axis at P and N. Tangents drawn to the circle at P and N meet at Q.

### a) Explain why the co-ordinates of D are (-4;0). (2)

### b) Show that the equation of the circle can be written as (x + 4) ^{2} + (y + 5) ^{2} = 25 (3)

### c) Find the co-ordinates of P. (4)

**D **

**P **

**N ** **M(-4;-5) **

**Q **

**x **

**x**

**y **

**y**

### d) Say what kind of quadrilateral MNQP is, giving reasons. (3)

### e) Find the co-ordinates of Q. (8)

**20 MARKS ** **QUESTION 7 **

### In the diagram below, the graphs of f(x)= tan x and g(x)= sin 2x have been drawn for x ∈ [-180 ^{0} ; 90 ^{0} ].

### a) Calculate the general solution for f(x) = g(x). (8)

### f(x) f(x)

### g(x)

### b) Hence, or otherwise, determine the values of x if f(x) = g(x) and x ∈ [-180 ^{º} ;0 ^{º} ). (3)

### (c) Use the solution(s) obtained in (b) to determine for which value(s) of x ∈ [-180 ^{º} ;90 ^{º} ) ^{g}

### / (x)

### f(x) ≥ 0. (3)

**14 MARKS **

**QUESTION 8 **

### In the diagram, an isosceles triangle MLN with LM = LN is inscribed in a circle. The length of LM is fixed at 2 units and ML̂N = 𝜃.

### a) Show that the area of ∆ LMN is 2 sin 𝜃 square units. (2)

### b) If the radius of the circle in (a) is allowed to vary, calculate the following:

### 1) the value(s) of 𝜃 when the area of the inscribed ∆ LMN is one square unit. (3)

### 2) the radius of the circle for which the area of the inscribed ∆ LMN is a maximum. (Leave your answer in surd form if necessary). (6)

**11 MARKS **

**QUESTION 9 **

### a) Prove the theorem which states that the angle between a tangent and a chord is equal to the angle in the alternate segment.

**Given: AC is a tangent to circle centre O at B. **

### E and D are points on the circle.

**Required to prove: ** CBD = BED ˆ ˆ

**Construction:. (1) **

**Proof: (4) **

### O D E

### C

### A B

### b) In the figure, AB and AC are tangents to the circle through B, C and F. AD is drawn parallel to FC and meets CD and BF produced in D. BC is drawn and CD is

### produced to E.

### 1) If CB ̂F = x, find two other angles each equal to x, giving reasons. (3)

### Hence, prove that ABCD is a cyclic quadrilateral, giving reasons. (2)

**10 MARKS **

### 3

### 2

### 2 2 2 1

### 1

### 1 1

### 1

*y* *x*

### F

### E C D

### B

### A

### 2)

### 2 x

### 3

**QUESTION 10 **

**Multiple Choice: Circle the letter that corresponds to the CORRECT answer. **

### a) In the figure, BA ̂C = 90 ^{0} and AD ⊥ BC, then

### (A) BD × CD = BC ^{2} (B) BD × CD = AD ^{2} (C) BA × CA = BC ^{2} (D) AB × AC = AD ^{2}

### b) If ΔABC ||| ΔEDF and ABC is not similar to DEF , then which of the following is NOT TRUE?

### (A) *AB.EF = AC.DE * (B) *BC.EF = AC.FD * (C) BC.DE = AB.EF (D) BC.DE = AB.FD

### c) If it is given that BC 1 area ΔPRQ

### ABC ||| PQR with then equal to

### QR 3 area ΔBCA

### =

### (A) 9 (B) 3 (C) ^{1} _{3} (D) ^{1} _{9}

**6 MARKS ** **A**

**A**

**B**

**B**

**C**

**C**

**D**

**D**

**QUESTION 11 **

### In the diagram below, DEFG is a parallelogram and AB ∥ DG.

**Prove that HC || GF. Give reasons. **

**7 MARKS ** **A**

**A**

**B** **C**

**B**

**C**

**H**

**H**

**D** **E**

**D**

**E**

**G**

**G**

**F**

**F**

**QUESTION 12 **

### In the diagram two circles overlap at B and C. Point A is chosen on one circle. Lines AB and AC are produced to meet the other circle at D and E respectively. DE is drawn.

### A second point P is chosen. Lines PC and PB are produced to meet the other circle at Q and R respectively. QR is drawn.

**PROVE that DE = QR **

**B**

**B**

**C**

**C**

**D** **A**

**D**

**A**

**E** **P**

**E**

**P**

**R**

**R**

**Q**

**Q**