Mathematics Grade 12 IEB Paper 2 2016 Question Paper

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE EXAMINATION NOVEMBER 2016 MATHEMATICS: PAPER II EXAMINATION NUMBER Time: 3 hours 150 marks PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY 1. This question paper consists of 27 pages and an Information Sheet of 2 pages (i–ii). Please check that your paper is complete. 2. Read the questions carefully. 3. Answer ALL the questions on the question paper and hand this in at the end of the examination. Remember to write your examination number on the space provided. 4. Number your answers exactly as the questions are numbered. 5. Diagrams are not necessarily drawn to scale. 6. You may use an approved non-programmable and non-graphical calculator, unless otherwise stated. 7. Ensure that your calculator is in DEGREE mode. 8. Round off your answers to one decimal digit where necessary, unless otherwise stated. 9. All the necessary working details must be clearly shown. 10. It is in your own interest to write legibly and to present your work neatly. FOR OFFICE USE ONLY: MARKER TO ENTER MARKS Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 TOTAL 8 24 11 16 8 8 12 10 16 11 9 17 /150 IEB Copyright © 2016 PLEASE TURN OVER

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 2 of 27 IEB Copyright © 2016

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 3 of 27 SECTION A QUESTION 1 In the diagram below, straight line AB makes an angle of 135° with the y-axis. AB cuts the y-axis at A. B is joined to C, a point on the x-axis, so that BC is parallel to the y-axis. (a) Explain why ABCO is not a cyclic quadrilateral. (1) (b) If OA = 8 units determine the equation of the line AB. (3) IEB Copyright © 2016 PLEASE TURN OVER

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 4 of 27 (c) If OC = 6 units then determine (1) the equation of BC. (1) (2) the area of OCBA. (3) [8] IEB Copyright © 2016

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 5 of 27 QUESTION 2 2sin 2 (180° − θ ) + sin 2θ 2sin θ If M = and P = cos 2θ cos θ − sin θ (a) (1) Prove that M = P. (5) IEB Copyright © 2016 PLEASE TURN OVER

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 6 of 27 (2) For what value(s) of θ will P be undefined if θ ∈ [ −180°;360°] ? (5) IEB Copyright © 2016

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 7 of 27 2 (b) If sin β = and cos β < 0 then 11 (1) In which quadrant is angle β? (Circle the correct number) (1) 2 1 3 4 (2) Without the use of a calculator, determine the value of tan β . (4) IEB Copyright © 2016 PLEASE TURN OVER

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 8 of 27 (c) (1) Prove that cos (α − 30° ) − cos (α + 30° ) =sin α . (4) (2) Hence, determine the general solution to the equation: cos(α − 30°) − cos(α + 30°) =2sin 2 α . (5) [24] IEB Copyright © 2016

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 9 of 27 QUESTION 3 In the diagram below: • Circle P touches the x-axis and the y-axis. • Circle P touches circle Q at one point. • Circle Q touches circle R at one point. • Circle P has a radius of 5 units. • PQ is parallel to the x-axis and RQ is parallel to the y-axis. • PR cuts the circles with centres P and R at A and B respectively. (a) If PQ = 9 units, determine the equation of circle centre Q. (4) IEB Copyright © 2016 PLEASE TURN OVER

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 10 of 27 (b) Find the length of line QR if the equation of circle centre R is ( x − p ) 2 + y 2 − 22 y = −117. (3) (c) Determine the length of AB, correct to two decimal digits. (4) [11] IEB Copyright © 2016

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 11 of 27 QUESTION 4 (a) Use the diagram below to prove the theorem that states that the opposite angles of a cyclic quadrilateral are supplementary. (5) IEB Copyright © 2016 PLEASE TURN OVER

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 12 of 27 (b) In the diagram below: • O is the centre of the circle. • A, B and C lie on the circumference of the circle. • Line DAE is a tangent to the circle at point A. • Â= 4 62°. • Â= 2 25°. Calculate the size of angle Ĉ1. (6) IEB Copyright © 2016

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 13 of 27 (c) (1) What single statement can be deduced from both of the following statements? M+N=D M+Q=D (1) (2) In the diagram below: • A, B, C and D lie on the circumference of the circle. B 1 C 2 2 A 1 2 1 D = A Prove that B̂ ˆ +C 1 ˆ . 2 (4) [16] IEB Copyright © 2016 PLEASE TURN OVER

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 14 of 27 QUESTION 5 = The graph of f ( x ) 3sin x + 1 with x ∈ [ 0°;360°] is drawn below. Line AB is parallel to the x-axis. A is a point on the y-axis. (a) Write down the coordinates of point B. (2) (b) Calculate the value(s) for x where f ( x) = −1 if x ∈ [ 0°;360°]. (4) (c) If g ( x) = k and k is a constant term, then for what value(s) of k will f ( x) = g ( x) have no real solutions if x ∈ [ 0°;180°] ? (2) [8] IEB Copyright © 2016

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 15 of 27 QUESTION 6 The manager of a hardware store records the number of staff that are in the store on a given day as well as the sales generated that day. Staff 20 5 17 7 4 8 15 1 10 12 23 14 Sales in 35 200 9 200 32 000 15 600 9 200 17 200 31 200 3 000 19 600 26 800 39 200 20 800 rand (a) Calculate the correlation coefficient for the data above and then comment on the strength of the relationship. (3) (b) Find the equation for the line of best fit in the form y = ... (2) (c) The manager decides to use 19 staff in his store. At the end of the day the sales is R23 000. Would the manager consider this a successful day? Explain your answer. (3) [8] Total for Section A: 75 marks IEB Copyright © 2016 PLEASE TURN OVER

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 16 of 27 SECTION B QUESTION 7 You develop a product and do some market research. The table below is a summary of the ages of people who say they will buy the product. Cumulative Age Mid-point Frequency Frequency 5 < x ≤ 15 10 200 200 15 < x ≤ 25 20 A 450 25 < x ≤ 35 30 20 470 35 < x ≤ 45 40 32 B 45 < x ≤ 55 50 23 525 55 < x ≤ 65 60 300 825 65 < x ≤ 75 70 475 1300 (a) Calculate the values of A and B in the table above. (2) (b) (1) Calculate the estimated mean age of the people who say they will buy the product. (2) (2) Find the modal class interval. (1) IEB Copyright © 2016

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 17 of 27 (c) Sketch the Ogive on the grid below. (3) (d) (1) Is the data normally distributed? Explain. (2) (2) You are developing a marketing strategy. Is the mean age a good indicator of how to advertise your product? Explain your answer. (2) [12] IEB Copyright © 2016 PLEASE TURN OVER

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 18 of 27 QUESTION 8 In the diagram below, circle centre T touches the x-axis at R. • AO is a tangent to the circle at P. • OT = 5 and TP = 3. (a) Determine the coordinates of T. (5) IEB Copyright © 2016

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 19 of 27 (b) Determine ˆ TOR. (2) (c) Determine yP , the y-coordinate of point P. (3) [10] IEB Copyright © 2016 PLEASE TURN OVER

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 20 of 27 QUESTION 9 In the diagram below, C and A are points on the x-axis and y-axis respectively. • OB ⊥ AC. • OB has a length of 20 units. • BC has a length of 80 units. (a) Determine the length of OC. (2) IEB Copyright © 2016

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 21 of 27 (b) Calculate the gradient of line AC. (4) (c) Calculate the coordinates of point B. (5) IEB Copyright © 2016 PLEASE TURN OVER

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 22 of 27 OB2 (d) Prove that ∆ABO / / / ∆OBC and hence deduce that AB = . BC (5) [16] IEB Copyright © 2016

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 23 of 27 QUESTION 10 The diagram below is the top view design of a new railway system. There are eight stations being built and these are labelled with letters from A–H. You have been asked to do some calculations for the railway company. As the engineer you know that: • AF//BE and AC//GD. AB 4 AG 9 • = and = . BC 7 AF 17 A B G H C F E D FE (a) Calculate . FC (3) CD (b) Calculate . DF (2) IEB Copyright © 2016 PLEASE TURN OVER

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 24 of 27 (c) If the straight line distance of the track from F to C is 374 kilometres and it takes 50 hours to build one kilometre of the track, determine the number of hours it will take to build the section from E to D. (6) [11] IEB Copyright © 2016

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 25 of 27 QUESTION 11 In the diagram below: • B, C, D, E and F lie on the circle centre O. • Lines AB and AF are tangents to the circle at B and F respectively. • Line BE passes through O. (a) ˆ +D Prove that C ˆ = 90°. 1 (4) (b) If D̂= 38°, determine the size of BAF. ˆ (5) [9] IEB Copyright © 2016 PLEASE TURN OVER

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 26 of 27 QUESTION 12 In the diagram below: • A, B, C and D lie on the circle centre O. • P is a point on BC so that OP ⊥ BC. • = DC AD = CP = 6 units. • ADCˆ =130 ° and BCD ˆ =θ. (a) Determine the area of ∆ADC. (2) (b) ˆ = 25°. Show that DBC (4) IEB Copyright © 2016

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 27 of 27 (c) Calculate the value of θ if θ < 90°. (6) (d) Given that A, D and C remain fixed points on the circle and that point B is lifted off the plane containing the circle and positioned at a point, T, which is 9 units ˆ vertically above point A. Determine TCA. (5) [17] Total for Section B: 75 marks Total: 150 marks IEB Copyright © 2016