Mathematics Grade 12 IEB Paper 2 2018 Question Paper

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE EXAMINATION NOVEMBER 2018 MATHEMATICS: PAPER II EXAMINATION NUMBER Time: 3 hours 150 marks PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY 1. This question paper consists of 24 pages and an Information Sheet of 2 pages (i–ii). Please check that your question paper is complete. 2. Read the questions carefully. 3. Answer ALL the questions on the question paper and hand it in at the end of the examination. Remember to write your examination number in the space provided. 4. Diagrams are not necessarily drawn to scale. 5. You may use an approved non-programmable and non-graphical calculator, unless otherwise stated. 6. Ensure that your calculator is in DEGREE mode. 7. All the necessary working details must be clearly shown. Answers only will not necessarily be awarded full marks. 8. It is in your own interest to write legibly and to present your work neatly. 9. Round off to one decimal place unless otherwise stated. FOR OFFICE USE ONLY: MARKER TO ENTER MARKS Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 TOTAL 17 10 10 23 9 9 8 19 9 10 13 13 /150 IEB Copyright © 2018 PLEASE TURN OVER

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 2 of 24 SECTION A QUESTION 1 In the diagram below M is the midpoint of the line that joins A(1; –1) and B(5; 3). B(5; 3) A(1; –1) (a) Find the coordinates of point M, the midpoint of line AB. (2) (b) Find the equation of a line perpendicular to AB that goes through M (the perpendicular bisector of line AB). (4) IEB Copyright © 2018

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 3 of 24 (c) Calculate the length of line AB. (Leave your answer in surd form.) (2) (d) Determine the equation of a circle that has a centre at M and goes through points A and B in the form of x2 + y2 + ax + by + c = 0. (3) (e) Determine the value(s) of k, if point H(4; k) lies on the circle. (3) (f) Determine the shortest distance of the circle from the y-axis. (3) [17] IEB Copyright © 2018 PLEASE TURN OVER

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 4 of 24 QUESTION 2 In the diagram below, circle centre O is drawn in the Cartesian plane. • MN is a tangent to the circle at M. • N is a point outside the circle with co-ordinates N (11; –5). (a) ˆ . Give a reason for your answer. Write down the size of OMN (2) IEB Copyright © 2018

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 5 of 24 If the equation of the circle above is x 2 − 8 x + ( y + 4 ) = 2 (b) 9 then: (1) Determine the coordinates of O. (2) (2) Determine the length of OM. (2) (c) Calculate the length of MN. (4) [10] IEB Copyright © 2018 PLEASE TURN OVER

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 6 of 24 QUESTION 3 On the set of axis below the graph of f ( x ) = cos 2 x with x ∈ 0o ;180o  has been sketched. 4 3 2 1 0 –1 –2 –3 –4 (a) On the set of axis shown above sketch the graph of g ( x ) = 3 sin2 x with x ∈ 0o ;180o  . (4) IEB Copyright © 2018

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 7 of 24 (b) Calculate the values for x where f ( x ) = g ( x ) if x ∈ 0o ;180o  . (4) g (x) (c) For what values of x will be undefined if x ∈ 0o ;180o  ? f (x) (2) [10] IEB Copyright © 2018 PLEASE TURN OVER

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 8 of 24 QUESTION 4 (a) Use the diagram below to prove the theorem that states: "The angle subtended by a chord at the centre of a circle is twice the size of the angle that it subtends at the circle." ˆ = 2 × ABC Required to prove: AOC ˆ Construction: (1) Proof: (5) IEB Copyright © 2018

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 9 of 24 (b) In the diagram below, circle centre O is drawn. • S, T, B and M are points on the circle. • Line MT cuts line OB at point E. • MSTˆ = 67o. • OBMˆ = 40o. (1) Find the size of Bˆ1. (2) (2) Determine the size of Tˆ2 . (4) (3) Find the size of Mˆ 2 . (2) IEB Copyright © 2018 PLEASE TURN OVER

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 10 of 24 (c) In the diagram below, P, Q, R and S are points on the circle. QS and PR intersect at point T. • The line from V is a tangent at P. • ˆ = 56o. QRP • ˆ = 70o. RPV (1) ˆ . Find the size of RST (5) (2) If Qˆ1 = 37o , then explain why QS is not the diameter of the circle. (4) [23] IEB Copyright © 2018

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 11 of 24 QUESTION 5 In the diagram below, A, B, C and E are points on a circle. AC and BE intersect at point F. D is a point outside the circle. CD and DE are drawn. • ˆ = x. ABE • ˆ = y. BEC • ˆ = θ. CDE (a) If θ= x + y then prove that FCDE is a cyclic quadrilateral. (5) IEB Copyright © 2018 PLEASE TURN OVER

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 12 of 24 (b) If AB = AE, then prove that line AE is a tangent to the circle that goes through F, C, D and E. (4) [9] IEB Copyright © 2018

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 13 of 24 QUESTION 6 Below is a cumulative frequency graph showing the average number of litres of water used per household per day in a town. 9 000 8 000 7 000 Cumulative frequency 6 000 5 000 4 000 3 000 2 000 1 000 0 0 50 100 150 200 250 300 350 Average litres of water used per household per day Other information given: • The lowest amount used by any household during a day is 20 litres of water. • The highest amount used by any household during a day is 280 litres of water. (a) Sketch a box and whisker plot to represent the average number of litres used per household per day. (5) IEB Copyright © 2018 PLEASE TURN OVER

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 14 of 24 (b) How many households use more than 100 litres of water per day? (1) (c) Describe the skewness of the data. (Give a reason for your answer.) (1) (d) As a result of a price increase, many of the households using in excess of 100 litres of water per day on average intend to reduce their water consumption substantially. How would this affect (1) standard deviation? (Explain your answer.) (1) (2) the skewness of the data? (Explain your answer.) (1) [9] 78 marks IEB Copyright © 2018

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 15 of 24 SECTION B QUESTION 7 Information about advertising via pamphlets and television separately is given in the table below. • The correlation between the expenditure (in rand) on advertising via pamphlets (x) and sales (y) is recorded in the first row. • The correlation between expenditure (in rand) on television advertising (x) and sales (y) is recorded in the second row of the table. The number of The number of Line of best fit Correlation outliers with high outliers with low (including outliers) Coefficient expenditure and expenditure and LOW sales HIGH sales Pamphlets y = 11 000 + 2x 0,95 1 0 Television y = 8 000 + 4x 0,88 0 3 (a) Which line of best fit would be the most accurate? (Give a reason.) (2) (b) If the outliers were removed, how would this affect the: (1) correlation coefficient of television advertising? (Give a reason.) (2) (2) gradient for each line of best fit? (Give a reason.) (2) (c) Using the information in the table above, explain which method of advertising you would use and why. (2) [8] IEB Copyright © 2018 PLEASE TURN OVER

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 16 of 24 QUESTION 8 (a) If sin 64° = p, determine the value of the following in terms of p: (1) cos334°.sin 244° (3) (2) 8 sin16°.cos16°.cos32° (3) (b) If sin 43o = A and cos(90o − k )cos 23o + cos 246o sin 23o B , = then find the value of k if A = B and 0 ≤ k ≤ 90 . o o (3) 2cos 2θ .cosθ 2 (c) Prove that: + 2 tanθ .sinθ = . cos θ − sin θ 2 2 cos θ (4) IEB Copyright © 2018

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 17 of 24 (d) In the diagram below, ∆FGW is drawn. • FW = 6 units. • FG = 116 units. • ˆ = 45o. GWH 116 units 45o 6 units Calculate the length of GH. (6) [19] IEB Copyright © 2018 PLEASE TURN OVER

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 18 of 24 QUESTION 9 In the diagram below, ∆AFC is drawn. B lies on AC with D and E on CF. • BD//AE and BE//AF. • CB:CA = 5:9. (a) If the length of CA is 18 units, find the length of BA. (2) Area of DBDC (b) Determine the . Area of DBED (3) (c) If the length of CF is 45 units, find the length of DE. (4) [9] IEB Copyright © 2018

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 19 of 24 QUESTION 10 In the diagram below, circle centre O passes through F, S, H and G. • DF and DS are tangents to the circle at F and S respectively. • DF//SG. • FOH intersects SG at R. (a) Prove that DDSF /// ∆OHG. (7) IEB Copyright © 2018 PLEASE TURN OVER

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 20 of 24 SF × FH (b) Show that 2 × DF = . HG (3) [10] IEB Copyright © 2018

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 21 of 24 QUESTION 11 In the diagram below, the given shape is enclosed by line AB, line BC and part of a circle with centre O. • AB = 13 units. • BC = 29 units. • AC = 8 units. • A, K and C are points on the circle. • EK is perpendicular to AC and goes through O. 13 units 29 units 8 units (a) If OE = 0,8 units, find the length of EK. (4) IEB Copyright © 2018 PLEASE TURN OVER

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 22 of 24 13 units 29 units 8 units (b) You cut out the shape above and fold it along the dotted line AC so that ∆ABC is on the vertical plane and the circular shape through the points A, K and C is on the horizontal plane. Calculate the straight line distance from B to K once it has been folded. (9) [13] IEB Copyright © 2018

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 23 of 24 QUESTION 12 A farmer uses a drone to check his fences for any damage. • The drone flies in a perfect circle and passes directly above points A, B and C. • Points A, B, C and D are on the same horizontal plane. • D lies on the x − axis. • At present the angle made between line AD and the x − axis is 45o. (a) Determine the current coordinates of point D. (2) IEB Copyright © 2018 PLEASE TURN OVER

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 24 of 24 (b) ˆ . Calculate the size of ABC (4) (c) How far to the right must D be moved along the x-axis so that the drone will fly directly above it on its circular flight path? (7) [13] 72 marks Total: 150 marks IEB Copyright © 2018