Mathematics Grade 12 IEB Paper 2 2019 Question Paper

Downloaded from hlayiso.com TOTAL MARKS NATIONAL SENIOR CERTIFICATE EXAMINATION NOVEMBER 2019 MATHEMATICS: PAPER II EXAMINATION NUMBER Time: 3 hours 150 marks PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY 1. This question paper consists of 32 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. Four blank pages (pages 29 to 32) have been included at the end of the exam paper. If you run out of space for a question, use these pages. If you use this extra space, make sure that you indicate this clearly at the question to ensure that your answer is marked in full. 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. Clearly show ALL calculations, diagrams, graphs etc. that you have used in determining your answers. Answers only will NOT necessarily be awarded full marks. 9. It is in your own interest to write legibly and to present your work neatly. 10. 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 14 14 8 18 10 11 8 21 17 12 7 10 /150 IEB Copyright © 2019 PLEASE TURN OVER

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 2 of 32 SECTION A QUESTION 1 In the diagram below: • P is the centre of the circle. • AB and CD pass through P. • A, B, C and D lie on the circle. • The equation of line AB is y = −2 x + 3. • The equation of line CD is y = 3 x − 7. (a) Determine the coordinates of A. (1) (b) Calculate the coordinates of point P, the centre of the circle. (3) IEB Copyright © 2019

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 3 of 32 (c) Calculate the length of AP. (2) (d) Write down the equation of the circle with centre P in the form ( x − p) + ( y − q ) r 2 2 =. 2 (2) (e) Calculate the coordinates of B. (2) (f) If Q is the x-intercept of AB, determine the equation of the line that is parallel to CD and passing through Q. (4) [14] IEB Copyright © 2019 PLEASE TURN OVER

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 4 of 32 QUESTION 2 (a) If cos 25° =m , then, without the use of a calculator, determine the value(s) of the following in terms of m : (1) sin25° (2) (2) cos50° (2) (3) cos55° (3) IEB Copyright © 2019

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 5 of 32 (b) Simplify the following expression into a single term: sin21° cos w + cos 21° sin w − tan2 β .cos2 β ( sin w + 21° ) (3) (c) In ∆ABC it is given that AB = AC = 5 units and Bˆ = 72°. Sketch ∆ABC and then calculate the area of ∆ABC. (4) [14] IEB Copyright © 2019 PLEASE TURN OVER

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 6 of 32 QUESTION 3 In the diagram below: The graphs of f ( x ) = 3 sin px and g ( x ) = 2sin x have been drawn for x ∈ [0°;360°] . (a) Write down the value of p. (1) (b) What is the period of f ( x ) ? (1) IEB Copyright © 2019

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 7 of 32 (c) Write down the coordinates of B, correct to one decimal digit. (5) (d) If k is a positive real number, then for what values of k will f ( x ) = k have no real solutions? (1) [8] IEB Copyright © 2019 PLEASE TURN OVER

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 8 of 32 QUESTION 4 (a) Use the diagram below to prove the statement that says, "The acute angle formed by a chord and a tangent at the point of contact is equal to the angle in the alternate segment." ˆ = ABC Required to prove that EAC ˆ . Construction: (1) Proof: (5) IEB Copyright © 2019

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 9 of 32 (b) In the diagram below, B, G, F and E are points on the circle. • EAG is a diameter. • CD is a tangent to the circle at B. • CBE ˆ = 55° . Determine the size of Eˆ1. (6) IEB Copyright © 2019 PLEASE TURN OVER

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 10 of 32 (c) In the diagram below, E, F, G and D lie on the circle with centre C. • Lines ED and GD are extended to H and J respectively. • HB and JB are drawn with HBJ = ˆ 100°. • ˆ FCE = 50° and ˆFGC = 35°. Prove that DHBJ is a cyclic quad. (6) [18] IEB Copyright © 2019

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 11 of 32 QUESTION 5 x (a) If g ( x ) = tan then: 2 (1) Determine all values of x (general solution) for which g ( x ) is undefined. (2) (2) On the set of axes provided below, sketch g ( x ) if x ∈ [ −180° ; 180°] . 3 y 2 1 x 0 -180 -150 -120 -90 -60 -30 0 30 60 90 120 150 180 -1 -2 -3 (3) IEB Copyright © 2019 PLEASE TURN OVER

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 12 of 32 (b) Prove the following identity: sin ( 90° + θ ) .sin2 (θ − 180° ) − cos3 (180° + θ ) − 2sin2 θ cos = 2θ cosθ (5) [10] IEB Copyright © 2019

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 13 of 32 QUESTION 6 (a) The table below is a cumulative frequency table that summarises the ages of people that attended a music concert. Class interval (age in years) Cumulative frequency 0 < x < 15 12 15 < x < 30 28 30 < x < 45 50 How many people who attended the music concert were older than 15 but younger than or equal to thirty years in age? (1) (b) Refer to the cumulative frequency curve representing test results below. Number of people Test results (1) Show on the graph where you would find the median test result. (1) (2) Determine the interquartile range. (2) IEB Copyright © 2019 PLEASE TURN OVER

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 14 of 32 (c) For the data in the table below the line of best fit is y= A + Bx . x 22 31 40 24 42 y 43 58 81 45 80 (1) Calculate the values of A and B: A =_____________ B=_____________ (2) (2) If the correlation coefficient is 0,99; explain why it is not a wise idea to make a prediction for y if x is equal to 230. (1) (d) (1) A box-and-whisker diagram is shown below. Draw in the median on the diagram above so that the plot represents data that is skewed to the right or positively skewed. (1) (2) The managing director of a sales department with 120 people gives an annual bonus to the top 40 performing individuals. Explain why the annual average income of his staff is positively skewed. (1) (e) You have a machine that cuts wooden poles into ten-metre lengths. You take a random sample of poles and measure their lengths in metres and calculate the standard deviation of the lengths to be 0,8. Do you think the machine needs a service? Explain. (2) [11] 75 marks IEB Copyright © 2019

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 15 of 32 SECTION B QUESTION 7 The diagram below shows the relationship between the time spent on an electronic device and the amount of time spent doing physical activity during the day. Time spent doing physical activity during the day (minutes) Time spent on an electronic device during the day (minutes) (a) Circle the correlation coefficient that best describes the data represented in the diagram above: = r −1 r 1 = r 0,8 = r −0,8 = (1) (b) If A and B were removed from the data set above, what would happen to the: (1) correlation coefficient? (1) (2) gradient of the line of best fit? (1) (3) Circle the line below which best describes the person represented by A. • A person who has just bought an i-pad and plays computer games. • A person who watches sport on television and likes to read books. • A person who plays professional sport and studies via the internet. (1) IEB Copyright © 2019 PLEASE TURN OVER

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 16 of 32 (c) Please refer to the information in the table below and answer the questions that follow. COFFEE SHOP A COFFEE SHOP B Days of the week Cups of coffee sold per day Cups of coffee sold per day Monday low fairly high Tuesday low fairly high Wednesday low fairly high Thursday low fairly high Friday high fairly high Saturday high fairly high Sunday high low MEAN 350 cups/day 350 cups/day Standard Deviation m cups/day p cups/day (1) Explain why the standard deviation at Coffee Shop B is smaller than the standard deviation at Coffee Shop A. (1) (2) If Coffee Shop A decides to sell coffee at a higher price on the weekends, then how would this affect the mean and standard deviation? (2) (3) What possible strategy could coffee shop B introduce so that the mean and standard deviation both increase? (Explain your answer.) (1) [8] IEB Copyright © 2019

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 17 of 32 QUESTION 8 (a) In the diagram alongside: • CD||NH. • NH intersects DE at G. • K is a point on DE with DG=GK. • HK||FE. • CN:CE = 2:5. EK (1) Determine . (Show all your working.) KG (3) Area of DDGH (2) Calculate . Area of DDEF (4) IEB Copyright © 2019 PLEASE TURN OVER

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 18 of 32 (b) In the diagram below, H; E; N and G lie on the circle with centre O. • NC is a tangent at N and passes through P. • B lies on HC with GB||NC. • HP and GB intersect at E. • HG = HE. • GO � N = 2x. (1) Prove that GN = NE. (6) IEB Copyright © 2019

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 19 of 32 (2) Prove that ∆GON ||| ∆GHE. (5) ON × GE HB × EP (3) Show that = . GN BC (3) [21] IEB Copyright © 2019 PLEASE TURN OVER

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 20 of 32 QUESTION 9 (a) A metal frame is built to help provide some shade to a triangular piece of land ABC. • A, B and C are on the same horizontal plane. • AC = 7 metres; CB = 8 metres and AB = 10 metres. • AF, BG and CH are vertical metal poles. • AF = BG = 3 metres and CH = 2 metres. • HF, FG and GH are metal poles that complete the metal frame. Calculate the area of ∆FGH. (The area of canvas required.) IEB Copyright © 2019

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Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 22 of 32 (b) In the diagram below, C and A are points that lie on the circle. • C and B lie on the x-axis. • AB is a tangent at point A(5; 3). • The equation of the circle is x 2 + y 2 − 6 x − 4 y + 8 =0. (1) Find the coordinates of C. (2) IEB Copyright © 2019

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 23 of 32 (2) Calculate the length of CB. (8) [17] IEB Copyright © 2019 PLEASE TURN OVER

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 24 of 32 QUESTION 10 In the diagram below, A; B and F lie on the circle. • The equation of line EA is 3 y − 2 x =8. • The gradient of line AF is –1. (a) ˆ . Calculate the size of EAF (5) IEB Copyright © 2019

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 25 of 32 (b) If EA = 52 and FB = 40 then calculate the length of CB if the centre of the circle lies on CB and CB ⊥ AF. (7) [12] IEB Copyright © 2019 PLEASE TURN OVER

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 26 of 32 QUESTION 11 In the diagram below, C, D and M are points on the circle. • MCD ˆ = x. • KD is a tangent to the circle at D. • E is a point on DK. • EM is another tangent to the circle at M. • KME ˆ =x + 45° and EKM ˆ = 2 x − 40°. Determine the value of x. [7] IEB Copyright © 2019

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 27 of 32 QUESTION 12 The diagram below is an aerial view of four wind turbines placed at A, D, E and B. • Line AB has equation 5 x + 12y = 60 . • A lies on the y -axis. • B lies on the x -axis. • E is the midpoint of DB. • C lies on AB and represents the control station. DD • The area of ADC : ECD is 8:9. (a) Calculate the distance of AB. (2) IEB Copyright © 2019 PLEASE TURN OVER

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 28 of 32 (b) Find the coordinates of C. (8) [10] 75 marks Total: 150 marks IEB Copyright © 2019

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 29 of 32 ADDITIONAL SPACE (ALL questions) REMEMBER TO CLEARLY INDICATE AT THE QUESTION THAT YOU USED THE ADDITIONAL SPACE TO ENSURE THAT ALL ANSWERS ARE MARKED. IEB Copyright © 2019 PLEASE TURN OVER

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