Mathematics Grade 12 P2 Western Cape Mock 2019 QP

Downloaded from hlayiso.com Metro North Education District Grade 12 Mathematics P2 September 2019 MARKS: 150 TIME: 3 hours INSTRUCTIONS AND INFORMATION Read the following instructions carefully before answering the questions. 1. This question paper consists of 10 questions. 2. Answer ALL the questions. 3. Clearly show ALL calculations, diagrams, graphs, etc. that you have used in determining your answers. 4. Answers only will not necessarily be awarded full marks. 5. You may use an approved scientific calculator (non-programmable and non-graphical), unless stated otherwise. 6. If necessary, round answers off to TWO decimal places, unless stated otherwise. 7. Diagrams are NOT necessarily drawn to scale. 8. An information sheet, with formulae, is included at the end of the question paper. Number the answers correctly according to the numbering system used in this question paper. 9. Write legibly and present your work neatly. This question paper consists of 13 pages and 1 information sheet and an answer book of 22 pages. 𝐶𝑜𝑝𝑦𝑟𝑖𝑔ℎ𝑡 𝑟𝑒𝑠𝑒𝑟𝑣𝑒𝑑 𝑃𝑙𝑒𝑎𝑠𝑒 𝑡𝑢𝑟𝑛 𝑜𝑣𝑒𝑟

Downloaded from hlayiso.com 𝑀𝑁𝐸𝐷/ 𝑆𝑒𝑝𝑡 2019 𝑀𝑎𝑡ℎ𝑒𝑚𝑎𝑡𝑖𝑐𝑠/𝑃2 2 QUESTION 1 1.1 A number of learners' marks for acertain exam were requested in a survey. Marks from the exam are represented in the cumulative frequency curve (ogive) below: EXAM MARK 55 50 50 48 44 45 40 NUMBER OF STUDENTS 35 35 30 25 23 20 15 15 10 6 5 3 0 0 0 10 20 30 40 50 60 70 80 90 100 110 MARK (%) 1.1.1 How many learners participated in this survey? (1) 1.1.2 Complete the frequency column in the table given in the ANSWER BOOK. Interval Frequency Cumulative frequency 20 ≤ 𝑥 < 30 3 30 ≤ 𝑥 < 40 6 40 ≤ 𝑥 < 50 15 50 ≤ 𝑥 < 60 23 60 ≤ 𝑥 < 70 35 70 ≤ 𝑥 < 80 44 80 ≤ 𝑥 < 90 48 (2) 90 ≤ 𝑥 < 100 50 1.1.3 Use the graph to determine the median mark. (2) 1.1.4 Write down the modal class. (1) 1.1.5 If a learner needs 70% to qualify for a reward, how many learners qualified? (2) 𝐶𝑜𝑝𝑦𝑟𝑖𝑔ℎ𝑡 𝑟𝑒𝑠𝑒𝑟𝑣𝑒𝑑 𝑃𝑙𝑒𝑎𝑠𝑒 𝑡𝑢𝑟𝑛 𝑜𝑣𝑒𝑟

Downloaded from hlayiso.com 𝑀𝑁𝐸𝐷/ 𝑆𝑒𝑝𝑡 2019 𝑀𝑎𝑡ℎ𝑒𝑚𝑎𝑡𝑖𝑐𝑠/𝑃2 3 1.2 The results from a test for students for the year 2000 and the year 2010 are illustrated in the box and whisker plot. The total mark for the test was 30. 1.2.1 Determine the interquartile range for 2010. (2) 1.2.2 In which year did students perform better in the test? Motivate your answer. (2) [12] QUESTION 2 A group of 12 learners have been asked to measure their resting heart rate (beats per minute) and the time (in minutes) that they exercise in a week. The data below was gathered. Minutes of exercise per week Resting heart rate (BPM) 30 82 40 77 60 75 90 70 140 68 180 67 270 60 350 58 360 52 420 50 440 48 500 45 2.1 Represent the data as a scatter plot on the grid provided (3) 2.2 Calculate the correlation coefficient for the given data. (2) 2.3 Use the data to calculate the equation of the least squares regression line. (3) 2.4 If a learner has a resting heart rate of 65 beats per minute, how many hours would you expect him to exercise per week? (3) [11] 𝐶𝑜𝑝𝑦𝑟𝑖𝑔ℎ𝑡 𝑟𝑒𝑠𝑒𝑟𝑣𝑒𝑑 𝑃𝑙𝑒𝑎𝑠𝑒 𝑡𝑢𝑟𝑛 𝑜𝑣𝑒𝑟

Downloaded from hlayiso.com 𝑀𝑁𝐸𝐷/ 𝑆𝑒𝑝𝑡 2019 𝑀𝑎𝑡ℎ𝑒𝑚𝑎𝑡𝑖𝑐𝑠/𝑃2 4 QUESTION 3 In the diagram A, B(−2; −4), C(4; −2) and D(3; 𝑝 ) are the vertices of a rectangle. The diagonals AC and BD intersect at M. 3.1 Given that the length of AC is √50 units, show that 𝑝 = 1. (4) 3.2 Determine the coordinates of M. (3) 3.3 Calculate the gradient of DC (2) 3.4 Determine the equation of line AB in the form 𝑦 = 𝑚𝑥 + 𝑐. (2) [11] 𝐶𝑜𝑝𝑦𝑟𝑖𝑔ℎ𝑡 𝑟𝑒𝑠𝑒𝑟𝑣𝑒𝑑 𝑃𝑙𝑒𝑎𝑠𝑒 𝑡𝑢𝑟𝑛 𝑜𝑣𝑒𝑟

Downloaded from hlayiso.com 𝑀𝑁𝐸𝐷/ 𝑆𝑒𝑝𝑡 2019 𝑀𝑎𝑡ℎ𝑒𝑚𝑎𝑡𝑖𝑐𝑠/𝑃2 5 QUESTION 4 In the diagram is the circle with equation (𝑥 + 1)2 + (𝑦 − 2)2 = 25 . DB is the diameter of the circle and A the centre of the circle. DE is a tangent to the circle at D(−5; −1). The angle E𝐷 ̂ C = 45°. The inclination angles of AD and BC is 𝜃 and 𝛼 respectively. B and C are points on the circumference of the circle. 4.1 Determine: 4.1.1 The coordinates of A, the centre of the circle. (2) 4.1.2 The coordinates of B (3) 4.1.3 The gradient of AD. (2) 4.1.4 The value of 𝜃, the inclination angle of AD. (2) 4.1.5 The equation of the tangent DE. (3) 4.2 Calculate the gradient of BC. (4) 4.3 Another circle with equation 𝑥 2 + 𝑦 2 − 6𝑥 + 2𝑦 = 8 is given. Show that: 4.3.1 The coordinates of the centre of the circle is M(3; −1). (4) 4.3.2 The two circles will intersect each other. Show all calculations. (4) [24] 𝐶𝑜𝑝𝑦𝑟𝑖𝑔ℎ𝑡 𝑟𝑒𝑠𝑒𝑟𝑣𝑒𝑑 𝑃𝑙𝑒𝑎𝑠𝑒 𝑡𝑢𝑟𝑛 𝑜𝑣𝑒𝑟

Downloaded from hlayiso.com 𝑀𝑁𝐸𝐷/ 𝑆𝑒𝑝𝑡 2019 𝑀𝑎𝑡ℎ𝑒𝑚𝑎𝑡𝑖𝑐𝑠/𝑃2 6 QUESTION 5 5.1 If sin40°. cos22° +cos40°. sin22° = 𝑘, determine without the use of a calculator, the value of the following in terms of 𝑘. 5.1.1 sin 62° (2) 5.1.2 tan 118° (4) 5.1.3 sin 14°. cos14° (3) 5.2 Prove the following identity: 1 − cos 2𝜃 =1 (4) sin 2𝜃 × tan 𝜃 5.3 For which value(s) of A will the following expression be real? √sin(180° + A) . cos(90° + A) − tan 45° (6) [19] 𝐶𝑜𝑝𝑦𝑟𝑖𝑔ℎ𝑡 𝑟𝑒𝑠𝑒𝑟𝑣𝑒𝑑 𝑃𝑙𝑒𝑎𝑠𝑒 𝑡𝑢𝑟𝑛 𝑜𝑣𝑒𝑟

Downloaded from hlayiso.com 𝑀𝑁𝐸𝐷/ 𝑆𝑒𝑝𝑡 2019 𝑀𝑎𝑡ℎ𝑒𝑚𝑎𝑡𝑖𝑐𝑠/𝑃2 7 QUESTION 6 In the diagram is the graph of 𝑓(𝑥) = sin(𝑥 + 𝑎) for the interval [-120°;120°] 6.1 Determine the numerical value of 𝑎. (1) 6.2 On the grid provided in the ANSWER BOOK, draw the graph of 𝑔(𝑥) = cos(3𝑥) for the interval 𝑥 ∈ [−120°; 120°]. Clearly show ALL intercepts with the axes, the turning point(s) and endpoint(s) of the graph. (4) 6.3 Determine the general solution for the following: 𝑓(𝑥) = 𝑔(𝑥) (5) 6.4 Determine the values of 𝑥 in the interval 𝑥 ∈ [0°; 120°], for which 𝑓(𝑥) > 𝑔(𝑥) . (2) 6.5 Describe the transformation from graph 𝑔 to the graph of 𝑘(𝑥) = cos(60° − 3𝑥). (2) [14] 𝐶𝑜𝑝𝑦𝑟𝑖𝑔ℎ𝑡 𝑟𝑒𝑠𝑒𝑟𝑣𝑒𝑑 𝑃𝑙𝑒𝑎𝑠𝑒 𝑡𝑢𝑟𝑛 𝑜𝑣𝑒𝑟

Downloaded from hlayiso.com 𝑀𝑁𝐸𝐷/ 𝑆𝑒𝑝𝑡 2019 𝑀𝑎𝑡ℎ𝑒𝑚𝑎𝑡𝑖𝑐𝑠/𝑃2 8 QUESTION 7 A mouse on the ground is looking up to an owl in a tree and a cat to his right on the ground. The angle of elevation from the mouse to the owl is (90° − 2𝜃). AM = 𝑘 𝑢𝑛𝑖𝑡𝑠, GC= 8 𝑢𝑛𝑖𝑡𝑠 , MG ̂ C= 150° and MĈG = 𝜃 7.1 ̂ G in terms of 𝜃. Give the size of MA (1) 7.2 Show that MG= 𝑘 sin 2𝜃 (2) 7.3 Show that MC= 𝑘 cos 𝜃 (4) 7.4 Show that the area of ∆MGC= 2𝑘 sin 2𝜃 (2) [9] 𝐶𝑜𝑝𝑦𝑟𝑖𝑔ℎ𝑡 𝑟𝑒𝑠𝑒𝑟𝑣𝑒𝑑 𝑃𝑙𝑒𝑎𝑠𝑒 𝑡𝑢𝑟𝑛 𝑜𝑣𝑒𝑟

Downloaded from hlayiso.com 𝑀𝑁𝐸𝐷/ 𝑆𝑒𝑝𝑡 2019 𝑀𝑎𝑡ℎ𝑒𝑚𝑎𝑡𝑖𝑐𝑠/𝑃2 9 QUESTION 8 8.1 In the diagram, AOD is a diameter of the circle centred at O. BC = CD and A ̂ 1 = 20°. Determine, with reasons, the size of each of the following angles: 8.1.1 ̂2 A (2) 8.1.2 Ĉ2 (2) 8.1.3 ̂C AB (3) 𝐶𝑜𝑝𝑦𝑟𝑖𝑔ℎ𝑡 𝑟𝑒𝑠𝑒𝑟𝑣𝑒𝑑 𝑃𝑙𝑒𝑎𝑠𝑒 𝑡𝑢𝑟𝑛 𝑜𝑣𝑒𝑟

Downloaded from hlayiso.com 𝑀𝑁𝐸𝐷/ 𝑆𝑒𝑝𝑡 2019 𝑀𝑎𝑡ℎ𝑒𝑚𝑎𝑡𝑖𝑐𝑠/𝑃2 10 8.2 In the diagram, O is the centre of the circle and EC is a tangent to the circle at C. DM = MC and OME is a straight line. Let ̂ O1 = 2𝑥. 8.2.1 Give, with reasons, THREE angles equal to 𝑥. (6) 8.2.2 ̂ 2 = 90° − 𝑥 Prove that O (3) 8.2.3 Prove that DOCE is a cyclic quadrilateral. (4) [20] 𝐶𝑜𝑝𝑦𝑟𝑖𝑔ℎ𝑡 𝑟𝑒𝑠𝑒𝑟𝑣𝑒𝑑 𝑃𝑙𝑒𝑎𝑠𝑒 𝑡𝑢𝑟𝑛 𝑜𝑣𝑒𝑟

Downloaded from hlayiso.com 𝑀𝑁𝐸𝐷/ 𝑆𝑒𝑝𝑡 2019 𝑀𝑎𝑡ℎ𝑒𝑚𝑎𝑡𝑖𝑐𝑠/𝑃2 11 QUESTION 9 9.1 ̂=D In the diagram, ∆ABC and ∆DEF are drawn such that A ̂, B ̂ and Ĉ = F̂. ̂=E Use the diagram in the ANSWER BOOK to prove the theorem which states that if two triangles are equiangular, then the corresponding sides are in proportion, that is DE DF = AB AC (6) 𝐶𝑜𝑝𝑦𝑟𝑖𝑔ℎ𝑡 𝑟𝑒𝑠𝑒𝑟𝑣𝑒𝑑 𝑃𝑙𝑒𝑎𝑠𝑒 𝑡𝑢𝑟𝑛 𝑜𝑣𝑒𝑟

Downloaded from hlayiso.com 𝑀𝑁𝐸𝐷/ 𝑆𝑒𝑝𝑡 2019 𝑀𝑎𝑡ℎ𝑒𝑚𝑎𝑡𝑖𝑐𝑠/𝑃2 12 9.2 AP is a tangent to the circle at P. CB ∥ DP and CB=DP. CBA is a straight line. ̂ = 𝑥 and Ĉ2 = 𝑦 . Let D Prove, with reasons that: 9.2.1 ∆APC lll ∆ ABP (4) 9.2.2 AP² =AB .AC (1) 9.2.3 ∆APC lll ∆ CDP (4) 9.2.4 AP² +PC² = AC² (4) [19] 𝐶𝑜𝑝𝑦𝑟𝑖𝑔ℎ𝑡 𝑟𝑒𝑠𝑒𝑟𝑣𝑒𝑑 𝑃𝑙𝑒𝑎𝑠𝑒 𝑡𝑢𝑟𝑛 𝑜𝑣𝑒𝑟

Downloaded from hlayiso.com 𝑀𝑁𝐸𝐷/ 𝑆𝑒𝑝𝑡 2019 𝑀𝑎𝑡ℎ𝑒𝑚𝑎𝑡𝑖𝑐𝑠/𝑃2 13 QUESTION 10 In ∆ 𝐴𝐵𝐶 in the diagram, D is a point on AB such that AD : DB = 5 : 4. P and E are points on AC such that DE ∥ BC and DP ∥ BE. BC is NOT a diameter of the circle. Given: BD̂ E = 120° , EC = 12 units and BC = 27units. 10.1 Determine, with reasons: 10.1.1 The length of AE (3) 10.1.2 area of ∆ AEB area of ∆ ECB (2) 10.2 Hence, determine the length of DP if ∆ADP lll ∆ABE (6) [11] TOTAL: 150 𝐶𝑜𝑝𝑦𝑟𝑖𝑔ℎ𝑡 𝑟𝑒𝑠𝑒𝑟𝑣𝑒𝑑 𝑃𝑙𝑒𝑎𝑠𝑒 𝑡𝑢𝑟𝑛 𝑜𝑣𝑒𝑟

Downloaded from hlayiso.com 𝑀𝑁𝐸𝐷/ 𝑆𝑒𝑝𝑡 2019 𝑀𝑎𝑡ℎ𝑒𝑚𝑎𝑡𝑖𝑐𝑠/𝑃2 14 MATHEMATICS INFORMATION SHEET:  b  b 2  4ac x 2a A  P (1  ni) A  P (1  ni) A  P (1  i ) n A  P (1  i ) n Tn  a  (n  1)d Sn  n 2a  (n  1)d  2 Tn  ar n 1 Sn    a r n 1 ; r  1 S  a ; 1  r  1 r 1 1 r F  x 1  i   1 n  P x[1  (1  i)n ] i i f ( x  h)  f ( x ) f ' ( x)  lim h 0 h  x  x2 y1  y 2  d  ( x 2  x1 ) 2  ( y 2  y1 ) 2 M  1 ;   2 2  y 2  y1 y  mx  c y  y1  m( x  x1 ) m m  tan  x 2  x1  x  a 2   y  b  2  r 2 a b c 1 In ABC:   a 2  b 2  c 2  2bc. cos A area ABC  ab. sin C sin A sin B sin C 2 sin      sin  . cos   cos  . sin  sin      sin  . cos   cos  . sin  cos     cos  . cos   sin  . sin  cos     cos  . cos   sin  . sin  cos 2   sin 2   cos 2  1  2 sin 2  sin 2  2 sin  . cos  2 cos 2   1  n 2  fx  xi  x  x  2  i 1 n n n( A) P( A)  P(A or B) = P(A) + P(B) – P(A and B) nS  yˆ  a  bx b  x  x ( y  y ) (x  x) 2 𝐶𝑜𝑝𝑦𝑟𝑖𝑔ℎ𝑡 𝑟𝑒𝑠𝑒𝑟𝑣𝑒𝑑 𝑃𝑙𝑒𝑎𝑠𝑒 𝑡𝑢𝑟𝑛 𝑜𝑣𝑒𝑟