Mathematics Grade 12 P2 EC June 2019 Question Paper (Eng)

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE GRADE 12 JUNE 2019 MATHEMATICS P2 MARKS: 150 TIME: 3 hours *JMATHE2* This question paper consists of 14 pages, including an information sheet and a special answer book of 19 pages.

Downloaded from hlayiso.com 2 MATHEMATICS P2 (EC/JUNE 2019) INSTRUCTIONS AND INFORMATION Read the following instructions carefully before answering the questions. 1. This question paper consists of TEN (10) questions. 2. Answer ALL the questions in the ANSWER BOOK provided. 3. Clearly show ALL calculations, diagrams, graphs, et cetera 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 off answers 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. 9. Write neatly and legibly. Copyright reserved Please turn over

Downloaded from hlayiso.com (EC/JUNE 2019) MATHEMATICS P2 3 QUESTION 1 Eastern High School compared the Term 1 percentages of 20 Grade 12 learners consisting of 10 boys and 10 girls. The following data was recorded: Boys’ marks 41 30 24 65 72 15 83 52 60 38 Girls’ mean mark = 51 Girls’ standard deviation = 15,95 1.1 Calculate the mean mark for boys. (1) 1.2 Calculate the standard deviation for the boys’ marks. (2) 1.3 Did boys or girls perform better? Give a reason for your answer. (2) 1.4 By what percentage must each of the boys’ marks be adjusted so that the mean of boys can be the same as that of the girls? (1) 1.5 Will the boys’ standard deviation increase, decrease or remain the same after the adjustment in QUESTION 1.4 above? (1) [7] Copyright reserved Please turn over

Downloaded from hlayiso.com 4 MATHEMATICS P2 (EC/JUNE 2019) QUESTION 2 The ages of people who were registering to vote at a voting station were recorded in the frequency table below: Ages (in years) Frequency Cumulative Frequency 18  x  28 4 28  x  38 14 38  x  48 28 48  x  58 17 58  x  68 12 68  x  78 3 2.1 Complete the frequency table. (2) 2.2 Draw the cumulative frequency graph (ogive). (3) 2.3 Write down the modal class. (1) 2.4 People who are 60 years and older are regarded as senior citizens and must not queue but be taken to the front. Estimate the number of senior citizens. (2) 2.5 Write down the lower (Q1), middle (Q2) and upper (Q3) quartiles. (3) 2.6 Draw a box and whisker diagram to represent the ages of the voters. (2) [13] Copyright reserved Please turn over

Downloaded from hlayiso.com (EC/JUNE 2019) MATHEMATICS P2 5 QUESTION 3 In the diagram below, K(0 ; 7), L(10 ; 2), M(7 ; – 4 ) and N(x ; y) are vertices of quadrilateral KLMN with MN || KL. θ and α are the angles formed by KM and ML with the x-axis respectively. 3.1 Determine: 3.1.1 The length of KL. Leave your answer in simplified surd form (2) 3.1.2 The gradient of KM (2) 3.1.3 The size of α, the angle of inclination of LM (3)  3.1.4 The size of LMK (4) 3.2 Determine the coordinates of N if KLMN is a parallelogram. Show ALL calculations. (4)  3.3 Is LMN a right angle or not? Justify your answer by calculation(s). (2) 3.4 Determine the area of ∆KNM. (5) [22] Copyright reserved Please turn over

Downloaded from hlayiso.com 6 MATHEMATICS P2 (EC/JUNE 2019) QUESTION 4 In the diagram below, circle with centre M, diameter GH with G(5 ; 0) and tangent HK with point of contact at H(– 1 ; 8) is given. 4.1 Write down the coordinates of M. (2) 4.2 Determine the equation of the circle in the form x  a    y  b   r 2 2 2 (3) 4.3 Determine the equation of the tangent HK. (4) 4.4 Determine the coordinates of J. (3) 4.5 Find the new coordinates of J if the circle is rotated 180o around the centre M. (2) 4.6 The equation of another circle is given as x 2  y 2  12 x  2 y  17  0 . Does the centre of the new circle lie on, inside or outside the originally given circle? Justify your answer with relevant calculations. (5) [19] Copyright reserved Please turn over

Downloaded from hlayiso.com (EC/JUNE 2019) MATHEMATICS P2 7 QUESTION 5 5.1 If sin 42  k , determine the following in terms of k . 5.1.1 tan 42° (2) 5.1.2 sin 84° (3) 5.1.3 sin 3° (4) 5.2 Simplify to a single trigonometric ratio: sin ( x  450 ). tan (180  x ).sin (90  x ) cos (  x ) (6) 5.3 Consider the identity: cos 3 4 cos 3 cos 5.3.1 Complete: cos (A + B) = … (1) 5.3.2 Prove the identity: 3 4 3 (4) 5.4 If cos   2 p and cos 2 = 7p , determine the possible value(s) of p. (5) [25] Copyright reserved Please turn over

Downloaded from hlayiso.com 8 MATHEMATICS P2 (EC/JUNE 2019) QUESTION 6 Given below is the graph of f(x) = sin (x – 45°), for x  [–90° ; 180°]. 6.1 Write down the range of f. (1) 6.2 On the same set of axes, sketch the graph of g(x) = tan x for x  [–90° ; 180°] in the SPECIAL ANSWER BOOK. Show ALL intercepts with the axes as well as asymptotes and end points. (3) 6.3 Write down the period of g. (1) 6.4 Write down the value(s) of x for which f(x) = g(x) for x  [–90° ; 180°]. (1) 6.5 For which value(s) of x is f(x).g(x)  0 for x  [0° ; 180°]? (2) 6.6 Write down the equation of h(x) if h(x) is a result of shifting f(x) such that its minimum value is zero. (1) [9] Copyright reserved Please turn over

Downloaded from hlayiso.com (EC/JUNE 2019) MATHEMATICS P2 9 QUESTION 7 In the diagram KN represents a vertical tower, of height h metres, standing on a horizontal   plane LMN. The angle of elevation of K, as seen from L, is w . NLM  y and NML  z . (NOTE: all angles are measured in degrees). h 7.1 Show that LN  tan w (1) h sin  y  z  7.2 Hence, prove that LM  tan w sin z (4) 7.3 Calculate LM if h  38 m, w  21o , y  52o and z  59o . (2) [7] Copyright reserved Please turn over

Downloaded from hlayiso.com 10 MATHEMATICS P2 (EC/JUNE 2019) Give reasons for your statements in QUESTIONS 8, 9 and 10. QUESTION 8 8.1 Complete: The perpendicular bisector of a chord passes through … (1) 8.2 In the diagram below, O is the centre of the circle, AB is a chord and OC  AB. OC produced, intersects the circle at D. AB = 20 cm, CD = 5 cm and OC= x cm. Determine, stating reasons: 8.2.1 The length of AC (2) 8.2.2 The radius of the circle (4) [7] Copyright reserved Please turn over

Downloaded from hlayiso.com (EC/JUNE 2019) MATHEMATICS P2 11 QUESTION 9 9.1 Complete: Exterior angle of a cyclic quadrilateral is equal to … (1) 9.2 In the diagram below, points P, Q, R and T lie on the circumference of a circle. MW and TW are tangents to the circle at P and T respectively. PT is produced to meet    RU at U. Furthermore, MPR  78 , PQT  41 and QTR  34 . 9.2.1 Write down, with reasons, THREE other angles that are each equal to 41°. (6) 9.2.2 Determine the following, stating reasons:  (a) T2 (2)  (b) Q2 (2)  (c) T4 (2)  (d) W (2) 9.2.3 Determine, with reasons, whether: (a) QR || PT or not (2) (b) PRTW is a cyclic quadrilateral or not (2) (c) TQ is a diameter or not (2) [21] Copyright reserved Please turn over

Downloaded from hlayiso.com 12 MATHEMATICS P2 (EC/JUNE 2019) QUESTION 10 10.1 In the diagram below, ∆KLM is given with R and S on KL and KM respectively such that RS || LM. KR KS Prove the theorem which states that (5) RL SM Copyright reserved Please turn over

Downloaded from hlayiso.com (EC/JUNE 2019) MATHEMATICS P2 13 10.2 In the diagram below, ∆ABC is drawn with D on BC and F and E on AC such that AB || FD, BF || DE, AB C and BF C. Furthermore, CA = 13 units and CB = 12 units. 10.2.1 Write down the length of AB (1) 10.2.2 Prove, stating reasons, that: (a) ∆CBA ||| ∆CFB (3) CB2 (b) CF = CA (3) 10.2.3 Hence, determine the length of CF, correct to the nearest unit. (2) 10.2.4 Give the length of AF. (1) 10.2.5 Determine the length of FE. a Leave your answer in the form . b (5) [20] TOTAL: 150 Copyright reserved Please turn over

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Downloaded from hlayiso.com 14 MATHEMATICS P2 (EC/JUNE 2019) INFORMATION SHEET: MATHEMATICS  b  b 2  4ac x 2a A  P (1  ni ) A  P (1  ni ) A  P (1  i ) n A  P (1  i ) n n Tn  a  ( n  1) d Sn  2a  (n  1)d  2 Tn  ar n 1 a r n  1 ; r  1 S  a ; 1  r  1 Sn  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  x y  y2  d  ( x 2  x1 ) 2  ( y 2  y1 ) 2 M  1 2 ; 1   2 2  y  y1 y  mx  c y  y1  m( x  x1 ) m 2 x 2  x1 m  tan  x  a 2   y  b  2  r 2 a b c InABC:   sin A sin B sin C a 2  b 2  c 2  2bc. cos A 1 area ΔABC  ab. 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   1 2 n 2 x  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

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