Mathematics Grade 12 P2 Gauteng June 2024 QP (Eng)

Downloaded from hlayiso.com JUNE EXAMINATION GRADE 12 2024 MATHEMATICS (PAPER 2) TIME: 3 hours MARKS: 150 13 pages + 1 information sheet and an answer book of 23 pages Please turn over

Downloaded from hlayiso.com MATHEMATICS 2 (PAPER 2) GR12 0624 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 in the SPECIAL ANSWER BOOK provided. 3. Clearly show ALL calculations, diagrams, graphs etc. which 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. Please turn over

Downloaded from hlayiso.com MATHEMATICS 3 (PAPER 2) GR12 0624 QUESTION 1 1.1 The data below shows the marks (out of 100), obtained by 16 learners in a Mathematics test (Test 1). 67 77 26 92 48 38 56 58 75 83 32 94 60 44 64 68 Calculate the: 1.1.1 Mean mark for the test (2) 1.1.2 Standard deviation of the data (1) 1.1.3 The number of learners whose marks are outside one standard deviation of the mean (2) 1.2 The 16 learners, referred to in QUESTION 1.1 above, wrote a second test (Test 2). The box and whisker diagram below shows the distribution of marks obtained by the 16 learners in the two tests. TEST 1 TEST 2 1.2.1 Describe the skewness of the marks for Test 1. (1) 1.2.2 Which test was easier for the learners? Give a reason for your answer. (2) 1.2.3 How many learners scored more than 84% in the second test? (2) [10] Please turn over

Downloaded from hlayiso.com MATHEMATICS 4 (PAPER 2) GR12 0624 QUESTION 2 2.1 In a survey, a group of people were asked about the total distance they had travelled from their homes in the last week of December 2023. The data collected is represented in the frequency table and ogive (cumulative frequency curve) below. Distance travelled Frequency (x kilometres) 50 ≤ x < 100 20 100 ≤ x < 150 30 150 ≤ x < 200 A 200 ≤ x < 250 B 250 ≤ x < 300 80 300 ≤ x < 350 70 350 ≤ x < 400 50 Cumulative frequency Distance travelled in kilometres 2.1.1 How many people participated in this survey? (1) 2.1.2 Determine the values of A and B in the table above. (2) 2.1.3 Use the ogive/table to estimate the number of people who travelled between 100 km and 300 km. (2) 2.1.4 If all people who travelled more than 350 km were removed from the survey, how would this affect the median of the data? (1) 2.2 If the estimated mean of the data below is 16,4, then what is the value of 𝑡 ? Class Interval Frequency 0 < 𝑥 ≤ 10 13 10 < 𝑥 ≤ 20 𝒕 20 < 𝑥 ≤ 30 12 30 ≤ 𝑥 < 40 4 (4) [10] Please turn over

Downloaded from hlayiso.com MATHEMATICS 5 (PAPER 2) GR12 0624 QUESTION 3 In the diagram below, the equation of line AD is 2 y  x  11 . M(1; -1) is the midpoint of the straight line joining B(a; b) and C(3;4). The angles of inclination of AD and BC are  and  respectively. E and F are the y-intercepts of AD and BC respectively. y A(–3 ; 7) E C(3 ; 4) 0 D x M(1 ; –1) F B(a ; b) 3.1 Determine the coordinates of B(a ; b). (3) 3.2 Determine the gradient of BC. (2) 3.3 Determine the size of 𝜃. (Correct to ONE decimal place) (4) 3.4 Determine the equation of line BC. (2) 3.5 Determine the area of ∆CEF. (4) [15] Please turn over

Downloaded from hlayiso.com MATHEMATICS 6 (PAPER 2) GR12 0624 QUESTION 4 The diagram below shows a circle with centre P and equation x 2  y 2  6 x  4 y  12. The circle cuts the y-axis at points R and S. RT is a tangent to the circle at R and cuts the x-axis at T. Angle 𝜃 is indicated. y R P x T O S 4.1 Determine the coordinates of P as well as the radius of the circle. (4) 4.2 Show that the coordinates of R is R(0 ; 6). (2) 4.3 Determine the equation of tangent RT in the form y mx c . (3) 4.4 Determine the size of 𝜃. (Correct to ONE decimal place) (3) 4.5 A vertical line is drawn as a tangent to circle P at Q(a ; b) where a > 0. Write down the coordinates of Q. (2) 4.6 For which values of k will y   3 x  k , be a secant to the circle? (4) 4 4.7 Another circle M with equation ( x  3) 2  ( y  2) 2  36 is drawn. Will circle M touch or cut circle P or will it neither touch nor cut circle P? Show all calculations to determine your answer. (4) [22] Please turn over

Downloaded from hlayiso.com MATHEMATICS 7 (PAPER 2) GR12 0624 QUESTION 5 5.1 Simplify the following expression, without using a calculator: 𝑠𝑖𝑛2𝑥 𝑡𝑎𝑛 (−𝑥) ∙ 𝑠𝑖𝑛 (90° + 𝑥) + 2𝑐𝑜𝑠(360° + 𝑥) (6) 5.2 Given that 𝑠𝑖𝑛27° = 𝑝, determine, without the use of a calculator, each of the following in terms of 𝑝. 5.2.1 𝑐𝑜𝑠27° (2) 5.2.2 𝑠𝑖𝑛 63° (2) 5.2.3 𝑐𝑜𝑠13,5° (3) 5.3 If 𝑐𝑜𝑠𝑥 + 𝑠𝑖𝑛𝑥 = 𝑘, express the following in terms 𝑘: 𝑐𝑜𝑠 (𝑥 − 45 °) (4) 5.4 Prove the identity: 𝑐𝑜𝑠2𝜃 + 1 1 + 𝑡𝑎𝑛𝜃 = (5) 𝑠𝑖𝑛2𝜃 𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜃 5.5 Determine the general solution of the equation: 4𝑠𝑖𝑛 𝜃 = 𝑐𝑜𝑠 (90° − 2𝜃) (6) [28] Please turn over

Downloaded from hlayiso.com MATHEMATICS 8 (PAPER 2) GR12 0624 QUESTION 6 In the diagram below, 𝑓(𝑥) = 𝑎𝑠𝑖𝑛𝑏𝑥 and 𝑔(𝑥) = 4𝑐𝑜𝑠𝑥 are drawn for the interval 𝑥 ∈ [−180°; 180°]. The graph of 𝑓(𝑥) passes through (90°; 5) and the graph of 𝑔(𝑥) passes through (90°; 0). A and B are points of intersection of the graphs of 𝑓 and 𝑔. A(−141,34°; −3,12). y (90° ; 5) B f(x) g(x) x A(–141,34° ; –3,12) 6.1 Determine the values of 𝑎 and 𝑏. (2) 6.2 The graph of 𝑔(𝑥) is shifted 30° to the right and 2 units vertically down to form the graph of ℎ(𝑥). Determine the equation of ℎ(𝑥). (2) 6.3 Calculate the minimum value of in the interval 𝑥 ∈ [−60°; 60°]. ( ) (1) 6.4 If A(−141,34°; −3,12) , determine the coordinates of B. (2) 6.5 For what values of 𝑘 will 𝑓(𝑥) = 𝑘 have no real solutions? (2) 6.6 Use the graph(s) to determine the values of 𝑥 in the interval 𝑥 ∈ [−180°; 180°] for which: 6.6.1 𝑓(𝑥) < 𝑔(𝑥) (2) 6.6.2 𝑓 (𝑥) ∙ 𝑔(𝑥) ≥ 0 (2) [13] Please turn over

Downloaded from hlayiso.com MATHEMATICS 9 (PAPER 2) GR12 0624 QUESTION 7 In the diagram below, TQR represents three points in a horizontal plane on a sportsfield. PQ represents a vertical flagpole. The angle of elevation of the top of the pole from R is equal to θ. T = 𝜃. TQ = TR = y. QR = x units. TQR = 𝛼. P T y 𝛼 y Q x R 7.1 Express 𝜃 in terms of 𝛼 and show that sin𝜃 = sin2𝛼 (2) 7.2 Hence, prove that in ∆PQR: PR = (4) 7.3 If 𝛼 = 49°; x = 20 m and y = 15 m, calculate the area of ∆TQR. (3) [9] Please turn over

Downloaded from hlayiso.com MATHEMATICS 10 (PAPER 2) GR12 0624 QUESTION 8 In the diagram below, O is the centre of the circle. A, B and D are points on the circle. BC is a tangent to the circle at B and DB̂C = 22,5°. Chord AD is produced to C such that CD = BD. C O D A B 8.1 Determine, giving reasons, the size of: 8.1.1 Â 1 (2) 8.1.2 D̂1 (2) 8.1.3 B̂ 2 (2) 8.1.4 Ô1 (2) 8.2 Prove, giving reasons, that OA ∥ CB. (2) 8.3 If it is further given that the radius of the circle is 12 units, calculate the length of BC. (4) [14] Please turn over

Downloaded from hlayiso.com MATHEMATICS 11 (PAPER 2) GR12 0624 QUESTION 9 In the diagram below, the diameter BE of circle O is produced to D. DA is a tangent to the circle and CD  BD. AC and BC cut the circle at E and F respectively. OF and EF are drawn. B O A F E C D 9.1 Prove, with reasons, that ABCD is a cyclic quad. (3) 9.2 Prove, with reasons, that BD bisects AB̂C. (3) 9.3 Prove, with reasons, that EC is a tangent to circle OEF. (4) [10] Please turn over

Downloaded from hlayiso.com MATHEMATICS 12 (PAPER 2) GR12 0624 QUESTION 10 10.1 In the diagram below, ΔABC is drawn with DE ∥ BC. A D E B C AD AE Prove the theorem which states that  . DB EC (5) Please turn over

Downloaded from hlayiso.com MATHEMATICS 13 (PAPER 2) GR12 0624 10.2 In the diagram below, D is the centre of a circle. AB and AE are tangents to the circle at B and E respectively. The diameter BG is produced and meets tangent AE at C. DG = CG. F is a point on AC such that DF ǁ AB. A F E C G D B AC 10.2.1 Find, with reasons, the ratio of . FC (3) 10.2.2 Prove, giving reasons, that ∆ABC /// ∆DEC. (3) AE.EC 10.2.3 Prove that DE 2  (5) 3 AreaFDC 10.2.4 Find the ratio of: (3) AreaABC [19] TOTAL: 150 END

Downloaded from hlayiso.com MATHEMATICS 14 (PAPER 2) GR12 0624 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 n Tn  a  (n  1)d Sn  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  y2  d  ( x 2  x1 ) 2  ( y 2  y1 ) 2 M  1 ;   2 2  y  y1 y  mx  c y  y1  m( x  x1 ) m 2 m  tan x 2  x1 x  a2   y  b2  r 2 a b c InABC:   sinA sinB sinC a 2  b 2  c 2  2bc. cos A 1 area ABC  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  sin2  2sin.cos  2 cos   1 2 n 2 x  x  x  i x   i 1 2 n n n(A) P(A)  P(A or B) = P(A) + P(B) – P(A and B) n S  yˆ  a  bx