MATHS P2 GR12 QP JUNE 2024 English hlayiso.com

NATIONAL SENIOR CERTIFICATE GRADE 12 JUNE 2024 MATHEMATICS P2 MARKS: 150 TIME: 3 hours This question paper consists of 15 pages, a formula sheet and an answer book of 25 pages.

2 MATHEMATICS P2 (EC/JUNE 2024) 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 the 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

(EC/JUNE 2024) MATHEMATICS P2 3 QUESTION 1 The maximum daily temperatures in Bloemfontein for the first 11 days in January were recorded as indicated in the table below. 27 32 35 36 30 27 17 26 34 37 40 1.1 Calculate the mean for the maximum daily temperatures for the first 11 days in January. (2) 1.2 Calculate the standard deviation. (1) 1.3 How many days were the temperatures more than one standard deviation of the mean? (3) 1.4 Determine the interquartile range of the data. (3) 1.5 Draw a box-and-whisker diagram in the grid provided on the ANSWER BOOK. (3) [12] Copyright reserved Please turn over

4 MATHEMATICS P2 (EC/JUNE 2024) QUESTION 2 In a certain school, the analysis of mathematics matric results in percentages were represented in the cumulative frequency graph (ogive) given below. Use the above graph to answer the following questions. 2.1 Complete the frequency table provided in the ANSWER BOOK. Percentage obtained Frequency Cumulative frequency 0  x  20 4 20  x  40 18 40  x  60 36 60  x  80 50 80  x  100 54 (2) 2.2 Write down the total number of matriculants who wrote mathematics in this school. (1) 2.3 Write down the modal class. (1) 2.4 Estimate the median percentage for mathematics of this school. (2) 2.5 If the requirement for a learner to be admitted in a certain institution is 70% and more in mathematics, determine how many matriculants will qualify for admission. (2) [8] Copyright reserved Please turn over

(EC/JUNE 2024) MATHEMATICS P2 5 QUESTION 3 In the diagram below, A (–2 ; p), B( –4 ; 0), C(–1 ; –3) and D(4 ; –1) are the vertices of a trapezium. AD||BC. Point G is the y-intercept of line AD. F lies on line AD. y A (–2 ; p) G θ F B (–4 ; 0) O x D (4 ; –1) C (–1 ; –3) 3.1 Determine the length of BC. (2) 3.2 Determine the gradient of BC. (2) 3.3 Determine the equation of line AD in the form y = mx + c. (3) 3.4 Calculate the value of p. (2) 5 1 3.5 If the coordinates of F are ( ; ), show that CF ⊥ AD . 2 2 (2) 3.6 Calculate the size of θ. (3) 3.7 Calculate the area of trapezium ABCD. (4) [18] Copyright reserved Please turn over

6 MATHEMATICS P2 (EC/JUNE 2024) QUESTION 4 In the diagram below, a circle with centre C(3;–1) and a radius of 10 units is drawn. PQR and PT are tangents to the circle at Q and T respectively. PT is parallel to the x-axis. R (k ; 21), C and P are vertices of ∆RCP. QR = 20 units. y R (k ; 21) Q x O C (3; − 1) P T 4.1 ̂ R. Write down the size of CQ (1) 4.2 Calculate the length of RC, and leave your answer in surd form. (2) 4.3 Calculate the value of k, if R lies in the first quadrant. (4) 4.4 Determine the equation of the circle with centre C, passing through T and Q. Write your answer in the form (x − a)2 + (y − b)2 = r2 . (2) 4.5 Determine the equation of PT. (2) Copyright reserved Please turn over

(EC/JUNE 2024) MATHEMATICS P2 7 4.6 The equation of line PR is given by 3y – 4x = 35. 4.6.1 Calculate the coordinates of P. (2) 4.6.2 Calculate the length of PQ with a reason. (2) 4.6.3 Is the area of ∆QRC = area of ∆QCP? Motivate your answer. (3) 4.7 Consider another circle with equation (x − 3)2 + (y + 16)2 = 16 and having centre M. 4.7.1 Write down the coordinates of the centre M. (1) 4.7.2 Write down the length of the radius of the circle with centre M. (1) 4.7.3 Prove that the circle with centre C and the circle with centre M, do not touch each other (intersect). (3) [23] Copyright reserved Please turn over

8 MATHEMATICS P2 (EC/JUNE 2024) QUESTION 5 5.1 If sin 1 4° = p, without using a calculator, determine the values of the following in terms of p: 5.1.1 cos 7 6° (2) 5.1.2 cos 4 4° (4) 5.1.3 2 sin 2 18° . cos 3 8° (3) sin(90° + θ). cos(θ − 360° ) 5.2 Given: 1 + sin(θ − 30° ). cos θ −sin θ. cos(θ − 30° ) 5.2.1 Simplify to a single trigonometric ratio of θ without using a calculator: sin(90° + θ). cos(θ − 360° ) 1+ sin(θ − 30° ). cos θ − sin θ. cos(θ − 30° ) (6) 5.2.2 Write down the maximum value of y = 1+ ( ) ( sin 90 0 +  . cos  − 360 0 ) ( ) ( sin  − 30 0 . cos  − sin  . cos  − 30 0 ) (1) sin 3x 5.3 Prove that = 3 − 4 sin 2 x sin x (5) 5.4 Given: sin2 x + sin 2 x − 3 cos2 x = 0. 5.4.1 Determine the general solution of the above equation. (5) 5.4.2 Hence, or otherwise, determine the values of x in the interval x ∈ [−90° ;180° ]. (3) [29] Copyright reserved Please turn over

(EC/JUNE 2024) MATHEMATICS P2 9 QUESTION 6 1 6.1 Sketch the graphs of f ( x) = 3sin x and g ( x) = tan x in the interval of 2   x  − 180 0 ;180 0 on the grid provided in the ANSWER BOOK. Clearly show all intercepts with the axes, turning points and the asymptotes. (6) 6.2  Use your graphs to answer the following questions for x  − 180 0 ;180 0  6.2.1 Write down the period of g. (1) 6.2.2 Write down the value(s) of x for which the graph of g is undefined. (2) 6.2.3 Write down the range of h if h( x) = f ( x) − 2 . (2) 6.2.4 How many solutions exists for f ( x) = g ( x) ? (1) [12] Copyright reserved Please turn over

10 MATHEMATICS P2 (EC/JUNE 2024) QUESTION 7 In the diagram below, PQ is a vertical pole having height h metres. R, Q and T are three points on the same horizontal plane. PR and PT are cables and the angle of depression from P to T is 30° . PR = 3h and angle RP̂T = 2 x . P 2x 30° h 3h Q R T 7.1 Write down the size of PT̂ Q . (1) 7.2 Determine the length of PT in terms of h. (3) 7.3 Calculate the size of x if RT = 7h . (5) [9] Copyright reserved Please turn over

(EC/JUNE 2024) MATHEMATICS P2 11 QUESTION 8 In the diagram below, the circle having centre O, passes through U, T, S and V. PUK is a tangent to the circle at U. TS|| PK. UOG is a straight line. TŜ U = 75° . T 2 P 1 1 G 2 1 75° S O 1 4 3 5 U 2 1 V K 8.1 Calculate with reasons the size of: 8.1.1 Ô1 (2) 8.1.2 Û 5 (2) 8.1.3 T̂1 (3) 8.1.4 V̂ (3) 8.1.5 Û 3 (2) 8.1.6 Ĝ 2 (3) 8.2 If it is further given that TS = 80 , calculate the length of TG with reasons, and leave your answer in simplest surd form. (2) [17] Copyright reserved Please turn over

12 MATHEMATICS P2 (EC/JUNE 2024) QUESTION 9 9.1 In ∆ABC below, M is a point on AB and N is a point on AC, such that MN || BC. A M N B C AM AN Prove the theorem which states that = . MB NC (5) Copyright reserved Please turn over

(EC/JUNE 2024) MATHEMATICS P2 13 9.2 In the diagram below, ∆PQR is drawn, EG || QF and EF is a straight line. QE : ER = 2 : 5 . PR = 49 units and FG = 10 units. Q E P F 10 G R 49 9.2.1 Calculate the length of GR with reasons. (4) 9.2.2 Prove that FE || PQ. (3) [12] Copyright reserved Please turn over

14 MATHEMATICS P2 (EC/JUNE 2024) QUESTION 10 In the diagram below, A, B, C and D are points on the circumference of the circle. PC and QC are drawn from P and Q respectively and intersect at C. QP is joined. DB||PQ. QB = 5BC . D P A 2 1 T 2 1 C B Q Prove that: CT 1 10.1 = PC 6 (3) 10.2 ΔQAC ||| ΔQBD (4) 10.3 QD.QA = 30BC 2 (3) [10] TOTAL: 150 Copyright reserved Please turn over

(EC/JUNE 2024) MATHEMATICS P2 15 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 + 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 In ABC: = = sin A sin B sin C a2 = b2 + c2 − 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 2  − 1  n 2  fx  (x − x ) i x=  = i =1 2 n n Copyright reserved Please turn over