Mathematics Grade 12 P2 North West June 2024 QP (Eng)

PROVINCIAL ASSESSMENT GRADE 12 MATHEMATICS P2 JUNE 2024 MARKS: 150 TIME: 3 hours This question paper consists of 11 pages, 1 information sheet and an answer book of 21 pages. Copyright reserved Please turn over

Mathematics/P2 2 NW/June 2024 Grade 12 INSTRUCTIONS AND INFORMATION Read the following instructions carefully before answering the questions. 1. This question paper consists of 9 questions. 2. Answer ALL the questions. 3. Write ALL the answers in the ANSWER BOOK provided. 4. Clearly show ALL calculations, diagrams, graphs, et cetera that you have used in determining the answers. 5. Answers only will NOT necessarily be awarded full marks. 6. You may use an approved scientific calculator (non-programmable and non- graphical), unless stated otherwise. 7. If necessary, answers should be rounded off to TWO decimal places, unless stated otherwise. 8. Diagrams are NOT necessarily drawn to scale. 9. Number the answers correctly according to the numbering system used in this question paper. 10. An information sheet with formulae is included at the end of the question paper. 11. Write neatly and legibly. Copyright reserved Please turn over

Mathematics/P2 3 NW/June 2024 Grade 12 QUESTION 1 In the figure below, A ( – 2; – 6), B(3; 6), C(9; 5) and D are joined to from a kite. y B (3; 6) C (9; 5) E x D A (– 2; – 6 ) 1.1 Calculate the length of AC and give your answer in simplest surd form. (2) 1.2 Calculate the gradient of AC. (2) 1.3 Determine the equation of AC. (2) 1.4 Give a reason why AE  BD. (1) 1.5 Determine the equation of BD. (3) 1.6 Determine the coordinates of E. (3) 1.7 Determine, with reasons, the coordinates of D. (3) 1.8 Calculate the size of AD̂C . (5) [21] Copyright reserved Please turn over

Mathematics/P2 4 NW/June 2024 Grade 12 QUESTION 2 In the diagram, the circle is centred at M (2; 2). Radius KM is produced to L, a point outside the circle, such that KML || y-axis. LTP is a tangent to the circle at T (–2; b). S  6 ;  4  is the midpoint of PK. 2.1 Given that the radius of the circle is 5 units, show that b = 5. (4) 2.2 Determine: 2.2.1 The coordinates of K. (2) 2.2.2 The equation of the tangent LTP in the form y  mx  c . (4) 2.2.3 The area of Δ LPK. (7) Another circle with equation ( x  2)  ( y  n)  25 is drawn. Determine, with 2.3 2 2 an explanation, the value(s) of n for which the two circles will touch each other externally. (4) [21] Copyright reserved Please turn over

Mathematics/P2 5 NW/June 2024 Grade 12 QUESTION 3 3.1 Simplify the expression below to a single trigonometric ratio: cos(90  x).sin( 540   x) (6) tan( x  180  ). cos( x) 3.2 Given cos(P  Q)  cosP.cosQ sinP.sinQ Use the above identity and express cos 2P in terms of cos P. (4) 4 3.3 If tan A  , where A  [90 ;360 ] 3 Use a sketch to determine the following: 3.3.1 sin A (3) 3.3.2 cos(A  30  ) (4) Given: cos12  p . With the aid of the diagram, determine the following without  3.4 using a calculator, in terms of p. 3.4.1 sin 78  (2) 3.4.2 sin 6  (4) [23] QUESTION 4 sin 2 x  cos 2 x  1 4.1 Prove that:  tan x (7) sin 2 x  cos 2 x  1 4.2 Hence, or otherwise, determine the value of tan 22,5 , without using the calculator. (Leave your answer in SURD form). (3) 4.3 Determine the general solution of the equation: 3 cos 2x  1  5 cos x . (7) [17] Copyright reserved Please turn over

Mathematics/P2 6 NW/June 2024 Grade 12 QUESTION 5 The sketch below, shows the graphs of f ( x)  a sin x and g ( x)  sin x  b , for x  [360  ;360  ]. Use the given graphs to answer the following questions: 5.1 Write down the values of a and b. (2) 5.2 What is the amplitude of g? (1) 5.3 Determine the value(s) of x, where x  [360 ;360 ] , for which:   5.3.1 f ( x)  g ( x) . (2) 5.3.2 f / ( x).g ( x)  0 . (3) 5.3.3 g decreases as x is increases. (2) [10] Copyright reserved Please turn over

Mathematics/P2 7 NW/June 2024 Grade 12 QUESTION 6 In the diagram below, a pole LY is placed at the corner of a triangular field XYZ. The area of the field is A m2 . XŶZ  2 , YẐL   and XY = p metres. LY = h metres and YZ = r metres and XZ = t metres 6.1 Determine, A, the area of ΔXYZ in terms of r, p and  . (1) 6.2 Express r in terms of A, p and  . (2) 6.3 Hence, show that the height of the pole, h, is given by: A h p cos2  (5) 6.4 If h = 20m, p = 10m and   60 .  Determine: 6.4.1 A, the area of ΔXYZ. (2) 6.4.2 The length of t correct to 2 decimal places (4) [14] Copyright reserved Please turn over

Mathematics/P2 8 NW/June 2024 Grade 12 QUESTION 7 7.1 In the diagram below, O is the centre of the circle. P, Q, S and R, are points on the circumference of the circle. PQ = QS and PQ̂S  130  . R O 2 1 2 P 1 2 1 S 1300 Q Determine, giving reasons, the sizes of: 7.1.1 Ŝ1 (3) 7.1.2 R̂ (2) 7.1.3 Ô1 (3) Copyright reserved Please turn over

Mathematics/P2 9 NW/June 2024 Grade 12 7.2 In the diagram below, O is the centre of the circle. AB is perpendicular to diameter DC. CM: MD = 2 : 7 and AB = 14 units. If CM = 2x: 7.2.1 Express DC in terms of x. (1) 7.2.2 Express OM in terms of x. (2) 7.2.3 Hence, or otherwise, calculate the length of the radius. (5) [16] Copyright reserved Please turn over

Mathematics/P2 10 NW/June 2024 Grade 12 QUESTION 8 8.1 In the diagram below,  MNC is drawn. A is a point on MN and B is a point on MC such that AB || NC. AC and NC are drawn. M A > B N > C Use the diagram to prove the theorem which states that: MA MB (7)  AN BC 8.2 The figure below shows  ABC with BC produced to D. RD is drawn with point T on AC and R on BA. CS is drawn. TC = 9 cm, AT = 27 cm, AR = 60 cm and AS = 80 cm. 8.2.1 Prove that SC || RT (3) 8.2.2 Determine the length of RS (1) 8.2.3 If AR : RB = 2 : 3 and BC = 30 cm, calculate the length of CD. (5) [16] Copyright reserved Please turn over

Mathematics/P2 11 NW/June 2024 Grade 12 QUESTION 9 In the figure below, KL is a tangent to the circle. L, J and R are points on the circle and MN is parallel to KL. J 1 2 M 3 T 1 2 N 1 2 3 R 1 1 2 2 K V L Prove that: 9.1 JMRN is a cyclic quadrilateral. (3) 9.2 ΔJNR ||| ΔLMK (3) 9.3 NL.JT LM.NR  (6) TV MK [12] TOTAL : 150 Copyright reserved

Mathematics/P2 NW/June 2024 Grade 12 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 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  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:   a 2  b 2  c 2  2bc. cos A sin A sin B sin C 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  cos2   sin 2   cos 2  1  2 sin 2  sin 2  2 sin  . cos 2 cos2   1  n 2  fx  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 b  x  x ( y  y ) (x  x) 2 Copyright reserved