Gr 12 Gauteng Maths P2 Sep 2020_hlayiso.com_.pdf

Downloaded from hlayiso.com Mathematics 1 (Paper 2) 10612/20 GAUTENG DEPARTMENT OF EDUCATION PREPARATORY EXAMINATION 2020 10612 MATHEMATICS PAPER 2 TIME: 3 hours MARKS: 150 14 pages + 1 information sheet P.T.O.

Downloaded from hlayiso.com Mathematics 2 (Paper 2) 10612/20 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 ANSWER BOOK provided. 3. Clearly show ALL calculations, diagrams, graphs etc. that you have used to determine 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. P.T.O.

Downloaded from hlayiso.com Mathematics 3 (Paper 2) 10612/20 QUESTION 1 The A-Rithmetic High School decided to compare the results of 31 Grade 12 learners in Mathematics and Physical Sciences in the 2019 Preparatory Examination.  The Mathematics results are recorded in the table below.  The box and whisker plot below illustrates the results of Physical Sciences.  Marks are recorded as percentages. Mathematics Results 7 11 15 19 19 23 28 28 31 38 39 40 41 48 48 52 53 55 57 59 59 64 67 72 76 83 85 87 89 92 96 - - Physical Sciences Results 5 40 50 70 100 1.1 Calculate the mean mark of the Mathematics learners. (2) 1.2 Comment on the skewness of the Mathematics data. (1) 1.3 Determine which subject performed better in the 2019 Preparatory Examination. Give a reason for your conclusion. (2) 1.4 Write down a possible mark for a learner who achieved the tenth lowest mark in Physical Sciences. (2) 1.5 A learner scored the fourth highest in both subjects. The learner obtained the GREATEST possible difference between both subjects. Calculate the learner’s mark in Physical Sciences. (2) [9] P.T.O.

Downloaded from hlayiso.com Mathematics 4 (Paper 2) 10612/20 QUESTION 2 A question raised by many educators is whether the results that a learner achieves in an examination is dependent on the time that the learner takes to complete the examination. The average time taken by each of the top 10 Mathematics learners was recorded. The data is represented in the table and scatter plot below. Average time 175 165 160 153 139 130 127 135 120 112 (in minutes) Learners’ marks 96 92 89 87 85 83 76 72 67 64 (in %) 100 SCATTER PLOT 95 90 Learners’ marks (in %) 85 80 75 70 65 60 55 50 100 105 110 115 120 125 130 135 140 145 150 155 160 165 170 175 180 Average time (in minutes) 2.1 Calculate the equation of the least squares regression line for the data. (3) 2.2 A learner completed the exam in 2,5 hours. Predict the mark that the learner achieved. (2) 2.3 Explain within the context why the regression line is not reliable. (1) 2.4 Calculate the standard deviation of the top 10 Mathematics learners. (2) 2.5 It is further given that  p ; 103,59  is the interval of 15 random learners’ marks within ONE standard deviation of the mean. If x  63,96 , calculate the value of p. (3) [11] P.T.O.

Downloaded from hlayiso.com Mathematics 5 (Paper 2) 10612/20 QUESTION 3 In the diagram below, points A (5 ; 0) , B(7 ; 6) and P ( x ; y) form a triangle. BP = AP and E is the midpoint of AB. 𝑦 E O 𝑥 3.1 Determine the coordinates of E. (2) 3.2 Determine the equation of line BA. (3) 3.3 Line BA is parallel to the straight line with equation rx  3 y  5  0. Calculate the value of r. (3) 3.4 If the area of AOP = 10 units2 and y  0 , calculate the coordinates of P. (7) [15] P.T.O.

Downloaded from hlayiso.com Mathematics 6 (Paper 2) 10612/20 QUESTION 4 The diagram below shows a circle with centre B(a ; b) . BP is parallel to the 𝑦-axis with P on the 𝑥-axis. AS is a tangent to circle B at A (2 ; 4) and intersects the 𝑥-axis at S and the 𝑦-axis at R. AE is a tangent to the smaller circle with centre D and touches the circle at E. ORSˆ = 45. 𝑦 A (2; 4) E B(a ; b) R 45 D P 𝑥 O S 4.1 Determine the equation of tangent AS. (4) 4.2 If OP = 4 units, determine the values of a and b, the centre of the larger circle. (4) 4.3 Determine the equation of the circle with centre B. (3) 4.4 The equation of the smaller circle with centre D is x2  2 x  y 2  2 y  0. Write this equation in the form ( x  a)2  ( y  b)2  r 2 . (3) 4.5 Write down the coordinates of D, the centre of the smaller circle. (1) 4.6 Calculate the length of AE, the tangent to circle D at E. (6) [21] P.T.O.

Downloaded from hlayiso.com Mathematics 7 (Paper 2) 10612/20 QUESTION 5 5.1 Calculate the value of 1  4sin 2 15 without the use of a calculator. (5) 5.2 Simplify without the use of a calculator: 3 sin x .sin 2 72  sin 2 198. 3 cos  x  90  (6) tan120. sin x 5.3 Determine the general solution of the following: 6sin x .cos x  3cos x  4sin 2 x  2sin x  0 (7) 5.4 Prove that:  cos A  1 1  tan A    (4)  cos 2A  cos A  sin A 5.5 If sin 2  k and 0  2  90, determine in terms of k: 5.5.1 cos 2 (2) sin 2 5.5.2 (5) tan  [29] P.T.O.

Downloaded from hlayiso.com Mathematics 8 (Paper 2) 10612/20 QUESTION 6 The sketch below shows the graphs of f ( x)  a sin x and g ( x)  cos dx for x 180 ; 180.  1 A  30 ;  is a point of intersection of f and g .  2 1 f D 0 1 6.1 Write down the values of a and 𝑑. (2) 6.2 Determine the coordinates of D. (1) 6.3 For which value(s) of 𝑥 is: 6.3.1 f decreasing for x 180 ; 180 ? (2) 6.3.2 f ( x) . g ( x)  0 for x 180 ; 0 ? (2) [7] P.T.O.

Downloaded from hlayiso.com Mathematics 9 (Paper 2) 10612/20 QUESTION 7 In the figure below, KM is a vertical flag post set in the centre of two circles which lie on the same horizontal plane. MKN ˆ  MLK ˆ = x. The radius of the inner circle ML  r units and the radius of the outer circle MN  2r units. K L M N 7.1 Calculate the value of 𝑥. (6) 7.2 ˆ  110, calculate the length of LN. If r  5 m and LMN (2) [8] P.T.O.

Downloaded from hlayiso.com Mathematics 10 (Paper 2) 10612/20 QUESTION 8 In the diagram below, points A, B, C and D lie on the circumference of a circle with AD || EC. CB is produced to E. GD is a tangent to the circle at D and DB = AD. ˆ  67. EBA E A 67º 1 B 2 3 G C 3 21 D 8.1 Calculate, with reasons, the size of the following angles: 8.1.1 AD̂C (2) 8.1.2 Ĉ (1) 8.1.3 Â (1) 8.1.4 D̂ 2 (3) 8.1.5 BD̂G (2) 8.2 Prove that AB = CD. (2) [11] P.T.O.

Downloaded from hlayiso.com Mathematics 11 (Paper 2) 10612/20 QUESTION 9 9.1 In the diagram below, A, B, C and D are points on a circle with centre O. OB intersects AC at M, the midpoint of chord AC. Let BD̂C  x . D A 1 21 R 2 1 2 B O 1 1= M C 9.1.1 Determine, with reasons, in terms of x: (a) Ô1 (1) (b) AB̂O (4) 9.1.2 Prove that AB is a tangent to the circle that passes through points A, D and R. (6) 9.1.3 Prove that AD2  4DO2  4AB2  4MB2 . (4) P.T.O.

Downloaded from hlayiso.com Mathematics 12 (Paper 2) 10612/20 9.2 In the diagram below, LM is a tangent to circle QNMWP at M. NW cuts QM and PM at U and V respectively. NQ = WP and NQ || MP. Q P N 2 1 = U V 1 2 4 3 2 1 1 W L M 9.2.1 State, with reasons, THREE angles equal to M̂ 2 . (3) 9.2.2 Prove that WMV||| QMN . (3) MV MN 9.2.3 Prove that  . (3) WV PW [24] P.T.O.

Downloaded from hlayiso.com Mathematics 13 (Paper 2) 10612/20 QUESTION 10 10.1 In ΔABC below, D and E are points on sides AB and AC respectively such that DE || BC. AD AE Prove the theorem which states that  . DB EC A D E B C (6) P.T.O.

Downloaded from hlayiso.com Mathematics 14 (Paper 2) 10612/20 10.2 ABCD is a parallelogram with diagonals that intersect at M. J is a point on BC. BJ : JC is 2 : 3. AJ meets BD at K. BD || JL and JL meets AC at L. Q is a point on AD such that AB || QM. A B K J Q M L D C 10.2.1 Determine, with reasons, the following ratios: ML (a) (2) LC AK (b) (3) KJ 2 10.2.2 If AB  10 units and BC  AB. 3 Calculate the length of AQ. (4) [15] TOTAL: 150 END

Downloaded from hlayiso.com Mathematics 15 (Paper 2) 10612/20 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 x [ 1  i   1] x [1  (1  i)n ] n F P i i f ( x  h)  f ( x ) f ' ( x)  lim h 0 h  x  x2 y1  y 2  d  ( x2  x1 ) 2  ( y2  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:   sin A sin B sin C a 2  b2  c 2  2bc. cos A 1 area ABC = ab.sinC 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 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