Mathematics Grade 12 P1 Gauteng Prelim 2024 Addendum (Eng)

Downloaded from hlayiso.com PREPARATORY EXAMINATION 2024 CANDIDATE'S NAME DATE BOOK NUMBER OF BOOKS TEACHER PAPER NUMBER 2 SUBJECT NAME MATHEMATICS (10612) ANSWER ALL THE QUESTIONS IN THE QUESTION PAPER. MARKER MODERATOR'S INITIALS IN RELEVANT RE-MARK/RE-CHECK BLOCK Marker's Code Question Marks Marks Question Marks Initials & Initials 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 TOTAL TOTAL READ THE INSTRUCTIONS ON THE NEXT PAGE. TIME: 3 hours MARKS: 150 34 pages + 1 information sheet P.T.O.

Downloaded from hlayiso.com MATHEMATICS (PAPER 2) 10612/24 2 INSTRUCTIONS AND INFORMATION Read the following instructions carefully before answering the questions. 1. This question paper consists of 10 questions. Answer ALL questions in the spaces provided. 2. Clearly show ALL calculations, diagrams, graphs, et cetera that you have used in determining your answers. 3. Answers only will NOT necessarily be awarded full marks. 4. You may use an approved scientific calculator (non-programmable and non-graphical), unless stated otherwise. 5. If necessary, round-off answers correct to TWO decimal places, unless stated otherwise. 6. Diagrams are NOT necessarily drawn to scale. 7. An INFORMATION SHEET with formulae is included at the end of the question paper. 8. No pages may be torn from this question paper. 9. Candidates may not retain a question paper or remove it from the examination room. Question papers must be returned to the invigilator at the end of the examination session. 10. Answers must be written in black/blue ink as distinctly as possible. Do not write in the margins. 11. Indicate the questions you have answered by drawing a circle around the relevant numbers on the front cover of the question paper where marks are to be recorded. 12. Draw a neat line through any work/rough work that must not be marked. 13. In the event that you use the additional space provided: 13.1 Write down the number of the question. 13.2 Leave a line and rule off after your answer. 14. Write neatly and legibly. P.T.O.

Downloaded from hlayiso.com MATHEMATICS (PAPER 2) 10612/24 3 KEEP THIS PAGE BLANK. P.T.O.

Downloaded from hlayiso.com MATHEMATICS (PAPER 2) 10612/24 4 QUESTION 1 The table below shows the monthly salaries of 100 employees at Adams Law. Monthly salary Number of employees Cumulative frequency (in thousand Rand) R0  x  R10 3 R10  x  R20 4 R20  x  R30 13 R30  x  R40 20 R40  x  R50 21 R50  x  R60 12 R60  x  R70 12 R70  x  R80 8 R80  x  R90 5 R90  x  R100 2 1.1 Draw an ogive (cumulative frequency graph) to represent the data given in the table. OGIVE 100 90 80 70 Cumulative Frequency 60 50 40 30 20 10 0 R0 R10 R20 R30 R40 R50 R60 R70 R80 R90 R100 Monthly salary (in thousand Rand) (4) P.T.O.

Downloaded from hlayiso.com MATHEMATICS (PAPER 2) 10612/24 5 1.2 Use the ogive to determine the number of employees that receive less than R35 000 per month. (1) 1.3 Determine the median of the data. (1) 1.4 The monthly salaries of the 5 employees in the interval R80 000  x  R90 000 are given below: R84 000 R85 000 R87 000 R89 000 R89 000 Two employees from the interval R70 000  x  R80 000 are promoted and receive monthly salaries of R83 000 and R84 000 respectively. 1.4.1 Determine the mean monthly salary in the interval R80 000  x  R90 000. (1) 1.4.2 Calculate the standard deviation in this interval. (1) 1.4.3 Determine the percentage of employees whose monthly salary lie within one standard deviation of the mean. (3) [11] P.T.O.

Downloaded from hlayiso.com MATHEMATICS (PAPER 2) 10612/24 6 QUESTION 2 During a netball tournament, there are several stalls selling food and refreshments. In a period of 10 minutes at 6 stalls, the following was observed: Stall Number of customers Amount of money spent (R) 1 10 163 2 7 138 3 9 185 4 12 220 5 8 147 6 13 245 SCATTER PLOT 270 250 230 Amount (R) 210 190 170 150 130 110 5 6 7 8 9 10 11 12 13 14 Number of customers 2.1 Determine the equation of the least squares regression line of the data. (3) 2.2 Predict the amount of money spent by 11 customers. (2) P.T.O.

Downloaded from hlayiso.com MATHEMATICS (PAPER 2) 10612/24 7 2.3 Determine the correlation coefficient of the data. (1) 2.4 The organisers of the event think that there is a very weak positive correlation between the number of customers and the amount of money received at stalls. Motivate whether you agree or not. (1) 2.5 At another stall, 6 customers spent a total amount of R195. If this point is included in the data, will the gradient of the least squares regression line increase or decrease? Motivate your answer without any further calculations. (2) [9] P.T.O.

Downloaded from hlayiso.com MATHEMATICS (PAPER 2) 10612/24 8 QUESTION 3 In the diagram below, P(9 ; 12), Q(9 ; 9) and R are vertices of ∆PQR. M is the midpoint of QR and N(a ; b) is a point on PM in the second quadrant. The equation of QR is given by 2y – 3x + 9 = 0. The angle of inclination of QR is θ. 𝑦 𝜃 𝑥 O M R 3.1 Calculate the coordinates of M, the x-intercept of line PM. (2) P.T.O.

Downloaded from hlayiso.com MATHEMATICS (PAPER 2) 10612/24 9 3.2 Determine the equation of PM in the form y  mx  c . (4) 3.3 Calculate the size of  . (2) 3.4 Show that b  3  a , if P, N and M are collinear. (1) P.T.O.

Downloaded from hlayiso.com MATHEMATICS (PAPER 2) 10612/24 10 3.5 Hence, determine the value of a and b if NQ  5 5 units. (5) 3.6 Determine the equation of a circle having centre at O, the origin, and passing through point R. (4) P.T.O.

Downloaded from hlayiso.com MATHEMATICS (PAPER 2) 10612/24 11 3.7 The acute angle between the lines QR and the line with equation y  mx  4 is 45  . Determine the possible value(s) of m. (4) [22] 𝑦 𝜃 𝑥 O M R P.T.O.

Downloaded from hlayiso.com MATHEMATICS (PAPER 2) 10612/24 12 QUESTION 4 In the diagram, the equation of the circle centred at B is given by ( x  1)2  ( y 1)2  20 . DF is a tangent to the circle at A with D and F, the x- and y-intercepts respectively. C (1; 6) is a point on DF with BC parallel to the y-axis. CBAˆ  ADO ˆ  . y F C A 𝜃 B 𝜃 x O D 4.1 Write down the coordinates of B. (1) 4.2 Show that AC  5 . (3) P.T.O.

Downloaded from hlayiso.com MATHEMATICS (PAPER 2) 10612/24 13 4.3 Write down the value of tan  . (1) 4.4 Show that the equation of AB is given by y  2 x  1 . (3) P.T.O.

Downloaded from hlayiso.com MATHEMATICS (PAPER 2) 10612/24 14 4.5 Determine the coordinates of A. (4) 4.6 Calculate the ratio of the area of  ABC to the area of  ODF. Simplify your answer. (6) [18] P.T.O.

Downloaded from hlayiso.com MATHEMATICS (PAPER 2) 10612/24 15 QUESTION 5 sin 2 180  x  .sin  x  5.1 Simplify  1 to a single trigonometric term.  sin 90  x . tan x (6) 5.2 Given: cos A  B   cosAcosB  sinAsinB 5.2.1 Use the above identity to deduce that sin A  B   sinAcosB  cosAsinB. (3) 5.2.2 Without using a calculator, simplify the following: cos 420 cos 15  sin 300 cos 105 (5) P.T.O.

Downloaded from hlayiso.com MATHEMATICS (PAPER 2) 10612/24 16 Given: tan 𝑥 −1 5.3 Prove that tan 𝑥 −1 = . 5.3.1 (3)  1  5.3.2 2 For what value(s) of x in the interval x  (0 ; 180) is tan x  2  1  tan x  undefined? (1) 5.4 Determine the general solution of the equation cos 2 x  cos x . (6) P.T.O.

Downloaded from hlayiso.com MATHEMATICS (PAPER 2) 10612/24 17 1 5.5 Given that sin   , calculate the numerical value of sin 3 , without using a 2 calculator. (7) [31] P.T.O.

Downloaded from hlayiso.com MATHEMATICS (PAPER 2) 10612/24 18 QUESTION 6 In the diagram below, the graph of f ( x)  2 cos x is drawn for the interval x  [180 ;180] . 6.1 On the grid below, draw the graph of g ( x)   cos 2 x for x  [180 ; 180] . Clearly show all intercepts with the axes and the turning point(s) of the graph. 3 2 𝒇 1 -180° -135° -90° -45° 0° 45° 90° 135° 180° -1 -2 -3 (3) 6.2 Write down the period of 2 g ( x  10). (1) 6.3 Use the graphs to determine the value(s) of 𝑥 in the interval x [0 ;180 ] for which   f ( x).g ( x)  0 . (2) 6.4 Write down the maximum value of f ( x)  g ( x). (1) P.T.O.

Downloaded from hlayiso.com MATHEMATICS (PAPER 2) 10612/24 19 6.5 Determine the range of y  22cos x2 (2) [9] P.T.O.

Downloaded from hlayiso.com MATHEMATICS (PAPER 2) 10612/24 20 QUESTION 7 The diagram below shows two vertical poles, AD and BC. Point E lies on the same horizontal plane as bases D and C of poles AD and BC. A Ê B  2 ; B Ê C  ; A B̂ E  90   3 and BC  y metres 7.1 Determine BE in terms of 𝛼 and y. (2) P.T.O.

Downloaded from hlayiso.com MATHEMATICS (PAPER 2) 10612/24 21 7.2 Show that AB = 2y. (5) 7 7.3 It is further given that AF  BC. 4 Determine BÂF , the angle of elevation of B from A. Give your answer to the nearest degree. (3) [10] P.T.O.

Downloaded from hlayiso.com MATHEMATICS (PAPER 2) 10612/24 22 QUESTION 8 In the diagram below, the circle passes through points A, B, C, D and G. AD is the diameter of the circle. BD and AC intersect at H and AĜC  58 . A 1 3 2 G 2 B 1 2 1 =3 H 2 1 D =C 8.1 Determine, giving reasons, the size of the following angles: 8.1.1 B̂2 8.1.2 B̂1 8.1.3 Â 2 8.1.1 (2) 8.1.2 (2) 8.1.3 (2) P.T.O.

Downloaded from hlayiso.com MATHEMATICS (PAPER 2) 10612/24 23 8.2 If it is given that AB = BC. Prove that AB is a tangent to the circle passing through A, H and D. (3) [9] P.T.O.

Downloaded from hlayiso.com MATHEMATICS (PAPER 2) 10612/24 24 QUESTION 9 9.1 In the diagram below, O is the centre of the circle. Points S, T and R lie on the circle. QS is a tangent to the circle at S. R T O Q S Use the diagram above to prove that the angle between the tangent and the chord is equal to the ˆ = Rˆ . angle in the alternate segment, that is, prove that QST (6) P.T.O.

Downloaded from hlayiso.com MATHEMATICS (PAPER 2) 10612/24 25 KEEP THIS PAGE BLANK. P.T.O.

Downloaded from hlayiso.com MATHEMATICS (PAPER 2) 10612/24 26 9.2 In the diagram below, D, E and F are points on the circle centred at Q. AGF is a tangent to the circle at F. ED is produced to meet the tangent at A. B and C are points on AE such that GB || FD. GC is joined. GCDF is a cyclic quadrilateral and FQ̂D  2 x . 2𝑥 9.2.1 Give a reason why Ê is equal to x . (1) P.T.O.

Downloaded from hlayiso.com MATHEMATICS (PAPER 2) 10612/24 27 9.2.2 Prove that GC || FE. (3) AB AC 9.2.3 Prove that  . BD CE (3) P.T.O.

Downloaded from hlayiso.com MATHEMATICS (PAPER 2) 10612/24 28 2𝑥 9.2.4 It is further given that AB = 12 units, CE = 22,5 units and 5AG = 4GF. Determine the length of CD. (5) [18] P.T.O.

Downloaded from hlayiso.com MATHEMATICS (PAPER 2) 10612/24 29 KEEP THIS PAGE BLANK. P.T.O.

Downloaded from hlayiso.com MATHEMATICS (PAPER 2) 10612/24 30 QUESTION 10 In the diagram below, K, H and G are points on the circle and FG is a tangent to the circle with centre E. Semi-circle with centre F is drawn. Points D and E lie on the semi-circle. CD and CE are tangents to the semi-circle at D and E respectively. CD || HE. 10.1 Give a reason why DC = EC. (1) 10.2 Prove that Δ DCE ||| Δ HEK . (6) P.T.O.

Downloaded from hlayiso.com MATHEMATICS (PAPER 2) 10612/24 31 10.3 Prove that DG = GE. (3) 2 10.4 Show that 2HE = DC × HK. (3) [13] P.T.O.

Downloaded from hlayiso.com MATHEMATICS (PAPER 2) 10612/24 32 Additional space P.T.O.

Downloaded from hlayiso.com MATHEMATICS (PAPER 2) 10612/24 33 Additional space P.T.O.

Downloaded from hlayiso.com MATHEMATICS (PAPER 2) 10612/24 34 Additional space TOTAL: 150 END

Downloaded from hlayiso.com MATHEMATICS (PAPER 2) 10612/24 35 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  x1  x2 y1  y2  d  ( x 2  x1 ) 2  ( y 2  y1 ) 2 M  ;   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 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  cos2   sin2   cos2  1  2 sin2  sin 2  2 sin  . cos 2 cos2  1  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 b x  x ( y  y) ( x  x ) 2

PREPARATORY EXAMINATION 2024 CANDIDATE'S NAME DATE BOOK NUMBER OF BOOKS TEACHER PAPER NUMBER 2 SUBJECT NAME MATHEMATICS (10612) ANSWER ALL THE QUESTIONS IN THE QUESTION PAPER. Downloaded from hlayiso.com