Mathematics Grade 12 P2 Mpumalanga Prelim 2024 QP

Downloaded from hlayiso.com education | MPUMALANGA PROVINCE REPUBLIC OF SOUTH AFRICA NATIONAL SENIOR CERTIFICATE GRADE 12 MATHEMATICS P2 SEPTEMBER 2024 MARKS: 150 TIME: 3 HOURS This question paper consists of 13 pages and 1 information sheet and an answer book is provided. Copyright reserved Please turn over a ie TA TTC TT TTT TT ETT a

Downloaded from hlayiso.com Mathematics/P2 2 MDE/Sep 2024 NSC INSTRUCTIONS AND INFORMATION Read the following instructions carefully before answering the questions. 1. The 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. 5. Ifnecessary, round off answers to TWO decimal places, unless stated otherwise. 7. Diagrams are NOT necessarily drawn to scale. 3. Write neatly and legibly. opyright reserved Please turn over

Downloaded from hlayiso.com MDE/Sep 2024 Mathematics/P2 QUESTION 1 3 NSC A Mathematics teacher wants to create a model by which she can predict a grade 12 learner’s final marks. She decided to use her 2015 results to create the model. Preparatory | <5 | 35 | 67 | 85 | 91 | 48| 78 | 72 | 15 | 75 | 69 | 37 exam(x) Final s7 | so | 74 | 80 | 92| 50] so | 81 | 23 | 80 | 75 | 42 exam(y) 1 Determine the equation of the least squares regression line in the form y = @ + bx. 3) 1.2 Draw a scatter plot and show the regression line. G3) 13 Predict the final mark for a learner who attained 46% in the preparatory examination. (2) 14 Determine the correlation coefficient of the data. qd) 1.5 Describe the relationship between preparatory and final exam results. qd) 1.6 Could you use this equation to estimate the preparatory exam mark for a learner who (2) attained 73% in the final exam? Give a reason for your answer. (12] QUESTION 2 The table below shows the results from a survey of cell phone expenditure for 100 leamers from a secondary school in Rustenburg. Expenditure(in rand) Frequency | Cumulative frequency 50 <x < 100 24 100 < x < 150 52 150 s x < 200 14 200 s x < 250 6 250 < x < 300 4 2.1 Complete the cumulative frequency table in the ANSWER BOOK. (2) 2.2 Draw an Ogive (cumulative frequency graph) for the data. (3) 2.3 Calculate the estimated mean of cell phone expenditure. (3) [8] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 4 MDE/Sep 2024 NSC QUESTION 3 3.1 ABC is a triangle with vertices A(1; 3), B(-2; 0) and C(p;—4) where p>0. The length of AC is 50 units. A” A(1;3) 142 « 4 2\1 > x B(-2;0)% E D 8 1 v C(p 3-4) 3.1.1 Determine the gradient of AB. (2) 3.1.2 Show, by calculation, that p=2. (4) 3.1.3. Determine the equation of the perpendicular bisector of AB. (4) 3.1.4 Write down the coordinates of D such that ABCD is a rectangle. 2) 3.1.5 Determine the equation of the circle passing through A, B and C. (4) 3.1.6 Calculate the size of © rounded off to the nearest whole number. (5) 3.2 Three straight lines AB, RS and x=—3 intersect each other. The equation of AB is 3x+by =—2 with b#0, and the equation of RS is ye-dr42, 4) 25] Calculate the value of 3. /opyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 5 MDE/Sep 2024 NSC QUESTION 4 Two circles in the diagram below represent two interlocking gears, which touch at the point Q(4;3). The circles have the following equations: x+y? =25 and x? -12x+y?-9y+50=0 t Px v 41 Show that the coordinates of P are(6; 44). (3) 4.2 Determine the equation of the common tangent AB. (4) 43 If the larger gear makes one full revolution, how many times wil! the smaller gear turn completely? (4 44 Find the area of AAOB . (3) 45 Another tangent to the circle with centre O, drawn from A, touches the circle at C. And C is the reflection of Q by the y axis. Determine the length of CQ. Q) {16} Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 6 NSC QUESTION 5 5.1 If cos 21° = p determine the following in terms of p. 5.1.1 tan 201° 5.1.2 sin 42° 5.1.3 cos 51° sin210°.cos510° Simplify ; S122 10".cosS10° implify > 235315°.sin(- 135°) Prove the identity: cos@—cos26+2 _1+cos@ 3sin@ — sin26 sind Determine the general solution of sin@ sin + cos cost = ee Given: sin@.cosB=-1 5.5.1 Write down the maximum and minimum value of cos# 5.5.2 Solve for 6 €[0°;270°] and £ €[-180°;90°]. opyright reserved MDE/Sep 2024 GB) GB) (3) es) (5) (4) dy @) 130] Please turn over

Downloaded from hlayiso.com Mathematics/P2 7 MDE/Sep 2024 NSC QUESTION 6 Given that y = f(x) =2cosx and y= g(x) =sin(x+30): 6.1 Sketch the graphs of f and g on the ANSWER BOOK on the same set of axes for (6) x @[—180° 180°]. 6.2 Read the following answers from your graphs: 6.2.1 Write down the period of /. (es) 6.2.2 Determine one value of x for which f(x)— g(x) =15. ( 6.2.3 Determine the positive values of x for which 2sin(x +30°).cosx< 0 2) [10] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 8 MDE/Sep 2024 NSC QUESTION 7 A mouse on the ground at point T is looking up to an owl in a tree at point U and a cat to his right on the ground at point V. The angle of elevation from the mouse to the owl is (90° — 20). TU =k units, GV = 8 units, TGV = 150° and TVG = @. U 71 Write down the size of TUG in terms of 6 qd) 72 Show that TG = k sin 29. Q) 73 Show that TV = k cos @ 4 74 Show that the area of ATGV = 2k sin 26. Q) [9] Jopyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 9 MDE/Sep 2024 NSC Give reasons for your statements and calculations in QUESTIONS 8, 9 and 10. QUESTION 8 8.1 A,B, C and D are points on the circumference of the circle with centre O. Prove the theorem which states that A+C =180°. (5) Copyright reserved Please tum over

Downloaded from hlayiso.com Mathematics/P2 10 MDE/Sep 2024 NSC 8.2 In the diagram below, PQRS isa cyclic quadrilateral and PR is a diameter of the circle. The line MRT is a tangent to the circle atR. §, =65° and R, =35°. M Determine the sizes of each of the following angles, with reasons: 82.1 & @) 8.2.2 Ry (2) 823 fF G3) [14] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 It MDE/Sep 2024 NSC. QUESTION 9 In the diagram, O is the centre of the circle and EC isa tangent to the circle at C. DM = MC and OME isa straight line. Let 0, = 2x. E 9,1 Write, with reasons, THREE angles equal to x. : (6) 9.2 Prove that 6, = 90° — x (3) 9.3 Prove that EC is a tangent to the circle passing through points, M,C and O. (4) 9.4 Prove that DOCE is a cyclic quadrilateral. GB) [16] ‘opyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 12 MDE/Sep 2024 NSC QUESTION 10 In the diagram below, two circles ABCD and BEFCG intersect at B, C and G. AC || BF. AC and BD intersect at G. BC and FG intersect at H. 10.1 Complete the following reasons: LOL Gye Bp eeeeeeesceeeeeeeseeeeneereeeeeneaen @) 10.1.2 BED=C, eeececeeeeeeeeeeeeeceeeeeenenenetin @) 10.2. Prove that 10.2.1 BH=FH 4) 10.2.2 ABEF || ADGF @) 10.2.3 FHL.BG =BH.FC (4) U3] TOTAL: 150 Copyright reserved Please turn over