Mathematics Grade 12 P2 North West Mock 2020 Question Paper

Downloaded from hlayiso.com education Lefapha la Thuto la Bokone Bophirima Noordwes Departement van Onderwys North West Department of Education NORTH WEST PROVINCE NATIONAL SENIOR CERTIFICATE GRADE 12 Pt aa lll ak clea Lalit ai a" MATHEMATICS P2 : u Uy a SEPTEMBER 2020 " MARKS: TIME: 3 hours ANI EMATI This question paper consists of 13 pages, 1 information sheet and an answer book of 21 pages. Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematits/P2scanaqec’. if NW/Septembet‘2020 : NSC 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. that you have used in determining your 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. 4 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

Downloaded from hlayiso.com Mathematics/P2ic0".-) NW/Septembef 202080): 2(lis'4 NSC QUESTION 1 During the 2019 Rugby World Cup the South African Springbok players played seven matches. The score obtained in each match by the Springbok players are given below: The average score obtained by the Springbok players during the seven matches they played in the 2019 Rugby World Cup, was 37,43. 1.1 Show that k= 13. 2) 1.2 Calculate the standard deviation of the data. (2) 1.3 The two highest scores lie outside p standard deviation of the mean. Calculate the maximum value of p. (3) 1.4 Suppose EACH score is increased by adding a value of y to each score. 1.4.1. Write down the interquartile range of the new data set. Qd) 1.4.2 Calculate the value of y if the lower quartile of the new data set is 30. qd) (9] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 NW/September,2020 NSC QUESTION 2 A town has two popular municipal swimming pools, namely the Madiba swimming pool and the Cronje swimming pool. The table below indicates the number of people that visited the Madiba swimming pool over a period of eight days. The corresponding maximum daily temperatures (in °C) for each day is also given in the table. Maximum daily temperature (°C) 18 24 22 28 34 30 26 20 Number of visitors 180 210 200 350 410 300 280 195 2.1 Determine the equation of the least squares regression line of the data. (3) 2.2 Calculate the value of the correlation coefficient. (1) 2.3 The manager of the Madiba swimming pool asks a grade 12 Mathematics learner, Jan-Breet, to predict how many visitors can be expected on a certain day with a maximum daily temperature of 32 °C. Jan-Breet predicts that 365 visitors can be expected on that day. Should the manager take this prediction seriously and prepare for 365 vistors? Motivate your answer. (3) 2.4 Over the very same period of eight days, with the exact same daily maximum temperatures, the Cronje swimming pool also record the number of visitors for each day. The equation for the least squares regression line for the Cronje swimming pool is given as p = k + 12,85x. Which swimming pool had the highest number of visitors over the period of eight days? Motivate, without any calculations, your answer. (3) [10] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematies/P2""’ 5 NW/September 2020 NSC QUESTION 3 In the diagram below, A(—20 ; 64), B and C are the vertices of A ABC, with Ca point on the x-axis. D(—12 ; p) and E(9 ; 26) are the midpoints of AC and BC. KC = 16 units. ¥ B EO; 26) x 3.1 The gradient of AK is undefined. Write down the equation of the line AK. qd) 3.2 Calculate the x-intercept of the line AC. qd) 3.3 Calculate the length of AC. (2) 3.4 Show that p = 32. qd) 3.5 Determine the equation of the line AB. (5) 3.6 Calculate the size of KAB (5) 3.7 If KAC = 38,67°, determine the area of A ABC. (6) [21] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematiés/P2dmisigae v4 6 NW/Septembéf12020 NSC QUESTION 4 In the diagram below, the equation of the circle with centre F is x+y? — 6x + 4y = 87. The equation of the circle with centre M is (x - 14)’ + (y - hy’ = 25. The point K(9 ; —10) lies on the circle with centre F. JN and KL are tangents to the circle F at the points J and K. The two circles touch externally. J x K(9 ; -10) 4.1 Write down the length of the radius of the circle with centre M. (1) 4.2 Determine the coordinates of F. (6) 43 Write down the length of the radius of the circle with centre F. (2) 44 Show, with calculations, that h = ¥104 - 2. (4) 4.5 Show that the equation of the tangent JN is y = as + = , if J(-3 ; 6) is given. (3) 46 A line NL, passes through the centre M, such that NL || JK. The circle with centre M, is translated alongside the line NL and between the two tangents of circle F, for the interval x ¢[p:q]. Give the minimum value of p, such that the circle M will not intersect the tangents to the circle with centre F, when translated alongside NL. (6) [19] Copyright reserved Please turn over es |

Downloaded from hlayiso.com Mathematics/P2¢ 7 NW/Septembéf!2020: ¢< 1/131 NSC QUESTION 5 5.1 In the diagram, P(x ; 1) is a point in the 24 quadrant and is 10 units from the origin. T is a point on the positive x-axis and obtuse POT = 6. 4 P(k3 1 J v0 \ o> x (@) T 5.1.1 Calculate the value of k. (2) 5.1.2 Without using a calculator, calculate the value of: (a) —cosé (1) (b) fl -sin@80°+ @).sin(360° — 8) (4) 5.1.3 Show, without using a calculator, that 6 6 V3 +3 5n( 2 _ 150 GF _ys50}e 5 sn( 5 15 }.cos($ 15 } aio (5) 5.2 Given: sin(@ + 60°) =d 5.2.1 Write down the values of d such that sin(@ + 60°) is defined. (2) 5.2.2 If d =sin6.cos60° + AF , the general solution of sin(@ + 60°) = d, is @=+m° + n.360°. Calculate, without the use of a calculator, the value of m. (5) 53 Itis given that A = ‘psin(a ~ m)sin(b ~ m)sin(c ~ m)..sin(x — m) Calculate the value of A. (3) [22] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathemati¢s/P2- "7" 8 NW/Septembef!2020 NSC QUESTION 6 In the diagram below, the graphs of f(x) = asinx and g(x) = tanbx are drawn for the interval x € [-180° ; 360°]. The point P(90° ; 3) is on the graph off, The function ix) =5 intersects the graphs of fand g. The asymptote of g goes through one of the turning points of /- A and B(154,69° ; 1,28) are points of intersections of the graphs fand g. g | P(90° ; 3) | | | x | 154,69° ; 1,38) | “ | *, | | y, | | | | | 6.1 Write down the values of a and b. (2) 6.2 Write down the period of g. qd) 6.3 Write down the coordinates of A. (2) 6.4 It is given that 2 tan 56,31° — 6sin168,93°= ¢+5 . Calculate in terms of ¢, the vertical distance between fand g, at the point where h intersects g. (4) 19] Copyright reserved Please turn over a

Downloaded from hlayiso.com Mathematics/P2 9 NW/September 2020 NSC QUESTION 7 In the diagram below, steel poles AF, BD and CE are equal in length and vertical to the horizontal plane FDE. The steel poles hold A ABC that forms the roof of a storage area. Gis the midpoint of DE. BCED is in the same vertical plane. AB = AC =x, BD=y and BAC=8. A 0 Se D 71 Show that BC=x,/2(1-cos@ ) G3) 7.2 Write down the size of BDG. (1) 2y? +x°(1- cosé 13 showtat BG = 22 OS") @) 74 It is given that y = ; metres and x = = metres . Calculate the maximum length of BG. (3) [10] Copyright reserved Please turn over TT TT

Downloaded from hlayiso.com Mathematics/P2 y 10 NW/Septembér!2020 NSC Give reasons for your statements in QUESTIONS 8, 9 and 10. QUESTION 8 In the diagram, O is the centre of the circle with radius OC. ABCD is a cyclic quadrilateral. N is another point on the circle such that ON || BC. Chord DN is drawn. OE bisects chord BC at E. GC is a tangent to the circle at C. ON and GC are produced to meet in F. OCE = 20° and DAB=150°. G F 8.1 Calculate, with reasons, the size of: 8.1.1 0, (1) 8.1.2 D, (2) 8.1.3 C, (2) 8.14 D, (4) 8.2 It is given that OF=/3y and EC=2x. The area of OECF is ay —x’, Determine the length of OE in terms of x and y. (5) [14] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematies7P2: ll f NW/Septembéf!2020.. i2\'8.1. NSC QUESTION 9 9.1 In the diagram, chords AB, BC and AC are drawn in the circle with centre O. DCE is the tangent to the circle at C. Prove the theorem which states that ACD = ABC (5) Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 12 NW/September 2630" NSC 9.2 In the diagram below, the centre of the circle is O with the diameter KL. MNP is a tangent to the circle at N. Chords KN and LN are drawn. R is a point on KN and ORSP forms a straight line. OP = 15 units. K oll AR P 2 A3 2. N L M 9.2.1 Give the length of OR if OR : RP=2:3. qd) 9.2.2 Itis given that KR* — KO* = 36. Prove that OR L KL. (2) 9.2.3 Prove that ORNL is a cyclic quadrilateral. (3) 9.2.4 MNP is translated to the left side of the circle, such that M’ falls outside of the circle and N’ inside of the circle. The point P’ coincides with the point S. The points O, L and S remain fixed as indicated on the original diagram and MNP || M’N’P’ . Is it possible for M'N’P’ to be a tangent to the circle drawn through O, L and S? Motivate your answer. (4) [15] Copyright reserved Please turn over

Downloaded from hlayiso.com 13 Mathematics/P2 NW/September 2020 NSC QUESTION 10 Two circles touch each other internally at B. O, the centre of the bigger circle, lies on the circumference of the smaller circle. FBG is a common tangent. AB and BC are chords of the bigger circle and intersect the smaller circle at D and E respectively. AHG forms a straight line. Chords OD, DE and OB of the smaller circle are drawn. AH:AG = 1:2. G H 23) 1 Oo Z IB A 2, E C F 10.1 Give a reason why: 10.1.1 B, = HCB (1) 10.1.2 B, =D, (1) 10.2. Prove that: 10.2.1 ADBE ||| ADBC (3) 10.2.2 D is the midpoint of AB (3) 10.2.3 AB* = BC.BE (4) 4 10.2.4 CE:BC =5:9 if DB =3 units and BE = 2 units. (3) 10.3 Calculate the ratio of "C24 AHD 6) area AHGB [21] TOTAL: 150 Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 NW/September 2020 NSC INFORMATION SHEET: MATHEMATICS ee DEVO ~4ae 2a A=P(l+ni) A= P(l-ni) A= P(l-i)" A= P(1+i)" T,=a+(n-ld_ S, = F[2a+(n-Da] T, =ar"! Ss _ alr" -1) sre 8, = a 3 -l<r<l3 "pH l-r ‘\n \—n _ p a2) 1 pawlca+)") (x)= lim S(x+h)- fe) i i h>0 h Xy)+X, Wp+y d= (x, -x)? +02-1" M| “L7*2 712 2 2 y=mx+c yoy, =m(x-x) ma 22-71 m = tan 27%] (<- a) +(y-8) =P" area AABC = sabsinc sin (a + B) = sin a.cos f + cosa.sin B cos(a + B)=cosa.cos f -sin a.sin B cos? a —sin? a cos 2a =41-2sin’ a 2cos*a—1 Copyright reserved a’ =b? +c% —2be.cos A sin (a - B) = sin a.cos f — cosa.sin B cos(a - f)= cos a.cos f +sin a.sin B sin 2a = 2sin a@.cosa P(A or B) = P(A) + P(B) — P(A and B)