Mathematics Grade 12 P2 NSC Supplementary 2018 QP (Eng)

Downloaded from nlayiso.com ae. basic education - Department: ) / Basic Education KO , WI REPUBLIC OF SOUTH AFRICA NATIONAL SENIOR CERTIFICATE GRADE 12 Pe " MATHEMATICS P2 a u ly ii i: FEBRUARY/MARCH 2018 " eee eer er err rr rr rrr Ter rrrrrrrerrTrrrt rrr ert fs MARKS: 150 TIME: 3 hours This question paper consists of 13 pages, 1 information sheet and an answer book of 27 pages. Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 2 DBE/Feb.—Mar. 2018 NSC INSTRUCTIONS AND INFORMATION Read the following instructions carefully before answering the questions. 1. 2. This question paper consists of 10 questions. Answer ALL the questions in the ANSWER BOOK provided. Clearly show ALL calculations, diagrams, graphs, et cetera that you have used in determining your answers. Answers only will NOT necessarily be awarded full marks. You may use an approved scientific calculator (non-programmable and non-graphical), unless stated otherwise. If necessary, round off answers to TWO decimal places, unless stated otherwise. Diagrams are NOT necessarily drawn to scale. An information sheet with formulae is included at the end of the question paper. Write neatly and legibly. Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 3 DBE/Feb.—Mar. 2018 NSC QUESTION 1 An organisation decided that it would set up blood donor clinics at various colleges. Students would donate blood over a period of 10 days. The number of units of blood donated per day by students of college X is shown in the table below. DAYS 1 2 3 4 5 6 7 8 9 10 UNITS OF BLOOD 45 | 59 | 65 | 73 | 79 | 82 | 91 | 99 | 101 | 106 1.1 Calculate: 11.1 The mean of the units of blood donated per day over the period of 10 days (2) 1.1.2 The standard deviation of the data (2) 1.1.3 How many days is the number of units of blood donated at college X outside one standard deviation from the mean? (G3) 12 The number of units of blood donated by the students of college X is represented in the box and whisker diagram below. e————_________ | T T A B 1.2.1 Describe the skewness of the data. qd) 1.2.2 Write down the values of A and B, the lower quartile and the upper quartile respectively, of the data set. (2) 1.3 It was discovered that there was an error in counting the number of units of blood donated by college X each day. The correct mean of the data is 95 units of blood. How many units of blood were NOT counted over the ten days? qd) [14] Copyright reserved Please turn over

Downloaded from hlayiso.com DBE/Feb.—Mar. 2018 Mathematics/P2 QUESTION 2 4 NSC The table below shows the number of hours that a sales representative of a company spent with each of his nine clients in one year and the value of the sales (in thousands of rands) for that client. NUMBER OF HOURS 30 50 80 | 100 | 120 | 150 | 190 | 220 | 260 VALUE OF SALES (IN THOUSANDS OF RANDS) 270 | 275 | 376 | 100 | 420 | 602 | 684 | 800 | 820 SCATTER PLOT 900 = ~ oo 2 6 ~ 9 3 a Q 6 nl wu fo} 6 b fo} a f, w S r=) N ° oO Value of sales (in thousands of rands) e 2 o 0 + =p ee 1 — (e) 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 Number of hours 21 Identify an outlier in the data above. ql) 2.2 Calculate the equation of the least squares regression line of the data. (3) 2.3 The sales representative forgot to record the sales of one of his clients. Predict the value of this client's sales (in thousands of rands) if he spent 240 hours with him during the year. (2) 2.4 What is the expected increase in sales for EACH additional hour spent with a client? (2) [3] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 3 DBE/Feb.—Mar. 2018 NSC QUESTION 3 In the diagram, P, Q(-7 ; -2), R and S(3 ; 6) are vertices of a quadrilateral. R is a point on the x-axis. QR is produced to N such that QR=2RN. SN is drawn. PTO =71,57° and SRN =0. 8(3 ; 6) T/71,57° D : oO R * QC7 ; -2) Determine: 3.1 The equation of SR (1) 3.2 The gradient of QP to the nearest integer (2) 3.3 The equation of QP inthe form y=mx+ce (2) 3.4 The length of QR. Leave your answer in surd form. (2) 3.5 tan@0°— 6) GB) 3.6 The areaof ARSN, without using a calculator “ [16] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 6 DBE/Feb.—Mar. 2018 NSC QUESTION 4 In the diagram, PKT is a common tangent to both circles at K(a ; 6). The centres of both circles lie on the line y = > The equation of the circle centred at O is x? +y* =180. The radius of the circle is three times that of the circle centred at M. X 4.1 Write down the length of OK in surd form. qd) 4.2 Show that K is the point (—12 ; -6). (4) 43 Determine: 43.1 The equation of the common tangent, PKT, in the form y=mx+c (3) 4.3.2 The coordinates of M (6) 4.3.3 The equation of the smaller circle in the form (x-—a)? +(y—b)? =r? (2) 4.4 For which value(s) of r will another circle, with equation x’ + y? =r’, intersect the circle centred at M at two distinct points? (GB) 45 Another circle, x? + y* +32x+16y+240=0, is drawn. Prove by calculation that this circle does NOT cut the circle with centre M(-16 ; -8). i Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 7 DBE/Feb.—Mar. 2018 NSC QUESTION 5 5.1 If cos20= -2, where 20 €[180°;270°], calculate, without using a calculator, the values in simplest form of: 5.1.1 sin20 (4) 5.1.2 sin’ 0 (3) 5.2 Simplify sin(180° — x).cos(—x) + cos(90° + x).cos(x—180°) to a single trigonometric ratio. (6) 5.3 Determine the value of sin3x.cos y+cos3x.siny if 3x+y=270°. (2) 5.4 Given: 2cosx =3tanx 5.4.1 Show that the equation can be rewritten as 2sin? x+3sinx—2=0. (3) 5.4.2 Determine the general solution of x if 2cosx =3tan x. (5) 5.4.3 Hence, determine two values of y, 144°<y<216°, that are solutions of 2cos5y =3tan5y. (4) 5.5 Consider: g(x) =—4cos(x + 30°) 5.5.1 Write down the maximum value of g(x). qd) 5.5.2 Determine the range of g(x) + 1. (2) 5.5.3 The graph of g is shifted 60° to the left and then reflected about the x-axis to form a new graph A. Determine the equation of hin its simplest form. G3) [33] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 8 DBE/Feb.—Mar. 2018 NSC QUESTION 6 PQ and AB are two vertical towers. From a point R in the same horizontal plane as Q and B, the angles of elevation to P and A are @ and 20 respectively. AQR =90°+6, QAR =6 and QR=x. 6.1 Determine in terms of x and 0: 6.1.1 QP Q) 6.1.2 AR (2) 6.2 Show that AB = 2xcos? @ 4 AB Determine — if @=12°. 6.3 termine OP i (2) [10] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 9 NSC QUESTION 7 DBE/Feb.—Mar. 2018 In the diagram, PQRT is acyclic quadrilateral in a circle such that PT = TR. PT and QR are produced to meet in S. TQ is drawn. SQP = 70° 7A 7.2 Calculate, with reasons, the size of: TAA f, 71.2 Q, If it is further given that PQ || TR: 7.2.1 Calculate, with reasons, the size of T, 722 Prove that IX _RQ TS RS Copyright reserved Please turn over (2) (2) (2) 2) [8]

Downloaded from hlayiso.com Mathematics/P2 10 DBE/Feb.—Mar. 2018 NSC QUESTION 8 In the diagram, PR is a diameter of the circle with centre O. ST is a tangent to the circle at T and meets RP produced at S. SPT=x and §=y. Determine, with reasons, y in terms of x. [6] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 i DBE/Feb.—Mar. 2018 NSC QUESTION 9 In the diagram, DEFG is a quadrilateral with DE=45 and GF = 80. The diagonals GE and DF meetin H. GDE=FEG and DGE=EFG. D 45 G E H 80 F 9.1 Give a reason why ADEG ||| AEGF. (1) 9.2 Calculate the length of GE. (3) 9.3 Prove that ADEH ||| AFGH. (3) 9.4 Hence, calculate the length of GH. (3) [10] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 12 DBE/Feb.—Mar. 2018 NSC QUESTION 10 10.1 In the diagram, O is the centre of the circle with A, B and C drawn on the circle. Prove the theorem which states that BOC = 2A. (5) Copyright reserved Please turn over

Mathematics/P2 10.2 In the diagram, the circle with centre F is drawn. Points A, B, C and D lie on the circle. Chords AC and BD intersect at E such that EC =ED. K is the midpoint Downloaded from hlayiso.com 13 DBE/Feb.—Mar. 2018 NSC of chord BD. FK, AB, CD, AF, FE and FD are drawn. Let B=x. 10.2.1 10.2.2 10.2.3 10.2.4 Copyright reserved Determine, with reasons, the size of EACH of the following in terms of x: (a) F, ) ¢é Prove, with reasons, that AFED is a cyclic quadrilateral. Prove, with reasons, that F, = x. If area AAEB = 6,25 x area ADEC, calculate ~. TOTAL: Q) 2) (4) (6) G) [24] 150

Downloaded from hlayiso.com Mathematics/P2 DBE/Feb.—Mar. 2018 NSC INFORMATION SHEET: MATHEMATICS _ —b+y52 ~4ac 2a A= P(+ni) A=P(-ni) A=P(1-i)" A=P(l+i)" T, =a+(n—N)d S, =F ea+(n—vd] T, =ar™! s, -ae"=!) sr#l S,=—);-ler<l r-l I-r pa 2+ =I) p2l-d+97") i i im FET f@) I'@)= x, +x. ¥, +y. Pe es rroees M| 12 2 2 = _y= _ YT = y=mxt+e yy, =m(x-x,) m= m=tan@ %y 7% (x-a) +(y-by =r? InAABC: ab _-¢ sinA sinB sinC a® =b* +¢7 -2be.cosA area N\ABC = 5 absinc sina +f) =sina.cosf +cosa.sinB sin(a— B) =sina.cosf —cosa.sinB cofat B) =cosa.cosf-sina.sing cofa— B) =cosa@.cosf+sina.sin£ cos’ a@—sin> a cos2a =41-2sin’@ sin2a@ = 2sina.cosa 2cos’?a—1 n 2 zeke >@-») ye o == An " (A) = P(A)= 78) P(A or B) = P(A) + P(B) — P(A and B) (x-¥y-9) paatbx Y@-z Copyright reserved