Mathematics P2 Nov 2016 Eng_hlayiso.com_.pdf

Downloaded from blayvica com ale basic education yy , Department: iy y ¥ rh y Basic Education SING hee REPUBLIC OF SOUTH AFRICA NATIONAL SENIOR CERTIFICATE GRADE 12 " MATHEMATICS P2 i NOVEMBER 2016 MARKS: 150 TIME: 3 hours This question paper consists of 14 pages, 1 information sheet and an answer book of 28 pages. | Copyright reserved Please turn over = —— — a

Mathematics/P2 Downloaded from hlayiso.com DBE/November 2016 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 DBE/November 2016 NSC QUESTION 1 A survey was conducted at a local supermarket relating the distance that shoppers lived from the store to the average number of times they shopped at the store in a week. The results are shown in the table below. Distance from the store in km 1 2 3 4 5 7 8 | 10 Average number of times shopped 12 | 10 7 7 6 2 3 2 per week SCATTER PLOT J ; Bee ORPNWAEUDINWWORN | Average number of times shopped per week ty) 1 2 3 4 5 6 7 8 9 10 Distance from the store in km 1.1 Use the scatter plot to comment on the strength of the relationship between the distance a shopper lived from the store and the average number of times she/he shopped at the store in a week. 1.2 Calculate the correlation coefficient of the data. 13 Calculate the equation of the least squares regression line of the data. 1.4 Use your answer at QUESTION 1.3 to estimate the average number of times that a shopper living 6 km from the supermarket will visit the store in a week. 1.5 Sketch the least squares regression line on the scatter plot provided in the ANSWER BOOK. Copyright reserved Please turn over (1) q) (3) (2) (2) 19]

Mathematics/P2 NSC QUESTION 2 Downloaded from hlayiso.com DBE/November 2016 The heights of 160 learners in a school are measured. The height of the shortest learner is 1,39m and the height of the tallest learner is 2,21 m. The heights are represented in the histogram below. 2.1 22 2.3 2.4 2:5 2.6 Histogram 80 Fi Number of learners 11 1,3 1,5 1,7 19 Heights (in m) Describe the skewness of the data. Calculate the range of the heights. dd) (2) Complete the cumulative frequency column in the table given in the ANSWER BOOK. Draw an ogive (cumulative frequency curve) to provided in the ANSWER BOOK. (2) represent the data on the grid (4) Fighty learners are less than x metres in height. Estimate x. (2) The person taking the measurements only had a 1,5 m measuring tape available. In order to compensate for the short measuring tape, he decided to mount the tape on a wall at a height of 1 m above the ground. After recording the measurements he discovered that the tape was mounted at 1,1 m above the ground instead of 1m. How does this error influence the following: 2.6.1 Mean of the data set 2.6.2 Standard deviation of the data set Copyright reserved I 1,5 m measuring tape { Distance above the ground GROUND () (1) [13] Please turn over

Mathematics/P2 Downloaded from hlayiso.com DBE/November 2016 NSC QUESTION 3 In the diagram, A(-7 ; 2), B, C(6;3) and D_ are the vertices of rectangle ABCD. The equation of AD is y= 2x +16. Line AB cuts the y-axis at G. The x-intercept of line BC is F(p: 0) and the angle of inclination of BC with the positive x-axis is @. The diagonals of the rectangle intersect at M. 3.1 3.2 3.3 3.4 3.5 3.6 337 3.8 D, ¥ y=2x+ l6—_y C(6 3) A(-7 ; 2) M a x O Fp; 0) G B Calculate the coordinates of M. (2) Write down the gradient of BC in terms of p. (1) Hence, calculate the value of p. (3) Calculate the length of DB. (3) Calculate the size of a Q) Calculate the size of OGB. (3) Determine the equation of the circle passing through points D, B and C in the form (x-a)? +(y-by =r’. (3) If AD is shifted so that ABCD becomes a square, will BC be a tangent to the circle passing through points A, M and B, where M is now the intersection of the diagonals of the square ABCD? Motivate your answer. (2) [19] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 DBE/November 2016 NSC QUESTION 4 In the diagram, M is the centre of the circle passing through T(3; 7), R and S(5;2). RT is a diameter of the circle. K(a ; 5) is a point in the 4" quadrant such that KTL is a tangent to the circle at T. + @ 0 _* K(a; d) 41 Give a reason why TSR =90°. (1) 42 Calculate the gradient of TS. (2) 43 Determine the equation of the line SR in the form y= mx +c. (3) 44 The equation of the circle above is (x—9)? + [> - 63] = 361. 4.4.1 Calculate the length of TR in surd form. (2) 4.4.2 Calculate the coordinates of R. (3) 4.4.3 Calculate sin R. (3) 4.4.4 Show that 6 = 124-29. (3) 4.4.5 If TK =TR, calculate the coordinates of K. (6) [23] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 DBE/November 2016 NSC QUESTION 5 5.1 Given: sin16° = p Determine the following in terms of p, without using a calculator. 5.1.1 sin196° 2) 5.1.2 cosl6° (2) 5.2 Given: cos(A — B) = cosAcosB + sinAsinB Use the formula for cos(A -B) to derive a formula for sin(A +B) (3) vi—cos? 2A completely, given that 0°< A <90°. 5 cos(—A).cos(90°+ A) Pet @) 53 Simplify 5.4 Given: cos 2B == and 0°<B<90° Determine, without using a calculator, the value of EACH of the following in its simplest form: 5.4.1 cosB (3) 5.4.2 sinB (2) 5.4.3 cos (B + 45°) (4) [21] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 8 DBE/November 2016 NSC QUESTION 6 In the diagram the graph of f(x) =2sin2x is drawn for the interval x € [-180° : 180°]. y —180° 6.1 On the system of axes on which / is drawn in the ANSWER BOOK, draw the graph of g(x)=—cos2x for x € [-180° ; 180°]. Clearly show all intercepts with the axes, the coordinates of the turning points and end points of the graph. (3) 6.2 Write down the maximum value of f(x) —3. (2) 6.3 Determine the general solution of f(x) = g(x). (4) 6.4 Hence, determine the values of x for which /(x)< g(x) in the interval x € [-180° ; 0°]. G) [12] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 9 DBE/November 2016 NSC QUESTION 7 Eis the apex of a pyramid having a square base ABCD. © is the centre of the base. EBA =@, AB=3m and EO, the perpendicular height of the pyramid, is x. Volume of pyramid = s (area of base) x (L height) 7.1 Calculate the length of OB. (3) 3 Ga Show that cos@ = 2,[x? + 2 2 (5) 73 If the volume of the pyramid is 15 m*, calculate the value of 6. (4) [12] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 10 DBE/November 2016 NSC Give reasons for ALL statements and calculations in QUESTIONS 8, 9 and 10. QUESTION 8 8.1 In the diagram below PQRT is a cyclic quadrilateral having RT || QP. The tangent at P meets RT produced at S. QP=QT and PTS = 70°. 8.1.1 Give a reason why P, = 70°. (dy 8.1.2 Calculate, with reasons, the size of: @ Q, G3) (b) P (2) Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 DBE/November 2016 NSC 8.2 A, B and C are points on the circle having centre O. S and T are points on AC and AB respectively such that OS 1 AC and OT L AB. AB=40 and AC =48. e O A LY B 8.2.1 Calculate AT. (1) 8.2.2 If OS= TOT, calculate the radius OA of the circle. (5) [12] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 2 DBE/November 2016 NSC QUESTION 9 ABC isa tangent to the circle BFE at B. From C a straight line is drawn parallel to BF to meet FE produced at D. EC and BD are drawn. E, =F, =x and C, =y. 9.1 Give a reason why EACH of the following is TRUE: 9.1.1 B, =x 9.1.2 BCD =B, 92 Prove that BCDE isa cyclic quadrilateral. 9.3 Which TWO other angles are each equal to x? 9.4 Prove that B, = C,. Copyright reserved Please turn over () (1) (2) 2) (3) 19]

Downloaded from hlayiso.com Mathematics/P2 DBE/November 2016 NSC QUESTION 10 10.1 In the diagram APQR is drawn. S and T are points on sides PQ and PR respectively such that ST || QR. PR S T Q > R Prove the theorem which states that Eee 4 (6) SQ TR Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 14 DBE/November 2016 NSC 10.2 In the diagram HLKF is a cyclic quadrilateral. The chords HL and FK are produced to meet at M. The line through F parallel to KL meets MH produced at G. MK =x, KF = 2x, ML=y and LH=HG. G 10.2.1 Give a reason why GFM =LKM. qd) 10.2.2 Prove that: (a) GH=y (3) (b) AMFH||| AMGF (5) (©) GF _3x FH 2y (2) y 3 10.2.3 Show that — =, /— x V2 (3) [20] TOTAL: 150 Copyright reserved

Downloaded from hlayiso.com Mathematics/P2 DBE/November 2016 NSC INFORMATION SHEET _-b+ vb? —4ac 2a A=P(l+ni) A= P(I-ni) A=P(l-i)" A=P(i+i)" T, =a+(n-ld S, = Sha+—na] T, =ar™ Ss ar" =I 3r#l S,=— 3 -lere<l n r-l l-r poaliti) -1 pal-d+i-") i i f(x) = lim S(x+h)- f(x) hood h Let Jy +y. d=V(8—¥)" +0 -¥,)" M| 2-1-2 2 2 y=mxte y-y, =m(x-x,) 227" m=tan@ ie a b c InAABC: - =— =— snA sinB~ sinC a? =b +02 —2be.cos A area KABC = 5 ab.sinC sin(a + f) =sina.cosf +cosa.sinZ coda: + f)=cosa.cosf-sina.sinB cos’ a-sin’ a cos 2a@ =41-2sin’a@ 2cos* a1 Copyright reserved sin(a — 8) = sina.cosf -cosa.sin# cota — f3)=cosa.cosf+sina.sinB sin2@ = 2sina@.cosa P(A or B) = P(A) + P(B) — P(A and B) p = Fy 9) Ye-37