Mathematics Grade 12 P2 NSC June 2016 Question Paper (Eng)

basic education Department: Basic Education REPUBLIC OF SOUTH AFRICA SENIOR CERTIFICATE EXAMINATIONS MATHEMATICS P2 2016 MARKS: 150 TIME: 3 hours This question paper consists of 15 pages, 1 information sheet and an answer book of 31 pages. Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 2 DBE/2016 SCE 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 SCE QUESTION 1 DBE/2016 On a certain day a tour operator sent 11 tour buses to 11 different destinations. The table below shows the number of passengers on each bus. 8 8 10 | 12 | 16 | 19 | 20 | 21 | 24 | 25 | 26 Ld Calculate the mean number of passengers travelling in a tour bus. 12 Write down the five-number summary of the data. 1.3 Draw a box and whisker diagram for the data. Use the number line provided in the ANSWER BOOK. 1.4 Refer to the box and whisker diagram and comment on the skewness of the data set. LS Calculate the standard deviation for this data set. 1.6 A tour is regarded as popular if the number of passengers on a tour bus is one standard deviation above the mean. How many destinations were popular on this particular day? Copyright reserved Please turn over 2) (3) (2) (1) (2) (2) [12]

Downloaded from hlayiso.com Mathematics/P2 4 DBE/2016 SCE QUESTION 2 On the first school day of each month information is recorded about the temperature at midday (in °C) and the number of 500 m£ bottles of water that were sold at the tuck shop of a certain school during the lunch break. The data is shown in the table below and represented on the scatter plot. The least squares regression line for this data is drawn on the scatter plot. Temperature at 7 midday (in °C) 18 | 21 | 19 | 26 | 32 | 35 | 36 | 40 | 38 | 30 | 25 Number of bottles of . water (500 mé) 12] 15 | 13 | 31 | 46 | 51 | 57 | 70 | 63 | 53 | 23 Scatter plot 75 — Number of bottles of water (500 mf) Temperature at midday (in °C) Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 5 DBE/2016 2.1 2.2 Copyright reserved SCE Identify an outlier in the data. Determine the equation of the least squares regression line. Estimate the number of 500 mé bottles of water that will be sold if the temperature is 28 °C at midday. Refer to the scatter plot. Would you say that the relation between the temperature at midday and the number of 500 mf bottles of water sold is weak or strong? Motivate your answer. Give a reason why the observed trend for this data cannot continue indefinitely. Please turn over (1) (3) (2) @) () [9]

Downloaded from hlayiso.com Mathematics/P2 6 DBE/2016 SCE QUESTION 3 In the diagram below (not drawn to scale) A(-8 :; 6), B(8; 10), C and D(-2; 0) are the vertices of a trapezium having BC || AD. T is the midpoint of DB. From B, the straight line drawn parallel to the y-axis cuts DC in r(s 3 33] and the x-axis in M. y B(8 ; 10) an AC8 ; 6) T c a F(s ; 33] - 3 C—| a D(-2:0) |O M % 3.1 Calculate the gradient of AD. (2) 3.2 Determine the equation of BC in the form y= mx +c. (3) 3.3 Prove that BD 1 AD. (3) 3.4 Calculate the size of BDM. (2) 3.5 If it is given that TC || DM and points T and C are symmetrical about line BM, calculate the coordinates of C. (3) 3.6 Calculate the area of ABDF. (5) [18] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 Z DBE/2016 SCE QUESTION 4 A circle having C(3 ; -1) as centre and a radius of 10 units is drawn. PTR isa tangent to this circle at T. R(k; 21), C and P are the vertices of a triangle. TR = 20 units. y R(k: 21) Oy T, K x La 5-1) P S 4.1 Give a reason why TC 1 TR. () 42 Calculate the length of RC. Leave your answer in surd form. (2) 43 Calculate the value of & if R lies in the first quadrant. (4) 4.4 Determine the equation of the circle having centre C and passing through T. Write your answer in the form (x-a)?+(y—b) =r? (2) 4.5 PS, a tangent to the circle at S, is parallel to the x-axis. Determine the equation of PS. (2) 4.6 The equation of PTR is 3y—4x = 35 4.6.1 Calculate the coordinates of P. (2) 4.6.2 Calculate, giving a reason, the length of PT. (3) 4.7 Consider another circle with equation (x-3) +(v +16)? =16 and having centre M. 4.7.1 Write down the coordinates of centre M. dd) 4.7.2 Write down the length of the radius of this circle. (1) 4.7.3 Prove that the circle with centre C and the circle with centre M do not intersect or touch. (3) [21] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 8 DBE/2016 SCE QUESTION 5 5.1 In the diagram PR 1 TS in obtuse triangle PTS. PT= 1/5; TR=2; PR=1; PS=i0 and RS=3 P v5 ; vio ml T 2 R 3 S 3.11 Write down the value of: (a) sinT (1) (b) cosS () S12 Calculate, WITHOUT using a calculator, the value of cos(T +8) (5) 3.2 Determine the value of: 1 2 > —tan* (180° + 8) cos(360° — @).sin(90° — 0) (6) 5.3 If sinx-—cosx= : , calculate the value of sin2x WITHOUT using a calculator. (5) [18] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 9 DBE/2016 SCE QUESTION 6 6.1 Determine the general solution of 4sinx+2cos2x = 2 (6) 6.2 The graph of g(x)=—cos2x for x €[-180° ; 180°] is drawn below. | | ty | ; 7° | | —— a_i _ & | =l id | 90° | O | | | | 4 | | | | - | | | | re | po] 6.2.1 Draw the graph of f(x) =2sinx—1 for x €[-180° ;180°] on the set of axes provided in the ANSWER BOOK. (3) 6.2.2 Write down the values of x for which g_ is strictly decreasing in the interval x €[-180° ; 0°] Q) 6.2.3 Write down the value(s) of x for which (x +30°)— g(x +30°) =0 for x €[-180° ; 180°] (2) [13] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 10 DBE/2016 SCE QUESTION 7 From point A an observer spots two boats, B and C, at anchor. The angle of depression of boat B from Ais @ D isa point directly below A and is on the same horizontal plane as B and C. BD=64m, AB=81m and AC=87m. 71 Calculate the size of @ to the nearest degree. (3) 7.2 If it is given that BAC= 82,6°, calculate BC, the distance between the boats. (3) 73 If BDC =110°, calculate the size of DCB. (3) [9] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 a DBE/2016 SCE Give reasons for ALL statements in QUESTIONS 8, 9, 10 and 11. QUESTION 8 8.1 In the diagram below, P, M, T and R are points on a circle having centre O. PR produced meets MS at S. Radii OM and OR and the chords MT and TR are drawn. T, = 148°, PMO=18° and § = 43° Calculate, with reasons, the size of: 8.1.1 Pp (2) B12. O, (2) 8.1.3 OMS (2) 8.1.4 — R, ifitis given that TMS= 6° (2) Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 12 DBE/2016 SCE 8.2 In the diagram below, the circle passes through A, B and E. ABCD isa parallelogram. BC is a tangent to the circle at B. AE=AB. Let G = % 8.2.1 Give a reason why B, =x qd) 8.2.2 Name, with reasons, THREE other angles equal in size to x. (6) 8.2.3 Prove that ABED is a cyclic quadrilateral. (3) [18] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 13 DBE/2016 SCE QUESTION 9 9.1 Complete the statement so that it is TRUE: The angle between the tangent to a circle and the chord drawn from the point of contact is equal to the angle ... dd) 9.2 In the diagram below, two unequal circles intersect at A and B. AB is produced to C such that CD is a tangent to the circle ABD at D. F and G are points on the smaller circle such that CGF and DBF are straight lines. AD and AG are drawn. Prove that: 921 B,=D,+D, (4) 9.2.2 AGCD is a cyclic quadrilateral (4) 92.3 DC=CF (4) [13] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 14 DBE/2016 SCE QUESTION 10 10.1. Inthe diagram below, AMVT and AAKF are drawn such that M=A, V=K and T=F Use the diagram in the ANSWER BOOK to prove the theorem which states that if two triangles are equiangular, then the corresponding sides are in proportion, MV MT that is —— = —— AF (7) Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 Is DBE/2016 SCE 10.2 In the diagram below, cyclic quadrilateral EFGH is drawn. Chord EH produced and chord FG produced meet at K. M isa point on EF such that MG || EK. Also KG = EF 10.2.1 Prove that: (a) AKGH||| AKEF (4) (b) EF? =KE.GH (2) (ce) KG? =EM.KF (3) 10.2.2 If itis given that KE = 20 units, KF = 16 units and GH = 4 units, calculate the length of EM. (3) [19] TOTAL: 150 Copyright reserved

Downloaded from hlayiso.com Mathematics/P2 DBE/2016 SCE INFORMATION SHEET _ —b+ Vb? -4ac 2a A=P(l+ni) A= P(1-ni) A=P(-i)" A=P(+i)" T, =a+(n—N)d S, =F 2a+(-na] T, =ar"" S _ alr” -1) sr#l S,=—-3-1erel Pd l-r pa alsa" = pala" i i f'(x) = lim f(xth)— f(x) h>0 h 7 Xp +X, yy +y d= (x, -x,)' +0,-»,)° 2-12 2 2 y=mxt+e Y-y, =mx-x,) m=22 1 m=tand a a (ea #8) =P In aaBc: 4— =P = sind sinB sinC a =b’ +0? —2be.cosd area AABC= jabsin sin(a + B) = sin@.cos# +cosa.sin£ cola + £)=cosa.cosf —sina.sin B 2 oo) cos @—sin° @ cos2@ =41—2sin’? a 2cos* a1 _ x x= n P(A) = 2A) n(S) Copyright reserved sin(a — £)=sina.cosB —cosa.sin B cosa ~ f)= cosa@.cos# +sina.sin B sin2a@ =2sina.cosa Yo, -x" o= n P(A or B) = P(A) + P(B) — P(A and B) _ de-xy-7) S(x-x)