Mathematics Grade 11 P2 National Exemplar November 2018 QP

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Downloaded from hlayiso.com Mathematics/P2 2 DBE/November 2018 CAPS — Grade 11 INSTRUCTIONS AND INFORMATION Read the following instructions carefully before answering the questions. 1. 2. This question paper consists of 11 questions. Answer ALL the questions in the SPECIAL ANSWER BOOK provided. Clearly show ALL calculations, diagrams, graphs, etc. that you used to determine the answers. Answers only will NOT necessarily be awarded full marks. If necessary, round off answers to TWO decimal places, unless stated otherwise. Diagrams are NOT necessarily drawn to scale. You may use an approved scientific calculator (non-programmable and non-graphical), unless stated otherwise. Write neatly and legibly. Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 3 DBE/November 2018 CAPS — Grade 11 QUESTION 1 A school held a sports day. One of the items on the programme was an obstacle race. Teams of 10 parents and learners participated in this race. The table below shows the time taken, in minutes, by each member of a particular team to complete the race. Ll 1.2 1.3 1.4 1.5 How long, in minutes, did it take for the fastest member of this team to complete the race? Determine the mean time taken by this team. Calculate the standard deviation for the data. How many members of the team completed the obstacle race outside of two standard deviations of the mean? It took another team a total time of x+5 minutes to complete the race. Calculate the value of x if the overall mean of the two teams combined was 18 minutes. Copyright reserved Please turn over (1) (2) (1) (3) G) [10]

Downloaded from hlayiso.com Mathematics/P2 4 DBE/November 2018 CAPS — Grade 11 QUESTION 2 2.1 A survey was conducted of the ages of players at a soccer tournament. The results are shown in the cumulative frequency graph (ogive) below. Cumulative frequency graph of the ages of soccer players 50 === = ====== LES = ae ae = : = Safnsss=as=e==> ==SS=S>e= 40 === ==s===== 7 » SEESE=E Sfcfass=s==- == == 530 = = z == & eo £ 20 s = 3 5 o 10 : 0 40 Ages of soccer players 2.1.1 How many players took part in the soccer tournament? (1) 2.1.2 Determine the number of players between the ages of 24 and 31 years old. (2) 2.1.3 Complete the frequency column of the table below in the ANSWER BOOK. CLASS FREQUENCY | CUMULATIVE INTERVAL FREQUENCY 1S<x<20 4 20<x<25 13 25<x<30 37 30<x<35 47 35<x<40 50 (3) 2.1.4 Use the grid provided in the ANSWER BOOK to draw a frequency polygon for the data. (4) Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 5 DBE/November 2018 CAPS — Grade 11 2.2 Two Grade 11 Mathematics classes have the same number of learners. The five- number summaries of the marks obtained by these classes for a test are shown below. CLASS A (30 ; 48 ; 65 ; 82 ; 90) CLASSB_ (50 ; 58 ; 65 ; 75 ; 90) The parents of learners in CLASS A and CLASS B observe that both classes have the same median and the same maximum mark and therefore claim that there is no difference in the performance between these classes. Do you agree with this claim? Use at least TWO different arguments to justify your answer. (3) [13] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 6 DBE/November 2018 CAPS — Grade 11 QUESTION 3 In the diagram, A(-2 ; 2), B(6; 5), C(0;-3) and D(x; y) are the vertices of a quadrilateral having AD||BC. BA produced has an x-intercept at E. BD and AC intersect at M. N(¢; 0) isa point on AD. x B(6; 5) Al-2 ; 2} E N(; 0) ie) y CO; -3) Dix; ») 3.1 Calculate the gradient of BC. (2) 3.2 Determine the equation of AD. (3) 3.3 Determine the value of f. (2) 3.4 Calculate the length of AN. (2) 3.5 If DC is defined by y= a3 ; determine the coordinates of D. (4) 3.6 Prove that ABCD isa parallelogram. @G) 3.7 Calculate the coordinates of M. (3) [19] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 7 CAPS ~ Grade 11 QUESTION 4 DBE/November 2018 In the diagram, F(2 ;-2), G(5; 2) and C(6;y) are the vertices of AFGC. FG FC. N isa point on FG such that the inclination of NC is 104,04°. The angle of inclination of FG is and FNC = 6. G(5;2) B 104,042 F(2;-2) 4.1 Calculate the gradient of FG. 4.2 Calculate the value of y. 43 Calculate the size of 0. 4.4 Calculate the length of NC. Copyright reserved CO6;y) (2) @) 3) (4) [12] Please turn over

Downloaded from hlayiso.com Mathematics/P2 8 CAPS — Grade 11 QUESTION 5 5.1 5.2 5.3 5.4 DBE/November 2018 In the diagram below, T(x;15) is a point in the Cartesian plane such that OT=17 units. P(-2;a) lies on OT. K is a point on the positive x-axis and TOK =8. T(x ; 15) P(-2;a) ty |e Determine, with the aid of the diagram, the following: 5.1.1 The value of x 5.1.2 tan6 5.1.3 cos(180°— 0) 5.1.4 sin’ 5.1.5 The value of a Simplify WITHOUT using a calculator: sin120°.cos210°.tan315°.cos27° sin 63°.cos540° Prove the identity: 1 cos? = tand cos@ 1+sin@ an Determine the general solution of 3sinx Copyright reserved [e} =2tanx ne Please turn over (2) qd) (2) 2) G) 7) 6) (6) [28]

Downloaded from hlayiso.com Mathematics/P2 9 DBE/November 2018 CAPS — Grade 11 QUESTION 6 The graphs of the functions f(x)=acosb@ and g(x)=ctan@ for x e[+180° ; 180°] are sketched below. The graphs intersect at P(58°; 1,6) and Q. y & | g = is 2. ~ =——s—e-e ee | PA G8"; 1,6 x -180° ~90° Qe 90° 180° nee 1 4 Q | a) i te af | | 6.1 Write down the range of f. (2) 6.2 If M(-45° ;-1) lies on g, determine the value of c. qd) 6.3 Write down the values of a and 5. (2) 6.4 Determine the coordinates of Q. (2) 6.5 K lies on f such that KM is parallel to the y-axis. Calculate the length of KM . (2) 6.6 If the system of axes is shifted 45° to the left and the graphs remain fixed, write down the equation that is now represented by graph /. (2) (14] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 10 DBE/November 2018 CAPS — Grade 11 QUESTION 7 71 In the diagram, p= 67°, PQ=3 cmand PR=9,2 cm. Determine the length of QR. 67° 3 cm, 2 cm Q R (3) 72 In the diagram below, DCB=a, AC=h units and ACB= 6. A Cc 7.21 Determine size of ACD in terms of 6 and a. qd) 72.2 Prove that AD = “8in(@— 2) (4) cosa 7.23 Determine the length of AD if A =17 units, 9 =58° and a = 23°. (2) 7.24 Calculate the area of AADC. (3) [13] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 11 DBE/November 2018 CAPS — Grade 11 QUESTION 8 The diagram below shows a cone with a perpendicular height of A cm, a radius of r cm and a slant height of 13 cm. Volume of cone = yk 130m Total surface area of the cone =ar’+ars 169xh—-nh° 8.1 Show that the volume of the cone is given by V= LL (4) 8.2 If h= 12 cm, determine the total surface area of the cone. @) (7] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 12 DBE/November 2018 CAPS — Grade 11 Give reasons for your statements and calculations in QUESTIONS 9, 10 and 11. QUESTION 9 91 In the figure, O is the centre of the circle. A, B, C, D and E lie on the circle such that chord AB and chord DC are equal in length and AEB = 39°. 9.1.1 Determine the size of O,. (2) 9.1.2 Determine the size of O,. @) Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 13 DBE/November 2018 CAPS ~ Grade 11 9.2 In the diagram, S is the centre of circle PQRT. PT isa diameter. RST =x-8° and PQR=2x—40°, =) Determine the value of x. (4) 93 In the diagram, O is the centre of the circle. Chord AC is perpendicular to radius OD at B. OB=2x units and AC = 8x units. Show that the length of BD is 2x(V5 -1) units. (6) [13] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 14 DBE/November 2018 CAPS — Grade 11 QUESTION 10 10.1 In the diagram below, O is the centre of the circle and PTSR is a cyclic quadrilateral. T Ss Prove the theorem that states that P+$ = 180°. (5) Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 15 DBE/November 2018 CAPS - Grade 11 10.2 In the figure, QS and PR are diameters of the circle with centre O such that PQ|ISR. PS is produced to T. N is a point on the circle such that Q, = Q,. SN is drawn. RS intersects QN at M. s, =48° 10.2.1 Determine, with reasons, the size of: @ ) (b) R (2) © ™M, 2) 10.2.2 Prove that ST is a tangent to the circle passing through M, N and S. (2) [14] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 16 DBE/November 2018 CAPS ~ Grade 11 QUESTION 11 O is the centre of the larger circle RTAP. OP is the diameter of the smaller circle PSO. NP isa tangent to both circles at P. RN 1 NP. Let P=x. 11.1 Prove that PR bisects ORN. (5) 11.2 Prove that ROS = PAT. (5) [10] TOTAL: 150 Copyright reserved