Mathematics Grade 11 P2 National Exemplar November 2019 QP

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Downloaded from hlayiso.com Mathematics/P2 2 DBE/November 2019 CAPS - Grade 11 INSTRUCTIONS AND INFORMATION Read the following instructions carefully before answering the questions. i, 2. co This question paper consists of ELEVEN 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. Copsieht reserved A A Please um over EASTERN CAPE

Downloaded from hlayiso.com Mathematics/P2 3 CAPS — Grade 11 QUESTION 1 DBE/November 2019 Mary wants to buy a car and visits a popular website. She finds a number of advertisements for the make of car that she would like to buy. She summarises the selling prices (in thousands of rands) of the cars on sale in the cumulative frequency table below. SELLING PRICE CUMULATIVE (IN THOUSANDS OF RANDS) | FREQUENCY FREQUENCY 50<x<60 3 3 60<x<70 4 7 70 <x <80 a 14 80<x<90 19 33 90 <x < 100 12 b 100 <x<110 5 50 1.1 Write down the values of a and b. 1.2 Draw a cumulative frequency graph (ogive) of the data on the grid provided in the ANSWER BOOK. 1.3 Mary wants to spend a maximum of R95 000. Use the cumulative frequency graph to estimate the number of cars that are on sale in the price range that Mary can afford. comers EASTERN CAPE (2) (3) (1) [6]

Downloaded from hlayiso.com Mathematics/P2 4 DBE/November 2019 CAPS - Grade 11 QUESTION 2 2.1 Two classes wrote a Mathematics test that had a maximum mark of 60. The results of each class are summarised in the box and whisker diagrams below. o——_ }-—_—* CLASS X | —— CLASS Y 2.1.1 Comment on the skewness of the results in class X. qd) 2.1.2 In which class is the standard deviation of the marks bigger? (1) 2.1.3 Comment on the average performance in the test of the two classes. Use relevant statistics to support your argument. (2) 2.2 The time, in minutes, that it took for the first goal to be scored in seven football games was recorded. The times, in ascending order, are represented by a, b, c, d, e, f and g in the table below. a b c d e f g The following observations were made about the data: All these goals were scored at different times. The minimum time for the first goal was 5 minutes. The range of the times was 48 minutes. The median time was 22 minutes. The difference between the time at the lower quartile and the minimum time was 7 minutes. The IQR of the times was 28 minutes. The mean time was 27 minutes. e=2c 2.2.1 Determine the values of a, b, c, d, e, f and g. (8) 2.2.2 If the standard deviation of the data set is 15,87 minutes, how many goals were scored within ONE standard deviation of the mean time? (3) (15] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 5 DBE/November 2019 CAPS — Grade 11 QUESTION 3 In the diagram, A(-2 ; 2), B(4 ; -6), C(4;-2) and E(6; y) are the vertices of a quadrilateral having AE||BC. D lies on BC such that AD BC and AC is drawn. M(x ; 2) isa point on EC. Ay E(; y) AC2 ; 2) M(x ; 2) x re) aa C4 5 -2) D BC4; -6) 31 Calculate the gradient of BC. (3) 3.2 If M is the midpoint of EC, determine the values of x and y. (3) 3.3 Calculate the length of BC. (2) 3.4 If it is further given that AE= 80, to which group of special quadrilaterals does ABCE belong? qd) 3.5 Determine the equation of AD inthe form y=mx+c. (3) 3.6 Calculate the coordinates of D. (5) 3.7 Determine the area of AAEC. (3) [20] Copaiah resend NT Pesce tum ver EASTERN CAPE

Downloaded from hlayiso.com Mathematics/P2 6 DBE/November 2019 CAPS - Grade 11 QUESTION 4 In the diagram, B isa point on the y-axis. A(4;y), B, C(3;0) and D(7; 6) are the vertices of rectangle ABCD. Diagonals BD and AC are drawn. The inclination of DC is @ and BAC =0. y A(4;y) & D(7 ; 6) B é x 0 CG ; 0) 4.1 Determine the gradient of CD. (2) 4.2 Calculate the size of a. (2) 4.3 Determine the value of y. (4) 44 Calculate the size of 0. (5) [13] Copsightreserved UN idan EASTERN CAPE

Downloaded from hlayiso.com Mathematics/P2 7 DBE/November 2019 CAPS - Grade 11 QUESTION 5 5.1 In the diagram below, P(k ; 24) is a point in the second quadrant such that OP=25 units. N isa point on the positive x-axis and PON = 8. fy P (k; 24) 25 AY x —_e—_—> [e} N WITHOUT calculating the size of 0, determine the value of the following: 5.1.1 k (2) 5.1.2 tand (1) 5.1.3 sina if 0+a=360° (3) 5.1.4 cos’6 —sin’a (3) 5.2 Simplify WITHOUT using a calculator: cos210°.tan135° sin(— 60°). cos420° 6) ays) Prove the identity: > cos'x 5-08’ x= —S (4) tan” x sin” x 5.4 Determine the general solution of V2 sinxcosx = cosx. (6) [24] Copsieh eserves I A Please ram over EASTERN CAPE

Downloaded from hlayiso.com Mathematics/P2 8 DBE/November 2019 CAPS — Grade 11 QUESTION 6 Sketched below are the graphs of the functions /(x)=cosx+q and g(x)=2sinbx for x €[-60° ; 120°]. y a & 1 £ |—T_ | | | . —60° =30° 0° 30° 90° 0° =H “2 6.1 Write down the coordinates of the maximum turning point of g in the given interval. 6.2 Determine the values of x where f is strictly increasing in the given interval. 6.3 Determine the values of g and 5. 6.4 Use your graphs to determine the values of x for which 2cosxsin3x—sin3x 2 0. cess AA EASTERN CAPE () (2) (2) (4) 19]

Downloaded from hlayiso.com Mathematics/P2 9 DBE/November 2019 CAPS — Grade 11 QUESTION 7 TA In the diagram A, C, E and B are the vertices of a quadrilateral. ABC =31°, BAC=106°, BCE = 46°, AC=4,6cm, CE=10cm and BC=x. fo Calculate the length of BC. (3) 712 Calculate the area of quadrilateral ACEB. (4) 7.2 In the diagram below, PTSR is a parallelogram. Q is the midpoint of PR. QS = 6units, PT = 3 units, TS = 8 units, QT = x units and T, =@. Ss ar 8 x? +28 7.2.1 Show that cosO = 6B) 6x qa Hence, determine the length of QT. (6) [16] comonesns AA) EASTERN CAPE

Downloaded from hlayiso.com Mathematics/P2 10 DBE/November 2019 CAPS — Grade 11 QUESTION 8 8.1 The cylindrical container shown below will be filled with spherical marbles. The container has a radius of Rm and a perpendicular height of # cm. The volume of the container is 300 cm*. The radius of each marble is 0,75 cm. aN Volume of sphere = on r Surface area of a sphere = 47 r? marble 300 8.1.1 Show that the height of the container, A, is given by h= R (2) 1 8.1.2 The container is filled with 100 marbles. Then water is poured into the container until the water reaches the top of the container. Calculate the volume of water that was poured into the container. (3) 8.2 In the diagram below, the right rectangular prism has a square base of x cm and aheight of (180 — 5x) cm. 180 — 5x x x Calculate the maximum surface area of the prism. (5) [10] comet AA EASTERN CAPE

Downloaded from hlayiso.com Mathematics/P2 11 DBE/November 2019 CAPS - Grade 11 Give reasons for your statements and calculations in QUESTIONS 9, 10, and 11. QUESTION 9 In the figure Q, P, T, S and R lie on the circle. M lies on QS such that PM || SR. PTS =145° 9.1 Calculate the size of: 9.1.1 Q 9.1.2 R 9.1.3 P, 9.2 Why is PR a tangent to the circle passing through P, M and Q? convenes: NAA EASTERN CAPE (2) Q) (1) (1) [6]

Downloaded from hlayiso.com Mathematics/P2 12 DBE/November 2019 CAPS - Grade 11 QUESTION 10 10.1 In the diagram below, O is the centre of the circle. P, T and S are points on the circumference of the circle. PS, PT and OT are drawn. Prove the theorem which states that SOT = 2xSPT. (5) Copsieh reseed HN A A Pease over EASTERN CAPE

Downloaded from hlayiso.com Mathematics/P2 13 DBE/November 2019 CAPS - Grade 11 10.2 In the diagram, O is the centre of the circle. Points A, B, C and D lie on the circumference of the circle. BOD is a diameter. AC and BD intersect at E. A, =51° and B, = 29°. 10.2.1 Determine the size of O,. (2) 10.2.2 Determine the size of A,. (2) 10.2.3. Determine the size of D. () 10.2.4 Determine the size of ACO. (3) [13] comes! AAA HH = EASTERN CAPE

Downloaded from hlayiso.com Mathematics/P2 14 DBE/November 2019 CAPS — Grade 11 QUESTION 11 In the diagram, O is the centre of the circle through the points A, B, C, D and T. HC and HT are tangents to the circle at C and T respectively. AD is produced to meet HT at R. OC bisects AD at J. Let C , = x. 11.1 Write down, with a reason, another angle equal to c O (2) 11.2 Show that CHRJ is a trapezium. (5) 1] Prove that OC bisects ACD. (3) 11.4 Write down, with a reason, ABD in terms of x. (2) is Determine R, in terms of x. (6) [18] TOTAL: 150 Conrigiras=t NA A EASTERN CAPE

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