KING DAVID HIGH SCHOOL LINKSFIELD GRADE 11 MATHEMATICS PAPER 2 NOVEMBER Writing Time: 2½ hours Total: 120 marks

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1 KING DAVID HIGH SCHOOL LINKSFIELD GRADE 11 MATHEMATICS PAPER 2 NOVEMBER 2017 Writing Time: 2½ hours Total: 120 marks NAME: INSTRUCTIONS AND INFORMATION Read the following instructions carefully before answering the questions: 1. This question paper consists of 20 pages (including the cover page). A separate information sheet is provided. 2. All the calculations must be clearly shown. You will NOT receive full marks if you merely write down the answer. 3. All the final answers must be rounded off to TWO decimal places, unless stated otherwise. 4. An approved calculator (non-programmable and non-graphical) may be used, unless stated otherwise. 5. Diagrams are not necessarily drawn to scale. 6. Write neatly and legibly. 7. Good luck. 1 [7] 2 [12] 3 [10] 4 [7] 5 [5] 6 [7] 7 [11] 8 [13] 9 [8] 10 [15] 11 [10] 12 [5] 13 [10] Total [120]

2 Q uestion 1 [7 marks] (a) B(8;2) and C(3;a) are points on the Cartesian plane. Determine the value(s) of a, if BC 5 units. (3) (b) Prove that the points N( 1;1), 2 Q(p;p ) and R(1;2p 1) are collinear. (4) Grade 11 November 2017 Paper 2 Page 2

3 Question 2 [12 marks] On a certain day, Jozi Tours sent 11 tour buses to 11 different destinations. The table below shows the number of passengers on each bus. Destination A B C D E F G H I J K No. of passengers (a) Calculate the mean number of passengers travelling on a tour bus. Give the answer correct to two decimal digits. (2) (b) Write down the five number summary. (3) (c) Draw a box and whisker diagram for this set of data. (2) (d) Refer to the above diagram and comment on the skewness of the data, giving a reason for your answer. (2) Grade 11 November 2017 Paper 2 Page 3

4 (e) Calculate the standard deviation for this data set. Give the answer correct to two decimal digits. (2) (f) A tour is regarded as popular if the number of passengers on a tour bus, is more than 1 standard deviation above the mean. How many destinations were popular on this day? (1) Question 3 [10 marks] The ages of teachers at a High School in Brisbane, are given in the table below: (a) Complete the table: (2) Age Frequency 25 < A < A < A < A < A < A < A < A 65 6 Cumulative Frequency Grade 11 November 2017 Paper 2 Page 4

5 (b) Draw an ogive on the set of axes provided, to represent the data in the table. (4) Ogive of ages of teachers (c) Use your ogive to find an estimate for the median age, clearly indicating this reading on your graph. (2) (d) The school would like to give all teachers older than 57 a special present. Use your graph to find an estimate for the percentage of teachers older than 57 years of age. (2) Grade 11 November 2017 Paper 2 Page 5

6 Question 4 [7 marks] The graph of f(x) sin2x for x [ 180 ;90 ] is sketched on the axes below: (a) State the period of f. (1) (b) Sketch g(x) cos(x 45 ) for x [ 180 ;90 ] on the same set of axes. Clearly indicate all intercepts with the axes, as well as the end-points. (3) (c) For what values of x 0, will f(x).g(x) 0? (2) (d) Determine the value of f( 180 ) g( 180 ). (1) Grade 11 November 2017 Paper 2 Page 6

7 Question 5 [5 marks] Determine the general solution for x, in each of the following, rounding answers to one decimal place: (a) 3cos x 1 (2) (b) tan2x 1,86 (3) Grade 11 November 2017 Paper 2 Page 7

8 Question 6 [7 marks] MNO is a tangent to the circle passing through the points N, P, Q, R and S. PQ//NS. ˆP 30 and Ŝ O (a) Determine, with reasons, the size of each of the following angles: (1) ˆN (2) 1 (2) ˆR (3) (b) Determine if PS is a diameter to the circle. Give a reason for your answer. (2) Grade 11 November 2017 Paper 2 Page 8

9 Question 7 [11 marks] (a) Simplify the following to a single trigonometric function: cos tan(180 ).cos(90 ) cos( ) (6) (b) Prove that: 1 sin 1 sin 4 tan 1 sin 1 sin cos (5) Grade 11 November 2017 Paper 2 Page 9

10 Question 8 [13 marks] In the diagram below PQRS is a quadrilateral, with Q(2;10) and S(6;2). The diagonals of PQRS bisect each other at M, at right angles. T and P are the intercepts of PR, with the respective axes. y Q(2;10) R _ M P T O S(6;2) x (a) Determine the gradient of PR. (2) Grade 11 November 2017 Paper 2 Page 10

11 (b) Show that the equation of PR is given by 2y x 8. (4) (c) Determine the co-ordinates of R. (4) (d) Calculate the size of ˆ RTO, correct to one decimal place. (3) Grade 11 November 2017 Paper 2 Page 11

12 Question 9 [8 marks] In the diagrams below, a leaning tower PQ is shown, which leans at an angle of, to the vertical. The stays (steel ropes), PR and PS, help keep the tower stable. From Q, the base of the tower, a vertical line QT is drawn. QS = 30m, SR = 10m, Rˆ 31 and P SQ 39, 7 (a) Give a reason as to why P 1 8, 7? (1) Grade 11 November 2017 Paper 2 Page 12

13 (b) Show that PS = 34,05m. (2) (c) Determine the size of, correct to the nearest degree. (5) Grade 11 November 2017 Paper 2 Page 13

14 Question 10 [15 marks] QC is the diameter of the circle with centre M. The tangent to the circle at A meets QC produced at P. E is the midpoint of AC and ME produced meets AP at D. (a) Complete the statements below: (1) ˆ 90 (1) E 4 (2) Aˆ Aˆ 90 (1) 2 3 (3) 90 tan radius (1) (b) Hence, prove the following: (1) MD // QA (2) Grade 11 November 2017 Paper 2 Page 14

15 (2) AMCD is a cyclic quadrilateral. (4) (3) ˆD 3 2Qˆ (3) (4) DC is a tangent to the circle. (3) Grade 11 November 2017 Paper 2 Page 15

16 Question 11 [10 marks] (a) O is the centre of the circle. DE is a tangent to the circle at C. A and B are points on the circumference of the circle. AB, AC and BC are joined. Prove that ECB ˆ BAC ˆ. (5) Grade 11 November 2017 Paper 2 Page 16

17 (b) In the figure below, P, Q, R and S are points on the circle with centre O. PRT and USTV are straight lines. USTV is a tangent to the circle at S. RST ˆ 50 and Rˆ TV Determine, with reasons, the size of PQ ˆ R. (5) Grade 11 November 2017 Paper 2 Page 17

18 Question 12 [5 marks] The Transamerica Pyramid is the tallest building in San Francisco. It is a square-based pyramid with height 260 m and base width 53 m. 260 m 53 m Decorative lights are installed over the festive season on each of the four the edges of the building (labeled VA, VB, VC, VD in the figure). Will a 1km length of lighting be sufficient to cover the length of the 4 edges? Show all calculations. (5) Grade 11 November 2017 Paper 2 Page 18

19 Question 13 [10 marks] (a) 1 If sin x cos x, determine the value of sin(180 x).sin(90 x), without using a 2 calculator. (5) (b) Determine the value of tan, if the distance between the points (cos ;sin ) and (3;4) is 26 units. (5) Grade 11 November 2017 Paper 2 Page 19

20 Extra lines (if needed): Grade 11 November 2017 Paper 2 Page 20