MATHEMATICS: PAPER II
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1 CLUSTER PAPER 2016 MATHEMATICS: PAPER II Time: 3 hours 150 marks PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY 1. This question paper consists of 28 pages and an Information Sheet of 2 pages(i-ii). Please check that your question paper is complete. 2. Read the questions carefully. 3. Answer ALL the questions on the question paper and hand this in at the end of the examination. 4. You may use an approved non-programmable and non-graphical calculator, unless otherwise stated. 5. All necessary working details must be clearly shown. 6. Round off your answers to one decimal digit where necessary, unless otherwise stated. 7. Ensure that your calculator is in DEGREE mode. 8. It is in your own interest to write legibly and to present your work neatly. 9. The last pages can be used for additional working, if necessary. If this space is used, make sure that you indicate clearly which question is being answered.
2 CLUSTER: PAPER II SEPTEMBER 2016 Page 2 of 28 SECTION A QUESTION 1 Refer to the sketch below: AD//BC. A is the point ( 2; 4) and D is the point (1; -5). Determine: 1.1 the coordinates of B. (2) 1.2 the equation of BC in the form ax + by + c = 0. (4)
3 CLUSTER: PAPER II SEPTEMBER 2016 Page 3 of the coordinates of C. (2) QUESTION 2 [8] Refer to the sketch below: ABC has vertices of A(0; 3), B(-3;-2) and C(5; 0). y A(0;3) C(5;0) x B(-3;-2) 2.1 Calculate the length of: (4) AB BC PLEASE TURN OVER
4 CLUSTER: PAPER II SEPTEMBER 2016 Page 4 of AC 2.2 Hence, show that ABC is a right angled isosceles triangle. (4) 2.3 Determine the area of ABC. (2) 2.4 Given that BC is a diameter of the circumscribed circle of ABC, show that the centre of this circle is M, the point (1; 1). (2) [12]
5 CLUSTER: PAPER II SEPTEMBER 2016 Page 5 of 28 QUESTION 3 Find the equation of the line that is a tangent to the circle (x + 1) 2 + (y + 2) 2 = 20 at (3; 4). (6) [6] PLEASE TURN OVER
6 CLUSTER: PAPER II SEPTEMBER 2016 Page 6 of 28 QUESTION 4 If 37cosθ + 10 = 2 and θ [180 ; 360 ], determine, with the use of a diagram, the value of 1 sin 2θ 4 cos θ. (4) [4]
7 CLUSTER: PAPER II SEPTEMBER 2016 Page 7 of 28 QUESTION Prove: 1 cos(90 2θ) tan (180 + θ) = cos 2θ (5) 5.2 For which values of θ is this identity in 5.1 undefined? (2) [7] PLEASE TURN OVER
8 CLUSTER: PAPER II SEPTEMBER 2016 Page 8 of 28 QUESTION 6 Evaluate without using a calculator: 6.1 cos 3. cos cos 267. sin 42 (5) 6.2 tan (1 + cos 120 ). sin (5) [10]
9 CLUSTER: PAPER II SEPTEMBER 2016 Page 9 of 28 QUESTION 7 The table gives the Olympic pole vault records for the past 12 games Olympic games. Year Height in metres , , , , , , , , , , , , Find the equation of the line of best fit for the height (y) against the year (x) in the form y = a + bx. Determine your values for a and b rounded to 2 decimal digits. (2) 7.2 Complete the scatter plot and draw the line of best fit on the scatter plot. (3) x x x x x x PLEASE TURN OVER
10 CLUSTER: PAPER II SEPTEMBER 2016 Page 10 of Determine the value of the correlation coefficient rounded to 2 decimal digits and explain clearly what can be deduced from this result. (3) 7.4 Use the model to predict the record pole vault height for the 2016 Olympic Games. (1) Use the model to predict the record pole vault height for the 2028 Olympic Games. (1) Do you think the actual record in 2028 will be higher or lower than this prediction? Give a reason for your answer. (2) [12]
11 CLUSTER: PAPER II SEPTEMBER 2016 Page 11 of 28 QUESTION 8 In the diagram, O is the centre of the circle passing through A, B, C and D. AB//CD and B = 20. Complete the following statements and reasons to prove that AOEC is a cyclic quadrilateral. No extra steps/calculations are allowed. (5) STATEMENTS C 1 = O 1 = 40 D = 20 E 1 = AOEC is a cyclic quad REASONS alt. angles ; AB//CD..... ext. angle of... [5] PLEASE TURN OVER
12 CLUSTER: PAPER II SEPTEMBER 2016 Page 12 of 28 QUESTION Complete the following statement: The line from the centre of the circle perpendicular to the chord. (1) 9.2 Refer to the figure below: In the circle with centre O. OT QP, OS PR, OT = 8 units, PQ = 30 units and PR = 23 units. Q T O x S P R Determine OS = x. (5)
13 CLUSTER: PAPER II SEPTEMBER 2016 Page 13 of 28 QUESTION 10 [6] Refer to the figure below: PQ and PS are tangents to circle QRST. ST, RS and RQ are chords and QS is joined. PQ//ST. Let S 2 = x. x 10.1 Calculate Q 1 in terms of x. (1) PLEASE TURN OVER
14 CLUSTER: PAPER II SEPTEMBER 2016 Page 14 of Prove: RSQ/// RQP (4) 10.3 Prove: QR 2 = RS. RP (2) [7] TOTAL FOR SECTION A = 77 MARKS
15 CLUSTER: PAPER II SEPTEMBER 2016 Page 15 of 28 SECTION B QUESTION 11 The circle, with equation x 2 10x + y 2 + 8y 40 = 0, is given. A point P(x; y) on the circumference of a NEW circle, is such that it is always 3 units from the circumference of the original circle, and outside the original circle. Determine the equation of this new circle. (6) [6]
16 CLUSTER: PAPER II SEPTEMBER 2016 Page 16 of 28 QUESTION 12 Solve for x correct to one decimal digit if θ [ 180 ; 180 ] sin(2θ 10 ) = cos (θ + 50 ). (8) [8]
![t)] where t is the time, in months (t](/docs-images/87/96968783/images/17-3.jpg "= 0 corresponds to 1 August 2015),")
17 CLUSTER: PAPER II SEPTEMBER 2016 Page 17 of 28 QUESTION 13 The company, Mega Ski, has sales of ski equipment as given by S(t) = 10[1 cos (30. t)] where t is the time, in months (t = 0 corresponds to 1 August 2015), and S(t) is in thousands of Rands Use the given graph paper below to sketch the function of the sales on a 12-month interval [0; 12]. (4) PLEASE TURN OVER
18 CLUSTER: PAPER II SEPTEMBER 2016 Page 18 of What is the period of the function? (2) 13.3 What is the minimum amount of sales? (1) 13.4 What is the maximum and when does the maximum amount of sales occur? (3) [10]
19 CLUSTER: PAPER II SEPTEMBER 2016 Page 19 of 28 QUESTION 14 At a particular time during the day, a tower of height 19,2 metres casts a shadow. At the same time, a person who is 1,65 metres tall casts a shadow 5 metres long. 19,2 m 1,65 m Shadow of tower 5 m What is the length of the shadow cast by the tower at that time? (6) [6] PLEASE TURN OVER
20 CLUSTER: PAPER II SEPTEMBER 2016 Page 20 of 28 QUESTION 15 A new fitness centre offers two different fitness classes. The attendance for each class for 12 sessions is represented in the box-and-whisker plot Determine the interquartile range of Class A. (1) 15.2 Compare the data from Class A and Class B. How are they alike? (1) 15.3 Compare the data from Class A and Class B. How are they different? (1) 15.4 Which class has more variability? Show calculations to support your answer. (3) 15.5 Which class has a higher mean deviation? Explain your answer. (2) [8]
21 CLUSTER: PAPER II SEPTEMBER 2016 Page 21 of 28 QUESTION 16 Refer to the sketch below: In the diagram O is the centre of the circle with AE and CE tangents to the circle. C 1 = 45 and O 1 = 127 A B D O E C Calculate the size of E. (5) [5] PLEASE TURN OVER
22 CLUSTER: PAPER II SEPTEMBER 2016 Page 22 of 28 QUESTION 17 In the diagram below, points R, P, A, Q and T lie on a circle. RA bisects R and AB = AQ. RA and TQ produced meet at B. Prove that: 17.1 AQ bisects PQ B. (3) 17.2 TR = TB. (2)
23 CLUSTER: PAPER II SEPTEMBER 2016 Page 23 of P = TR P (3) [8] PLEASE TURN OVER
24 CLUSTER: PAPER II SEPTEMBER 2016 Page 24 of 28 QUESTION 18 Refer to the sketch: In the circle below, AB and CD are chords intersecting at E. A C E D B If AE = 5, BE = 12 and CE = 6, what is the length of DE? (6) [6]
25 CLUSTER: PAPER II SEPTEMBER 2016 Page 25 of 28 QUESTION 19 Daniel decided to participate in this year s Tour de France bicycle race. One of the routes are around the Annecy Lake which is a circular lake that has a diameter of 4 km, as shown in the diagram below. Points B and D are the opposite sides of Annecy Lake and lie on a straight line through the centre of the lake, with each point 5 km from the centre. The course of the race is ABCDE, where AB and DE are tangents to the lake. A E 4 km B 5 km C 5 km D Annecy Lake Determine the length of the route. (6) PLEASE TURN OVER
26 CLUSTER: PAPER II SEPTEMBER 2016 Page 26 of 28 [6]
27 CLUSTER: PAPER II SEPTEMBER 2016 Page 27 of 28 QUESTION 20 The diagram shows triangles ABC and ABD with AD parallel to BC. The sides AC and BD intersect at Y. The point X lies on AB such that XY is parallel to AD and BC. C D Y A X B 20.1 Prove ABC /// AXY. (4) 20.2 Hence, or otherwise, prove that 1 = XY AD BD (6) PLEASE TURN OVER
28 CLUSTER: PAPER II SEPTEMBER 2016 Page 28 of 28 [10] TOTAL FOR SECTION B = 73 MARKS TOTAL: 150 marks
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