Department of Mathematics

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Department of Mathematics TIME: 3 Hours Setter: DAS DATE: 07 August 2017 GRADE 12 PRELIM EXAMINATION MATHEMATICS: PAPER II Total marks: 150 Moderator: GP Name of student: PLEASE READ THE FOLLOWING

Answer all the questions on the question paper and hand this in at the end of the examination. Remember to write your name on the paper. 4. Diagrams are not necessarily drawn to scale. 5.

1 Department of Mathematics TIME: 3 Hours Setter: DAS DATE: 07 August 2017 GRADE 12 PRELIM EXAMINATION MATHEMATICS: PAPER II Total marks: 150 Moderator: GP Name of student: 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. Remember to write your name on the paper. 4. Diagrams are not necessarily drawn to scale. 5. You may use an approved non-programmable and non-graphical calculator, unless otherwise stated. 6. All necessary working details must be clearly shown. 7. Round off your answers to one decimal digit where necessary, unless otherwise stated. 8. Ensure that your calculator is in DEGREE mode. 9. It is in your own interest to write legibly and to present your work neatly. Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 TOTAL

2 Page 2 of 28 SECTION A QUESTION 1 (a) The diagram shows the points A(0; 6), B(8; 6) and C(4; 0). AB C = θ. Point P is the midpoint of AC. (1) Calculate the length of BC, leaving your answer in simplified surd form. (2) (2) Determine the size of angle θ. (3)

3 Page 3 of 28 (3) Use analytical methods to show that O, P and B are NOT collinear. (4) (4) Determine the area of triangle ABC. (2)

4 Page 4 of 28 (b) The diagram shows a line with gradient 2 intersecting the x-axis at A and the y-axis at B. Point P(5; 3) is indicated. (1) If the area of AOB is 16 units 2, determine the coordinates of point A. (4) (2) The line through P perpendicular to AB cuts the x-axis at C. Determine the coordinates of point C. (4) [19]

5 Page 5 of 28 QUESTION 2 PLEASE ENSURE THAT YOUR CALCULATOR IS IN DEGREE MODE (a) Simplify as far as possible: sin(360 θ).cos(90 +θ) 1 cos 2 ( θ) (4) (b) Prove the identity: sin 2θ sin θ = sin θ cos 2θ + cos θ cos θ + 1 (5)

6 Page 6 of 28 (c) The diagram shows the graphs of f(x) = sin 2x and g(x) = cos 2x for 0 x 180. Point P is a point of intersection of the two graphs. (1) Write down the period of g. (1) (2) Calculate the coordinates of point P. Clearly show your working. (5) (3) For what values of x is f(x). g(x) < 0? (2)

(2) (2) On the same set of axes given above, draw the graph of g(x) = 3 sin x 1.

7 Page 7 of 28 (d) The graph shows the curve of f(x) = a tan x + b for x [0 ; 360 ]. (1) Determine the values of a and b. (2) (2) On the same set of axes given above, draw the graph of g(x) = 3 sin x 1. (4) (3) Without solving the equation, write down the number of solutions there are for f(x) = g(x) for x [0 ; 360 ]. (1) [24]

8 Page 8 of 28 QUESTION 3 (a) In the diagram below, TAN is a tangent to the circle with centre O at point A. Use the diagram to prove the theorem that states that CÂN = B. (5)

9 Page 9 of 28 (b) (1) Complete the following statement: Equal chords subtend (1) (2) In the diagram below, ABCD is a cyclic quadrilateral with AD = BC. Prove, giving reasons, that AB // DC. (4)

10 (c) In the diagram, AB=AC and D and E are points on BC. AD and AE are produced to meet the circumscribed circle of ABC at F and G respectively. Page 10 of 28 (1) Prove that Ĝ = B + Â 1 (4)

11 Page 11 of 28 (2) Hence prove that DFGE is a cyclic quadrilateral. (3) [17]

12 Cumulative Frequency Page 12 of 28 QUESTION 4 (a) The cumulative frequency graph shows the monthly salaries, in thousands of rands, of a sample of 500 shoppers in Sandton City. The lowest salary is R5 000 and the highest salary is R Salary (thousands of rands) (1) How many shoppers earned more than R per month? (1) (2) Use the cumulative frequency graph to draw a box-and-whisker plot for the data. (4)

13 Page 13 of 28 (3) Comment on the skewness of the distribution of the data. (1) (4) An outlier is defined as any value which is more than 1,5 times the interquartile range above the upper quartile, or more than 1,5 times the interquartile range below the lower quartile. (i) How high must a salary be in order to be classified as an outlier? (2) (ii) Show that none of the salaries is low enough to be classified as an outlier. (2)

14 Page 14 of 28 (b) Consider the following pairs of points: x y (1) Calculate the equation of the line of best fit in the form y = A + Bx. Give A and B correct to 3 decimal places. (3) (2) Calculate the correlation coefficient correct to 3 decimal places and describe the correlation. (2) (3) An interesting feature of the line of best fit is that it always passes through the point (x ; y ), where x is the mean of the x-values and y is the mean of the y-values. Confirm that this is true for the given data. (3) [18] 78 marks

15 Page 15 of 28 SECTION B QUESTION 5 (a) The diagram shows a circle with centre P passing through A(0; 2), B(0; 6) and point C on the x-axis. AC = 5 3 and AB C = 60. (1) Giving a reason, write down the size of angle AP C. (2) (2) Hence determine r, the radius of the circle. (3) (3) Hence determine the equation of the circle in the form (x a) 2 + (y b) 2 = r 2. (3)

16 (b) The diagram shows a circle with centre C and equation x 2 + y 2 + 6x 4y = 12. The circle cuts the y-axis at points A and B. AD is a tangent to the circle at A and cuts the x-axis at D. Angle θ is indicated. Page 16 of 28 (1) Determine the coordinates of C as well as the radius of the circle. (4)

17 Page 17 of 28 (2) Determine the size of the angle θ. (6) [18]

18 Page 18 of 28 QUESTION 6 Given: 6 tan θ tan θ = 2 sin 2θ (a) Show that the equation can be expressed as 4 cos 4 θ + 3 cos 2 θ 1 = 0. (5)

19 Page 19 of 28 (b) Hence solve the equation 6 tan θ tan θ = 2 sin 2θ for θ (0 ; 180 ). (4) [9]

20 Page 20 of 28 QUESTION 7 (a) The diagram shows triangle ABC. D, K and E are points on the sides of the triangle. AD: DB = 2: 3 BE: EC = 4: 3 CD and AE intersect at P DK AE Determine, giving reasons, the numerical value of CP: PD. (6)

21 (b) The diagram shows a circle passing through points A, B and D. BE and DF are tangents to the circle at B and D respectively. The tangents intersect at point C such that FĈE = θ. Page 21 of 28 If it is further given that BÂD = FĈE, determine, giving reasons, the value of θ. (5)

22 Page 22 of 28 (c) In the diagram, AB is a diameter of the circle with centre O. BP = BT = BO = AO PT is a tangent to the circle at T EP AP Let Â = x Giving reasons: (1) Prove that TBPE is a cyclic quadrilateral. (3) (2) Use similarity to prove that PT 2 = PB. PA. (5)

23 Page 23 of 28 (3) Prove that PT = PE. (2) (4) Hence, or otherwise, determine the numerical value of PT : AE. (5) [26]

24 Page 24 of 28 QUESTION 8 (a) Two metal strips with different widths need to be welded together to form an angle of 37 as shown below. In order to accomplish this, the two strips need to be cut at angles α and θ as shown. (1) Express α in terms of θ. (2) (2) Prove that 10 = 15. cos α cos θ (3)

25 Page 25 of 28 (3) Hence, or otherwise, determine the values of α and θ. (6)

26 (b) A frustum is a cone with its top section cut off parallel to the base. In the frustum shown below, the area of the circular base is twice that of the upper surface. The vertical distance between the upper and lower circular faces is 4 cm. Page 26 of 28 If it is given that the frustum has a volume of 83,206 cm 3, determine the radius of the circular base. Given: Volume of cone = 1 3 πr2 h

27 Page 27 of 28 (8) [19] 72 marks Total: 150 marks

28 Page 28 of 28 SPACE FOR ADDITIONAL WORKING (clearly number your answers)

P1 REVISION EXERCISE: 1

P1 REVISION EXERCISE: 1 1. Solve the simultaneous equations: x + y = x +y = 11. For what values of p does the equation px +4x +(p 3) = 0 have equal roots? 3. Solve the equation 3 x 1 =7. Give your answer