Department of Mathematics

Transcription

1 Department of Mathematics TIME: 3 Hours Setter: DS DATE: 09 August 2016 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 27 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 Q9 TOTAL

2 Page 2 of 27 SECTION A QUESTION 1 (a) In the diagram, AB is the diameter of the circle with centre O(7; p). A(3; 8), B(t; 4) and C lie on the circle. The straight line BCDE has equation y = 1 x 3, with D on 2 2 the x-axis and E on the y-axis. The angle θ is shown. (1) Determine the values of p and t. (2) (2) Show that D lies vertically below A. (3)

3 Page 3 of 27 (3) Write down, giving a reason, the size of angle AC B and hence determine the equation of line AC. (5) (4) Determine, correct to one decimal digit, the size of θ. (3)

4 Page 4 of 27 (b) A(3; 7), B(t; 4) and C(3; 5) are points in the Cartesian plane. (1) If BC = 106, show that t = 2. (4) (2) Hence determine the area of ABC. (3) [20]

5 Page 5 of 27 QUESTION 2 PLEASE ENSURE THAT YOUR CALCULATOR IS IN DEGREE MODE (a) (1) Prove the following identity: sin 2θ 1 + cos 2θ = tan θ (4) (2) Hence, or otherwise, determine the values of θ [0 ; 360 ] for which 1 + cos 2θ is undefined. sin 2θ (4)

6 Page 6 of 27 (b) Simplify as far as possible: sin(180 + θ). sin(90 + θ) cos( θ). tan(180 θ) (4) (c) Solve for θ if sin θ = cos 250 and 180 < θ < 360. (4)

7 Page 7 of 27 (d) The graph shows the curves of f(x) = a cos x and g(x) = tan(bx) + c for x [0 ; 180 ]. (1) Find the values of a, b and c. (3) (2) Write down the co-ordinates of points A and B. (2) (3) For what values of x is f(x). g(x) < 0? (2) (4) If f was a representation of a sine curve, write down a possible equation to define f. (2) [25]

8 Page 8 of 27 QUESTION 3 (a) Use the diagram below to prove the theorem that states that the opposite angles of a cyclic quadrilateral are supplementary. (5)

9 Page 9 of 27 (b) In the diagram below: Points A, B, C and D lie on the circle. EDF is a tangent to the circle at D. ED A = θ, A = 2θ, DB C = 40 and AD = BC. Determine, giving reasons, the value of θ. (6)

10 Page 10 of 27 (c) In the diagram, AD is a diameter of the larger circle with centre C and AC is the diameter of the smaller circle with centre B. FD is a tangent to the smaller circle at E while FA is a tangent to both circles at A. (1) Prove that DEB DAF. (4)

11 Page 11 of 27 (2) Hence show that DF = 3AF. (4) (3) If it is further given that the diameter of the larger circle is 8 cm, determine the area of DEB. (4) [23]

12 Number of pupils Page 12 of 27 QUESTION 4 A group of Matric pupils wrote a Maths test and were asked to indicate how long they had spent preparing for the test the previous day. The data is presented below as an ogive Time (minutes) (a) How many pupils wrote the test? (1) (b) How many pupils spent more than an hour preparing for the test? (1) (c) Use the ogive to estimate the lower quartile, median and upper quartile. (3)

13 Page 13 of 27 (d) If it is further given that the shortest and longest times spent preparing for the test were 15 min and 120 min respectively, describe the skewness of the time distribution. Explain your answer. (2) [7] 75 marks

14 Page 14 of 27 SECTION B QUESTION 5 (a) The graph below shows the points A(3; 35), B(6; 53), C(8; 44), D(14; 48), E(17; 65), F(21; 61) and G(28; 80). The dotted line represents the line of best fit, y = ax + b. (1) Determine the values of a and b. (3) (2) Calculate the correlation coefficient and comment on the validity of the line of best fit. (2) (3) Point H is added to the data, causing the value of a to decrease. Clearly indicate on the graph a possible position for point H. (1) (4) Point I is added to the data, causing no change to the values of either a or b. Clearly indicate on the graph a possible position for point I. (1)

15 Page 15 of 27 (b) In order to evaluate a point (x i ) in terms of its relative standing to the rest of the data, we calculate its z-score. The z-score gives the number of standard deviations the data point (x i ) is from the mean. z = x i x σ (1) Explain why a negative z-score indicates that the data point is to the left of the mean. (1) (2) Consider the following stem-and-leaf plot: Calculate the z-score of the highest data point, and interpret your answer. (6) [14]

16 Page 16 of 27 QUESTION 6 (a) Determine the general solution to: sin θ cos θ + cos θ sin θ = 4 (6)

17 Page 17 of 27 (b) If sin(x + 45 ) = 2 sin x, prove: tan x = (6) [12]

18 Page 18 of 27 QUESTION 7 (a) Four identical circles (with centres E, F, G and H) just touch each other as well as the sides of square ABCD as shown. The circle with centre H has equation x 2 + y 2 6x 8y + 21 = 0. (1) Determine the equation of the circle with centre F. Give your answer in the form (x a) 2 + (y b) 2 = r 2. (6)

19 (2) Determine the equation of the circle passing through the points A, B, C and D. Give your answer in the form (x a) 2 + (y b) 2 = r 2. Page 19 of 27 (5)

20 Page 20 of 27 (b) The diagram below shows a semicircle with diameter OB along with the line 4x 3y = 45. Prove that the line 4x 3y = 45 is a tangent to the semicircle. (6) [17]

21 Page 21 of 27 QUESTION 8 (a) In the diagram: A, B and D lie on the circle BC touches the circle at B Ĉ = AB D = x CDA is a straight line Prove that AB is a diameter of the circle. (5) (b) Consider the following diagram: Prove that PC is not parallel to QD. (3)

22 Page 22 of 27 (c) The diagram shows ABC. ED BC and GH CA. AD: DC = 3: 2 BG: GC = 2: 1 Determine, giving reasons, the value of HK HG. (6)

23 Page 23 of 27 (d) In the diagram below, ADF is a tangent to the larger circle, passing through B, C and D, at D. CBA and CDE are straight lines with A and E on the smaller circle. BD is a common chord. Prove that AD = AE. (4)

24 (e) In the diagram, AB is a diameter of the circle. CA is a tangent to the circle at A. Chord BE produced meets the tangent at F. FB A = x. Page 24 of 27 Prove, giving reasons, that CDEF is a cyclic quadrilateral. (5) [23]

25 Page 25 of 27 QUESTION 9 The three circles in the diagram have radii of 4, 5 and 6 cm respectively, and just touch each other. Determine, correct to 2 decimal places, the shaded area enclosed between the three circles.

26 Page 26 of 27 [9] 75 marks Total: 150 marks

27 Page 27 of 27 SPACE FOR ADDITIONAL WORKING (clearly number your answers)