St. Anne s Diocesan College. Grade 12 Core Mathematics: Paper II September Time: 3 hours Marks: 150

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1 St. Anne s Diocesan College Grade 12 Core Mathematics: Paper II September 2018 Time: 3 hours Marks: 150 Please read the following instructions carefully: 1. This question paper consists of 21 pages and an Information sheet. Please check that your question paper is complete. 2. Answer all the questions on the question paper. 3. You may use an approved non-programmable and non-graphical calculator, unless otherwise stated. 4. All necessary working details must be clearly shown. 5. Round off answers to 1 decimal digit where necessary, unless otherwise stated. 6. Ensure that your calculator is in DEGREE mode. 7. Diagrams are not drawn to scale. SECTION A SECTION B Question Mark Total Total Percentage Name: Teacher: Page 1 of 21

2 SECTION A Question 1 Kite ABCD is drawn below and the points P, Q and R are x-intercepts. y A( 2; 4) 2 1 B M P Q R x C(6; 2) D( 4; 7) (a) Calculate the coordinates of M, the midpoint of the diagonal AC. (2) (b) Show that the gradient of the diagonal AC is 3. 4 (1) (c) Prove that the diagonals are perpendicular to each other. (3) Page 2 of 21

3 (d) Determine the equation of the diagonal BD. (3) (e) Calculate the magnitude of  1. (4) [13] Page 3 of 21

4 Question 2 The circle that passes through the points A( 1; 6), B(0; 1) and C( 7; 2) is given below. AC is a straight line. y D A( 1; 6) M θ x C( 7, 2) B(0, 1) (a) Show that the coordinates of M, the midpoint of AC, are M( 4; 2). (1) (b) Show that MA = MB. (3) (c) Hence, give the equation of the circle. (2) Page 4 of 21

5 (d) Show, by calculation, whether the coordinate P( 8,5 ; 2) lies inside or outside the circle circumference. (3) (e) Calculate the coordinates of D if the line BM is produced to a point on the circumference at D. (2) (f) Explain why the parallelogram ABCD is a rectangle (2) (g) Determine the equation of the tangent to the circle at B in the form y =... (3) (h) (i) Give the new equation of the circle in (c) if it is reflected over the x-axis. (1) (ii) Give the new equation of the circle in (c) if it is shifted 5 units left and 3 units down. (2) [19] Page 5 of 21

6 Question 3 A record is kept of the number of times a netball team sinks a ball into the hoop in 10 separate matches (a) Give the range of the results. (1) (b) (i) Determine the 5-number summary from the given data above. (3) (ii) Use the 5-number summary to draw a box-and-whisker above the number line below. (c) Calculate the mean. (2) (d) Show, by calculation, that the data is skewed to the left. (2) (e) Calculate the standard deviation. (1) (f) Determine the percentage of data that lies within one standard deviation of the mean. (3) (g) If the netball team played one more match and did not score, would the standard deviation increase, decrease or stay the same? (1) [14] Page 6 of 21 (1)

7 Question 4 Give full reasoning for the following questions. (a) In the given diagram, A, B, C and D lie on the circumference of the circle centre O. AÔB = 115 and AB = CD. B A D x 115 O O 2 1 C Determine the size of x: (3) (b) In the diagram below, O is the centre of the given circle and A, B, C and D lie on the circumference. Ĉ = 73 and BÔE = DÔE = x. B C 73 A E O x x Prove that quadrilateral ABOE is cyclic. D (4) Page 7 of 21

8 (c) AD and DC are tangents to circle centre O. Make a construction to help determine the size of x. x B A O 68 D C (4) [11] Question 5 (a) If cos 50 = p, determine the following in terms of p: 1. cos 100 (2) 2. sin 230 (3) Page 8 of 21

9 (b) Prove that cos 15 sin 15 + tan 15 = 4 without the use of a calculator. (6) (c) Give the general solution for x, correct to one decimal place if: 2 sin x. cos x = 0,822 (3) [14] [71] Page 9 of 21

10 Cumulative frequency SECTION B Question 6 An events company markets their venue on Instagram and is able to see the age range of their audience. Age Range Frequency Cumulative Frequency 7 < x < x < x < x < x < x < x (a) Complete the table by calculating the cumulative frequency. (2) (b) Use the completed table to draw a cumulative frequency graph below. (3) Cumulative frequency graph showing age of audience on Instagram (c) Determine the median age. Show your working on the graph. (2) (d) What percentage of viewers were older than 57years? (2) [9] Page 10 of Age of audience in years

11 Question 7 Use the diagram below to prove that: a sin A = b sin B = c sin C B c a A b C [5] Page 11 of 21

12 Question 8 In the sketch below, AB is the vertical building. B, C and D are points on the same horizontal plane. The angle of elevation of C to A at the top of the building is θ. DB C = 30, DĈB = θ and BD = 12m. A B 30 θ θ C 12m (a) Give the size of BD C in terms of θ. D (1) (b) Show that BC = 6 cos θ sin θ (5) Page 12 of 21

13 (c) Hence, determine that the height AB of the building in terms of tan θ. (4) [10] Page 13 of 21

14 Question 9 In the given diagram, PAT is a tangent to the circle ABC at A. AC is produced to D and BD PT 1 2 B D C 3 A T (a) Prove that B 2 = D. P (2) (b) Hence show that AB 2 = AC. AD (5) (c) If AB = 30, AC = x and AC: CD = 1: 2, calculate the length of AC. Leave you answer in surd form. (3) [10] Page 14 of 21

15 Question 10 AOB is a diameter of circle centre O. BC is a tangent and AC cuts the circle at D. E is the midpoint of AD B 2 1 O A 1 2 E 1 2 D C (a) List three angles, at different vertices, that equal 90 with reasons. (3) (b) Prove that OE BD (1) (c) Prove that 4OE 2 = AD. DC (4) [8] Page 15 of 21

16 Question 11 (a) Solve the equation 1 sin θ = cos 2θ where θ [0 ; 180 ] (6) (b) Hence, the graph f(x) = cos 2θ is drawn below in the interval θ [0 ; 180 ] y f(x) x Draw the graph g(x) = 1 sin θ where θ [0 ; 180 ] on the same set of axes and label the coordinates where g(x) = f(x). (3) (c) (i) Give the period of the graph f: (1) (ii) Give the amplitude of the graph g: (1) (iii) Give the new equation of g if is translated 30 left: (1) [12] Page 16 of 21

17 Question 12 (a) Determine the radius and the coordinates of the centre of the circle: x 2 + 6x + y 2 16y = 11 (4) (b) Does the circle (x + 5) 2 + (y 2) 2 = 18 touch the circle (x 1) 2 + (y + 2) 2 = 10 externally? Show working to justify your answer. (4) Page 17 of 21

18 (c) The straight line y = 3x + 5 is a tangent to the circle (x a) 2 + (y b) 2 = r 2 at the point P( 2; 1). The equation of a diameter of the circle is y = 3x + 1. y P( 2; 1) x Determine the values of a, b and r. (7) [15] Page 18 of 21

19 Please turn over for final Question 13. Page 19 of 21

20 Question 13 The diagram shows a cyclic quadrilateral where AB = BC and AD = DC = x. B = 107 and AB = 7cm. A 7cm 107 B x C D x (a) Show that the area of the cyclic quadrilateral is 66 units 2 (to the nearest unit). (7) Page 20 of 21

21 The diagram shows a pyramid sitting inside a cylinder. The base of the pyramid is the cyclic quadrilateral discovered in question 13(a). The radius is r and the height of the cylinder is 4r. The volume of a pyramid is given by the formula: V = 1 (area of base)h 3 4r A B D r C (b) What is the remaining volume inside the cylinder if the radius of the circle is 6cm and if the pyramid fitted inside the cylinder is solid? (3) [10] [79] Page 21 of 21