Grade 11 November Examination 2015 Mathematics: Paper 2 Time: 3 hours Marks: 150

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1 Grade 11 November Examination 2015 Mathematics: Paper 2 Time: 3 hours Marks: 150 Instructions and Information: Read the following instructions carefully before answering the questions. 1. This question paper consists of 12 questions and 17 pages. Answer ALL the questions. 2. Number the answers correctly according to the numbering system used in this question paper. 3. Clearly show ALL calculations, diagrams, graphs etc that you have used in determining your answers. All working should be shown in its proper place. 4. Answers only will not necessarily be awarded full marks. 5. An approved scientific calculator (non-programmable and non-graphical) may be used, unless stated otherwise. 6. If necessary, answers should be rounded off to TWO decimal places, unless stated otherwise. 7. Diagrams are NOT necessarily drawn to scale. 8. A diagram sheet is provided for your convenience. Please detach and write your name and your teachers name on it and staple this to the front of your answer script. Question ; 6.2; 8.1 and 9 should be answered on the diagram sheet. 9. It is in your own interest to write legibly and to present your work neatly. Page 1 of 17 November 2015

2 Question A (-2 ; -1), B (3 ; -3) and C (2 ; 4) are vertices of a triangle in a Cartesian Plane. y x Calculate the length of AC. (2) Determine the gradients of AC and BC. (2) Calculate the magnitude of AC B. (4) If ADCB is a parallelogram, write down the coordinates of D, the 4 th vertex of ADCB. (2) Page 2 of 17 November 2015

3 1.2 In the diagram below, right-angled triangles ABC and ODC are drawn. O is the origin. A and C lie on the y-axis. C is the midpoint of OA. D is the point ( 2 ; 4). y A B C D( 2;4) O x Determine the equation of OD. (2) Determine the coordinates of C. (3) Determine the equation of AB in the form ay + bx + c = 0. (2) [17] Page 3 of 17 November 2015

4 Question 2 In the diagram given below, circle A has centre (3 ; 0) and a radius of 5 units. 2.1 State the equation of circle A. (2) 2.2 If the centre of circle B is (7 ; 8) and it shares a y-intercept with circle A, determine the equation of circle B. Show all working. (4) 2.3 Circle C has equation x 2 2x + y 2 12y + 32 = 0. Determine the coordinates of the centre of circle C and give its radius. (6) 2.4 Find the equation of the tangent to circle B at the point of intersection of the three circles. (3) [15] Page 4 of 17 November 2015

5 Question 3 The following table shows the weights of 100 rugby players in the World Cup. Weights (kg) in intervals Number of players 70 < x < x < x < x < x < x < x Cumulative frequency Answer Question on the attached DIAGRAM SHEET. 3.1 Complete the copy of the cumulative frequency table on the diagram sheet. (2) 3.2 Draw an ogive (cumulative frequency graph) to represent the data on the grid provided on the diagram sheet. (2) 3.3 Show on your graph where you would read off the median (use the letter A) and write the value of the median down in the space provided. (2) 3.4 Use your ogive to determine the value of the Interquartile Range. Show all working. Do NOT answer this question on the diagram sheet. (3) Question 4 A business recorded the following figures as percentage profit made each month for a year: [9] Find the five number summary for this set of data and sketch a box-andwhisker plot. (5) 4.2 Explain what the box and whisker plot tells us about the spread of the data. (2) Page 5 of 17 November 2015

6 4.3 In the first month of the next year, the business records a percentage profit of p. When p is included in the figures, it changes the mean to 52. Calculate the value of p. (3) 4.4 One of the employees discovered that p was incorrect. He recalculated the actual percentage profit to be q. When added to the data, instead of p, as the 13 th data value, q did not alter the five number summary in any way. Write an inequality for all possible values of q. (2) Question 5 Anton wants to make a bedside lampshade. He first makes a cone with dimensions as shown in diagram A and cuts the top off. He covers the remaining piece with material. [12] Diagram A Diagram B 8cm 16cm Surface area of an open cone = πrs (s = slant height) 5.1 Find the surface area of the open cone in Diagram A. (Leave the answer in terms of π in the simplest form). (3) 5.2 Determine the amount of fabric needed to cover the exterior part of the lampshade in Diagram B. (Round the answer to two decimal places). (4) [7] Page 6 of 17 November 2015

7 Question The diagram below shows the graphs of f(x) = cos bx and g(x) = sin(x + c) for the interval 180 x 180. Use the graph to answer the following questions Determine the values of b and c. (2) Write down the period of graph f. (1) What is the maximum value of g? (1) For which value(s) of x will g increase as x increases? (2) Find the equation of h if h is the resulting graph of g if it is reflected about the x-axis and then shifted 40 to the right. (2) 6.2 Sketch the graph of y = cosx 1 for the interval x [ 180 ; 90 ] on the grid provided on the attached DIAGRAM SHEET. Clearly indicate intercepts with the axes, turning points and endpoints of the graph. (3) [11] Page 7 of 17 November 2015

8 Question If sin 25 = k, express tan 115 in terms of k. Show all calculations. (4) 7.2 Simplify the following without the use of a calculator: tan(360 A). sin(90 +A) cos 180 sin( 180 +A) (6) cos cos 120 cos ( ) (5) 7.3 Determine the general solution: sin 2x = cos(x 20 ) (5) 7.4 Consider the following identity: tan θ + cos θ = 1 sin θ 1 cos θ Prove the identity. (5) For which value(s) of θ is the above identity undefined? Show all steps. (3) [28] Page 8 of 17 November 2015

9 Question Use the following diagram to prove that in any ABC: Area ABC = 1 bc sin A 2 (answer this question on the DIAGRAM SHEET provided, making use of the diagram if necessary.) B c a A b C (4) 8.2 Refer to the diagram below to answer the questions that follow: x x In BEA, find BE in terms of α and x. (2) Show that BD = x sin α sin β. (1) Hence, or otherwise find the area of BED in terms of α, β, θ and x. (4) Calculate the area of BED if x = 2cm; α = 40 ; β = 37 and θ = 10. (1) Page 9 of 17 November 2015

10 8.3 The diagram below shows a tower at A and a building at C. Mr Smarty Pants needs to approximate the straight line distance between the tower at A and the building at C. He is, however, unable to measure that distance because of other buildings situated between the two. He stands at a point B, where he is able to observe both the tower at A and the building at C. From this point he measures that AB C = 18 and the angle of elevation of T from B is 37. At point C, he measures that AC B = 28. The height of the tower, AT, is 269 m. If B, A and T are in the same vertical plane and B, A and C are in the same horizontal plane: Show that the distance from B to A is approximately 357 m. (2) Determine the straight line distance from A to C (use AB = 357 m). (3) [17] Page 10 of 17 November 2015

11 Question 9 ANSWER THIS QUESTION ON THE DIAGRAM SHEET In the diagram below, O is the centre of the circle. Prove the theorem which states that: BO C = 2BA C. [6] Question 10 In the figure, O is the centre of the circle. NP is a tangent to the circle at M. NP//KL and MO L = 152 Giving reasons, find the size of each of the following angles: 10.1 K (2) 10.2 M 1 (2) 10.3 L 2 (5) [9] Page 11 of 17 November 2015

12 Question 11 In the diagram below, PQ is a tangent to circle SRQWT at Q. PRS is a straight line. RW cuts SQ and QT at K and L respectively. PS//QT, RS = TW, Q 2 = x and Q 1 = y 11.1 Give a reason why each of the following angles are equal to x: S (1) Q 4 (1) W 1 (1) 11.2 Prove that R 1 = L 3. (4) 11.3 Hence, or otherwise prove that PRKQ is a cyclic quadrilateral. (4) [11] Page 12 of 17 November 2015

13 Question 12 Two circles intersect at A and B. Chords AF and BF of the larger circle meet the smaller circle at D and C respectively. CD produced meets the larger circle at E. AE and EF are joined. EC//GH. GH is not necessarily a tangent. AB C = x. Prove that: 12.1 E 1 + E 2 = D 4 (4) 12.2 Prove that GH is indeed a tangent to the big circle at F. (4) [8] TOTAL = 150 Page 13 of 17 November 2015

14 Grade 11 Paper 2 ANSWER SHEET November 2015 Name: Teacher: Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Analyt geom Analyt geom (Cricle) Data Data TSA Trig graphs Trig Trig triangle formulae Circle geom Circle geom Circle geom Circle geom Cumulative frequency TOTAL Question 3 Weights (kg) in intervals Number of players 70 < x < x < x < x < x < x < x TOTAL 100 Cumulative frequency Ogive representing the weights of 100 World Cup rugby players Median = Weight (kg) Page 14 of 17 November 2015

15 Question Question In ABC prove that: area ABC = 1 bc sin A 2 (4) Page 15 of 17 November 2015

16 Question 9 Prove the theorem which states that: BO C = 2BA C (6) Question 10 Page 16 of 17 November 2015

17 Question 11 Question 12 Page 17 of 17 November 2015

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