SUBJECT Mathematics PAPER 2 GRADE 11 DATE 21 NOV 2017 EXAMINER Mrs Sillman MARKS 150 NAME MODERATOR Gr 11 Teachers TEACHER DURATION 3 hours

Transcription

1 SUBJECT Mathematics PAPER 2 GRADE 11 DATE 21 NOV 2017 EXAMINER Mrs Sillman MARKS 150 NAME MODERATOR Gr 11 Teachers TEACHER DURATION 3 hours QUESTION NO DESCRIPTION MAXIMUM MARK ACTUAL MARK 1-3 Analytical Geometry 27 4 Euclidean Geometry 10 5 Data Handling Trigonometry D Trigonometry Data Handling Analytical Geometry 7 12 Trig Functions Measurement Euclidean Geometry 22 TOTAL 150

2 INSTRUCTIONS: 1. This paper consists of 15 questions and 25 pages (which includes the diagram sheet and formula sheet) 2. Read the questions carefully. 3. Answer all questions. 4. Number your answers clearly and use the same numbering as in the question paper. 5. You may use an approved non-programmable and non-graphical calculator, unless otherwise stated. 6. Round off your answers to two decimal digits where necessary. 7. All necessary working details must be shown. Answers only, without the relevant calculations will not necessarily be given marks. Equations may not be solved solely with a calculator. 8. It is in your interest to write legibly and present your work neatly. 9. You may detach the Information Sheet and Diagram sheet at the back of the question paper. Grade 11 2 of 25 Mathematics Paper 2

3 QUESTION 1 9 MARKS Given quadrilateral ABCD, with corners A(2; 2), B(4; 3), C(7; 4), D(5; 1). Diagram is NOT drawn to scale. a) Calculate the gradient of BC (2) b) Calculate the length of CD (2) c) Calculate the midpoint of diagonal AC (2) d) Prove that ABCD is a parallelogram (3) Grade 11 3 of 25 Mathematics Paper 2

4 QUESTION 2 8 MARKS a) A straight line has the equation y = 2x 3. i) Give the inclination of the line with the x axis. (2) ii) Give the equation of a line that is perpendicular to y = 2x 3 and passes through the point (6; 2). (3) b) A line has an angle of inclination of 60. What is the gradient of the line? (1) c) Determine the equation of a line perpendicular to the x axis passing through the point (4 ; 1) (2) Grade 11 4 of 25 Mathematics Paper 2

5 QUESTION 3 K (-3 ; 5), L (t ; 2) and M (1 ; -5) are points in the Cartesian plane. Calculate t if: 10 MARKS a) K, L and M are collinear (5) b) KL ML (5) Grade 11 5 of 25 Mathematics Paper 2

6 QUESTION 4 In the diagram, O is the centre of the circle. YI E = 50. AB is a tangent at Y. 10 MARKS Determine, with reasons the size of the following angles: a) Y b) E c) Y d) Y e) R (10) Grade 11 6 of 25 Mathematics Paper 2

7 QUESTION 5 7 MARKS The AP Maths marks of 23 boys are expressed in percentages in the following stem-and-leaf diagram: Determine the: a) median mark (1) b) modal mark (1) c) inter-quartile range of the marks (3) d) standard deviation of the marks to one decimal place (2) QUESTION 6 26 MARKS a) Solve for θ: 3tanθ = 3 if xε[0 ; 360 ] (3) Grade 11 7 of 25 Mathematics Paper 2

8 b) Simplify without the use of a calculator: 1) ( ). ( ). ( ) ( ). ( ) (7) 2) tan120. cos210 + (7) Grade 11 8 of 25 Mathematics Paper 2

9 c) Simplify to a single trigonometric ratio for θε[0 ; 90 ] (1 sinα)(1 + sinα) (3) d) Prove that:.. = tan x (6) Grade 11 9 of 25 Mathematics Paper 2

10 QUESTION 7 7 MARKS In the diagram below, P(x; 24) is a point such that OP = 25 and RO P = β, where β is an obtuse angle. a) Calculate the value of x (2) b) Determine the value of each of the following WITHOUT the use of a calculator: 1) sinβ (1) 2) cos (180 β) (2) 3) tan ( β) (2) Grade of 25 Mathematics Paper 2

11 QUESTION 8 The diagram represents a small crane lifting a light pole. AC = 1,5m AB = 2,2m BC = 2,9m Determine A and C correct to two decimal places. 6 MARKS Grade of 25 Mathematics Paper 2

12 QUESTION 9 Snow skiers pass a point on a ski slope at a speed in km/h 10 MARKS The speeds are in the table: Speed (km/h) Frequency (f) Cumulative frequency 0 x < x < x < x < x < a) Complete the table (2) b) Draw a cumulative frequency curve (ogive) representing the information on the next page (4) c) Indicate clearly on the cumulative frequency curve the lower quartile (Q ) and the upper quartile (Q ). Mark where these points would be read off with and A and B respectively. (2) d) Use your graph to estimate the number of skiers who passed the point with speeds greater than 35km/h (2) Grade of 25 Mathematics Paper 2

13 Grade of 25 Mathematics Paper 2

14 QUESTION 10 8 MARKS The following table shows the Term 2 marks for English and Maths for a random sample of 10 Grade 11 boys English (x) Maths (y) a) Calculate the correlation coefficient to 3 decimal places. Does this represent a strong or weak correlation? (2) b) Determine the equation of the least squares regression line. (3) c) Use your line to predict the Maths mark, to the nearest whole number, of a boy who scores 30% in English (1) d) Comment on the reliability of your prediction in (c) and give one reason for your answer (2) Grade of 25 Mathematics Paper 2

15 QUESTION 11 7 MARKS Determine the size of the acute angle between the lines 5y = x 5 and y = 5x 1. Show that the gradient of the bisector of this angle is 1. From this, determine the equation of the bisector. (Hint: a diagram may be of help) Grade of 25 Mathematics Paper 2

16 QUESTION 12 The sketch below shows the graphs of: f(x) = sin ax and g(x) = cos(x + θ) for xε[ 180 ; 180 ] 12 MARKS a) Determine the period of f(x) = sin ax (1) b) Write down the range of g(x) (1) c) Write down the values of θ and a (2) d) Determine the values of x, if xε[ 180 o ; 0 o ] for which: 1) f(x) g(x) > 0 (2) 2) g(x). f(x) 0 (3) e) Determine the minimum value of: 3 ( ) (3) Grade of 25 Mathematics Paper 2

17 QUESTION 13 8 MARKS A cylindrical aerosol can has a lid in the shape of a hemisphere that fits exactly on top of the can. The height of the can is 16cm and the radius of the base of the can is 2,9cm. a) Calculate the surface area of the can with the lid in place, as shown in Figure 1 (5) b) If the lid is 80% filled with liquid, as shown on Figure 2, calculate the volume of the liquid in the lid. (3) Grade of 25 Mathematics Paper 2

18 QUESTION 14 On the diagram below, AB is a tangent to the circle (centre O) at X. W, X, Y and Z are points on the circle. OY W = 35 8 MARKS a) Prove (with reasons) that XZ Y = 35 (3) Grade of 25 Mathematics Paper 2

19 b) If X = 30, calculate (with reasons) the value of Y (3) c) Hence, calculate (with reasons) the value of X (2) Grade of 25 Mathematics Paper 2

20 QUESTION MARKS O is the centre of the bigger circle. O lies on the circumference of the smaller circle. OWVU are points on the circumference of the smaller circle. TSVU are points on the circumference of the bigger circle a) Determine O and S in terms of x (2) Grade of 25 Mathematics Paper 2

21 b) Name THREE angles equal to x, with reasons. (4) c) Determine the size of W in terms of x, with reasons (3) d) Prove that WS = WV (5) Grade of 25 Mathematics Paper 2

22 EXTRA PAPER FOR WORKING OR WHERE MORE SPACE WAS REQUIRED IF ANYTHING ON THIS PAGE NEEDS TO BE MARKED, LABEL IT CLEARLY WITH THE QUESTION NUMBER AND ALSO STATE THAT YOU HAVE WORKED HERE ON THE QUESTION ITSELF Grade of 25 Mathematics Paper 2

23 DIAGRAM SHEET QUESTION 12 QUESTION 14 Grade of 25 Mathematics Paper 2

24 QUESTION 15 Grade of 25 Mathematics Paper 2

25 FORMULA SHEET Grade of 25 Mathematics Paper 2