Name: Teacher: GRADE 11 EXAMINATION NOVEMBER 2016 MATHEMATICS PAPER 2 PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY

GRADE 11 EXAMINATION NOVEMBER 2016 MATHEMATICS PAPER 2 Time: 3 hours Examiners: Miss Eastes; Mrs Rixon 150 marks Moderator: Mrs. Thorne, Mrs. Dwyer PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY 1. Read the questions carefully. Answer all the questions on the Question Paper. 2. You may use an approved, non-programmable, and non-graphical calculator, unless otherwise stated. 3. Round off your answers to ONE DECIMAL PLACE, where necessary unless otherwise indicated. All the necessary working details must be clearly shown. 4. It is in your own interest to write legibly and to present your work neatly. 5. Diagrams are not drawn to scale. 6. Please note that there is an information sheet provided. 7. Give reasons for all geometry statements. Name: Teacher: Marking Grid (for Educators use only) Section A: Marks Earned Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Total Marks 14 10 7 7 27 3 5 5 Section B: Q9 Q10 Q11 Q12 Q13 Q14 Q15 Q16 Q17 TOTAL Marks Earned Total Marks 11 7 7 4 7 11 9 9 7 150

FORMULAE: Section A:

Question 1: [14] 1.1 A triangle has the following vertices: A(0 ; 2) B(4 ; 5) C(4 ; 4) 1.1.1 Find the equation of the line through B and C. (1) 1.1.2 Find the midpoint of AC. (2) 1.1.3 Find the gradient of AC. (2) 1.1.4 Find the equation of the perpendicular bisector of AC. (3) 1.2 P(5 ; t) ; (6 ; 10) ; R( 5 ; 12) are 3 points on the Cartesian Plane. Find the value of t if: 1.2.1 the points are collinear. (3) 1.2.2 the triangle PQR is a right angled triangle with Q = 90. (3) Question 2: [10]

2.1 If sin 37 = a, express each of the following in terms of a. 2.1.1 cos 53 (2) 2.1.2 cos 323 (3) 2.2 If 5 tan A = 12 and 90 < A < 360. Using a diagram calculate 3cos 2 A without using a calculator. (5)

Question 3 [7] The diagram below represents a semi-circle and a triangle. All working details should be shown. A B 19cm 21 11cm C 3.1 Determine the length of AB (2) 3.2 Determine the area of the shape. (5) Question 4 [7]

Given f(x) = a. sinx + q and g(x) = d. tan b. x 4.1 Give the values of a, q, d and b. (4) 4.2 For which values of x will f(x) > 0, given that x [180 ; 180 ] (2) 4.3 f(x) is reflected around the x axis. Give the new equation. (1) Question 5: [27]

5.1 8 Students wrote a Maths test and their results are listed below: 35% 26% 49% 58% 57% 87% 71% 51% 5.1.1 Determine the mean (2) 5.1.2 Determine the standard deviation (2) 5.1.3 What percentage of test results lie within one standard deviation of the mean? (4) 5.1.4 The teacher found that the marks were too low. He added 10 to each mark. Write down the mean and the standard deviation of the new set of scores. (2) 5.1.5 Identify if there are any possible outliers in the set of data if applicable. (4) UF = Q 3 + 3 2 (Q 3 Q 1 ) and LF = Q 1 3 2 (Q 3 Q 1 )

5.2 The following frequency table shows the distribution of the marks of 200 students in a Mathematics Test. Mathematics Mark Frequency Cumulative frequency 1-10 20 11 20 40 21 30 a 120 31 40 50 41 50 b 51 60 10 5.2.1 Complete the cumulative frequency table above and subsequently, determine the values of a and b. (3) 5.2.2 Draw the ogive on the grid provided below. (5)

5.2.3 Estimate the interquartile range from your graph. (2) 5.2.4 Determine the cut-off mark for the top 20% of students for this test from your graph. (3) Question 6: [3] In the figure, AB is a tangent at T on the circle. P, Q and R are points on the circle. PR and TQ are drawn to intersect at M. PQT ˆ 25 and QMR ˆ 110. Determine, with reasons, the values of the angles marked with a, b and c. T A c P M a Q b B R

Question 7: [5] In the diagram, AB is the diameter of circle ABCD. A 2 = x. Express, with reasons, ˆD in terms of x. C 2 1 D B x 2 1 A Question 8: [5] In the diagram, AB is a chord of a circle with centre O. OMP AB, AB = 14, MP = 3 and OM = x Determine, with reasons, the length of the radius of the circle. A O x M 3 P B

Section B: Question 9: [11] 9.1 Prove: (tan 2 θ sin 2 θ)( cos2 θ+sin 2 θ sin 2 ) θ sin 2 θ cos 2 θ = 1 (5) 9.2 For which values of x will the identity in 9.1 be undefined? (6)

Question 10 [7] Without the use of a calculator, solve for α in the following: sin 2 α = 3 sinα. cos α and 90 < α < 180

Question 11 [7] Use the diagram below and answer the following questions giving reasons where necessary. 11.1 Determine R in terms of α and β. (2) 11.2 Prove that: PS = d sin(α+ β) cos θ sin α (5)

Question 12 [4] A pentahedron (a solid shape with five faces) is in the form of a pyramid on a square base. The height is twice as long as one side of its base. Its Volume is V = 446 3 cm3 Determine the dimensions of the square base. Volume of pyramid = 1 area of the base Height 3

Question 13: [7] Given that (k 3)y = (k + 2)x + 8 is a straight line. 13.1 Find the value of k if the line is horizontal. (3) 13.2 Find the value of k if the line has an inclination of 135. (4)

Question 14: [11] A(1 ; 6) B(3 ; 0) C(12 ; 3) and D are the vertices of a trapezium with AD BC. The angle of inclination of the straight line BC is θ, as shown in the diagram below. A straight line FH passing through vertex A does not pass through any of the sides of the trapezium. This line makes an angle of 45 with the side AD of the trapezium. H x D 45 0 C(12 ; 3) A(1 ; 6) P F B(3 ; 0) θ G y 14.1 Determine the gradient of the straight line FAH. Use reasons where necessary. (6)

14.2 Determine the co-ordinates of D such that ABCD forms a rectangle. (5)

Question 15: [9] In the figure ABCD is a cyclic quadrilateral and FAS is a tangent to the circle at A. C x and D1 y. 1 F A 1 2 3 4 S B 1 2 x 1 2 C 15.1 Giving reasons express F E and in terms of x and y. (4) 2 1 y D 15.2 Express Â2 in terms of x and y (2) 15.3 Calculate the value of y if AC is a tangent to circle AFB. (3)

QUESTION 16 [9] In the figure DC is a diameter of centre O. CE is a tangent to the circle at C. The diagonals of cyclic quadrilateral ABCD intersect at F. C 3 1 2 O E F 2 1 3 B D 1 2 1 A 2 3 2 1 P 16.1 Prove FAPB is a cyclic quadrilateral. (4) 16.2 If ACE x, name three other angles equal to x. (3) 16.3 What can you deduce about FP and CE? (2)

Question 17: [7] Consider the following equilateral triangle with sides of length 10 units. AE is the perpendicular height of the triangle. E Determine the coordinates of the points B and C. Give reasons where necessary.