Grade 11 November Examination 2016 Mathematics: Paper 2 Time: 3 hours Marks: 150

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Grade November Examination 06 Mathematics: Paper Time: 3 hours Marks: 50 Instructions and Information: Read the following instructions carefully before answering the questions.

1 Grade November Examination 06 Mathematics: Paper Time: 3 hours Marks: 50 Instructions and Information: Read the following instructions carefully before answering the questions.. This question paper consists of 3 questions. Answer ALL the questions.. 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.; 5.3 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 of 7 November 06

2 Question The marks achieved by 0 Grade Eleven learners in their June Mathematics examinations are shown in the following table. Mark frequency Cumulative frequency 0 x < x < x < x < x < x < x < Draw an ogive (cumulative frequency curve) using the above results on grid provided on the DIAGRAM SHEET. (3). The Mathematics teacher believes that the median mark achieved by the grade will increase by 0 marks in the November examinations. What must the median mark in November be to achieve this increase? Show the working you used to determine your answer. ().3 How many students were above the 80 th percentile in the June examinations? Show working. () [7] Page of 7 November 06

3 Question The table below shows the distances (in kilometres) travelled by car daily, by a sales representative for 4 working days Calculate the mean distance travelled by the sales representative. (). Calculate the standard deviation for the distances travelled. ().3 Draw a box-and-whisker diagram for the distances travelled. (5).4 Comment on the skewness of this data. ().5 The sales representative discovered that her odometer was faulty. The actual reading on each of the 4 days was q km more than that which was indicated. Write down, in terms of q (if applicable), the:.5. actual mean. ().5. actual standard deviation. () [] Page 3 of 7 November 06

4 Question 3 In the diagram below, P ( 9; ), Q (9; 9) and R ( 3; 9) are the vertices of PQR. M is the midpoint of QR. P ( - 9 ; ) Y y Q ( 9 ; 9 ) N ( a ; b ) O θ M(3 ; 0) X x R ( - 3 ; - 9 ) 3. Calculate the gradient of QR. () 3. Calculate θ, the angle of inclination of QR with the positive x-axis. () 3.3 Determine the equation of median of PQR from point P. (3) 3.4 Determine the co-ordinates of S if PQSR is a parallelogram. () 3.5 N (a; b) is a point in the second quadrant (Note: N is NOT drawn to scale) Write down an expression for the length of NQ in terms of a and b. () 3.5. Hence determine the co-ordinates of N (ie. The values of a and b) if P, N and M are collinear and NQ = 5 5. (6) [6] Page 4 of 7 November 06

5 Question 4 4. The diagram below shows two circles touching at point E( ; ). The smaller circle has its centre at the origin and a radius of 5. The bigger circle s centre is at D(a ; 3). The length of the radius of the larger circle is twice the length of the radius of the smaller circle. AB is a common tangent to the circles at E. y E ( ; -) x (a ; -3) 4.. Show that a = 6. (5) 4.. Determine the equation of the common tangent which passes through E. Show all calculations. (4) 4. A circle with equation x 4x + y + 8y + k = 0 has a diameter of length 4 units. 4.. Calculate the coordinates of the centre of the circle. (3) 4.. Calculate the value of k. (3) [5] Page 5 of 7 November 06

6 Question 5 Sketched below is the graph: f(x) = cos x for x [ 90 ; 70 ]. 5. What is the range of f? () 5. What is the period of f? () 5.3 Sketch the graph of g if g(x) = sin(x 60 ) on the grid provided on the DIAGRAM SHEET, for x [ 90 ; 70 ]. Clearly label all intercepts with the axes, turning points and the end points. (4) 5.4 For which values of x for x [ 90 ; 70 ] is f(x). g(x) 0? (4) [] Page 6 of 7 November 06

7 Question 6 6. If 3 sin α 5 = 0 and cos α < 0, find WITHOUT using a CALCULATOR and with the aid of a suitable sketch, the value of: 4 tan α + 6 cos α. (4) 6. Simplify without a calculator: sin 0.cos 50 cos 35.sin 35 (6) 6.3 Consider the following identity: sin (80 +θ)+cos( θ).sin(90 +θ) sin θ + tan θ = +cos θ sin θ 6.3. Prove the identity. (6) 6.3. For which values of θ is the above identity undefined? () 6.4 Determine the general solution of the following: sin( θ + 57 ) = 0,8 (4) 6.5 Given: sin n θ cos n θ tan n θ = cos n θ 6.5. Prove the above identity. (Ignore any restrictions) (3) 6.5. Hence calculate, correct to 4 decimal places: sin 06 8 cos 06 8 tan 06 8 tan 03 8 sin 03 8 cos 03 8 (3) [8] Page 7 of 7 November 06

8 Question 7 Z 7. In the diagram alongside, XYZ is drawn with T on XY. X = 5, 8 YZ = 8 units, TX = 4 units, and TY = 3 units. X 5 4 T 3 Y 7.. Calculate, showing your workings and giving reasons where necessary, the size of Ŷ, correct to the nearest degree. (4) 7.. If Ŷ = 84, determine the length of TZ. (3) 7. In the diagram below, Q is the foot of a vertical cell phone tower PQ. Q, R and S are points on the same horizontal plane. The angle of elevation of P from R is x. SQ R = y and SQ = t metres. P t y Q x S R If the area of RQS = A square metres, show that: PQ = A tan x t sin y (5) [] Page 8 of 7 November 06

9 Question 8 A necklace is made by using 0 wooden spheres and 0 wooden cylinders. The radii of the spheres and the cylinders are exactly the same and are 5mm. The height of each cylinder is 9mm. The wooden spheres and cylinders are to be painted. (Ignore the holes in the spheres and cylinders.) V = πr h S = πr + πrh V = 4 3 πr3 S = 4πr 8. Determine the volume of wood used to make the necklace. Leave your answer rounded off to decimal places. (3) 8. Determine the surface area that would need to be painted. Leave your answer in terms of π. (4) [7] Page 9 of 7 November 06

10 Question 9 ANSWER THIS QUESTION ON THE DIAGRAM SHEET In the diagram below, PRST is a cyclic quadrilateral of circle O. Use the copy of this diagram on the DIAGRAM SHEET, to prove the theorem which states that: P + S = 80. [5] Page 0 of 7 November 06

11 Question 0 In the diagram alongside, AB and DC are chords of a circle. E is a point on AB such that BCDE is a parallelogram. DE B = 08 and DA E = x Calculate, giving reasons, the value of x. [5] Question In the diagram alongside, ED BC. EB and FA are tangents to the circle. E F x B 3 4 EAD is a straight line. Let ˆB = x A S 3 C G D. Find, giving reasons, 5 other angles equal to x. (0). Show that FBSA is a cyclic quadrilateral. (5) [5] Page of 7 November 06

12 Question In the diagram below, O is the centre of circle MPQ. MQ is extended to R and PR is joined. MP = RP and QP = QR. R Q x x O M 3 P. Determine Ô in terms of x if R = x. (5). Prove that RP is a tangent to the circle. (3) [8] Page of 7 November 06

13 Question 3 AC is a tangent to the circle at B. O is the centre of the circle and KB is a diameter of the circle. R, K and D are points on the circumference of the circle. BE = ER and B = x. 3. Prove that DC KR. (5) 3. Prove that B 3 = x. (4) [9] TOTAL = 50 Page 3 of 7 November 06

14 Grade Paper DIAGRAM SHEET November 06 Name: Teacher: Q Q Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q0 Q Q Q3 Data Data Analyt geom Analyt geom (Cricle) Trig graphs Trig Trig triangle formulae Volume/ surface area Circle geom Circle geom Circle geom Circle geom Circle geom Cumulative frequency TOTAL Question. Ogive showing Grade Mathematics marks Marks Page 4 of 7 November 06

15 Question Question 9 (THEOREM) Page 5 of 7 November 06

16 Question 0 Question E F x B 3 4 A S 3 C G D Page 6 of 7 November 06

17 Question R Q x O M 3 P Question 3 Page 7 of 7 November 06

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

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