Mathematics Grade 11 P2 Eastern Cape November 2016 QP (Eng)

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE GRADE 11 NOVEMBER 2016 MATHEMATICS P2 MARKS: 150 TIME: 3 hours This question paper consists of 14 pages and a special answer book.

2 Downloaded from hlayiso.com MATHEMATICS P2 (EC/NOVEMBER 2016) INSTRUCTIONS AND INFORMATION 1. This question paper consists of 10 questions. 2. Answer ALL the questions in the SPECIAL ANSWER BOOK provided. 3. Clearly show ALL calculations, diagrams graphs, et cetera which you have used in determining the answers. 4. Answers only will NOT necessarily be awarded full marks. 5. If necessary round off your answers to TWO decimal places, unless stated otherwise. 6. Diagrams are not necessarily drawn to scale. 7. You may use an approved scientific calculator (non-programmable and non- graphical) unless stated otherwise. 8. Write neatly and legibly. Copyright reserved Please turn over

3 Downloaded from hlayiso.com MATHEMATICS P2 (EC/NOVEMBER 2016) QUESTION 1 Mr Ngwane is the sales manager for a furniture shop. Every month his 15 staff members report on the number of customers who visited during the previous month. The results were given as follows: 12 15 15 19 22 23 26 26 32 33 33 33 33 35 35 1.1 Determine the: 1.1.1 median of the data (1) 1.1.2 interquartile range (3) 1.1.3 mean of the data (2) 1.1.4 standard deviation of the data. (2) 1.2 Determine the percentage of customers who visited the furniture shop that are outside one standard deviation of the mean. (3) [11] Copyright reserved Please turn over

4 Downloaded from hlayiso.com MATHEMATICS P2 (EC/NOVEMBER 2016) QUESTION 2 A group of learners wrote a standardised English test that was scored out of 60. The results were represented in a cumulative frequency graph below. 2.1 How many learners wrote the test? (1) 2.2 How many learners scored at least 20 out of 60? (2) 2.3 Using the graph, estimate the median test score. (2) Copyright reserved Please turn over

5 Downloaded from hlayiso.com MATHEMATICS P2 (EC/NOVEMBER 2016) 2.4 Complete the frequency table below using the SPECIAL ANSWER BOOK provided. Marks obtained Frequency 0<x≤10 10<x≤20 20<x≤30 30<x≤40 40<x≤50 50<x≤60 (5) 2.5 Write down the modal group. (1) [11] Copyright reserved Please turn over

6 Downloaded from hlayiso.com MATHEMATICS P2 (EC/NOVEMBER 2016) QUESTION 3 In the diagram below, ∆ PQS is drawn with vertices P (-2 ; 3), Q (3 ; 6) and S in a Cartesian plane. Line QS passes through the origin at O. PS║QR. PŜQ  20,230 . 3.1 Calculate the gradient of QS. (2) 3.2 Calculate the size of  . (2) 3.3 Determine the: 3.3.1 gradient of PS, round off to the nearest number. (4) 3.3.2 equation of PS (3) 3.4 If it is further given that the equation of QS is y  2 x , determine the coordinates of S. (4) 3.5 Calculate the length of QS, in simplified surd form. (3) Copyright reserved Please turn over

7 Downloaded from hlayiso.com MATHEMATICS P2 (EC/NOVEMBER 2016) 3.6 Calculate the area of ∆PQS, rounded off to two decimal digits. (5) 3.7 If it is further given that PQRS is a parallelogram, determine the coordinates of R. (3) 3.8 A(5 ; 4), B(0 ; -1) and C(t ; 2) are collinear points, determine the value of t. (4) [30] Copyright reserved Please turn over

8 Downloaded from hlayiso.com MATHEMATICS P2 (EC/NOVEMBER 2016) QUESTION 4 4.1 Given: p. sin   4  0 and p. cos   3  0 where p  0 4.1.1  Explain why   90 0 ;180 0 .  (3) 4 4.1.2 Show that: tan    3 (2) 4.1.3 Determine the numerical value of p. (2) 4.2 Simplify without using a calculator: sin 1430. cos 127 0  sin 530. cos 37 0 (5)  1  4.3 Prove the identity:  tan y     1  cos 2 y  tan y  tan y  (6) 1 5 4.4 Determine the general solution of: cos    cos  6 (6) [24] Copyright reserved Please turn over

9 Downloaded from hlayiso.com MATHEMATICS P2 (EC/NOVEMBER 2016) QUESTION 5 5.1 On the same set of axes draw f ( x)   cos(450  x) and g ( x)  tan( x) in the interval of  90 0  x  180 0 . Make use of the grid provided in the SPECIAL ANSWER BOOK. 5.2 For which values of x is f ( x)  g ( x)  0 for x ∈ [−90° ; 90°] (2) 5.3 Write down the equation of h(x) if h( x)   f ( x  450 ) . (2) [10] Copyright reserved Please turn over

10 Downloaded from hlayiso.com MATHEMATICS P2 (EC/NOVEMBER 2016) QUESTION 6 6.1 Complete:   a 2  b 2  2ab cos 𝐶̂ (1) 6.2 TRS is a secant of the circle, and SU is a tangent at U. TU = 16 cm, TR = 12 cm and T̂  410 . Calculate the: 6.2.1 Length of UR, correct to two decimals. (3) 6.2.2 Size of Û 2 (3) 6.2.3 Length of secant TRS (5) [12] Copyright reserved Please turn over

11 Downloaded from hlayiso.com MATHEMATICS P2 (EC/NOVEMBER 2016) QUESTION 7 Surface Area  2πrh Surface Area  4r 2 1 Volume  lbh Volume  lbh 3 The picture below shows a storage tank in which a farmer stores his grain. The tank is made up of a right cylinder with a hemisphere on top. The perpendicular height of the tank to the top is 7 500 cm and the radius of the tank is 10 m. 7 500 cm 10 m 7.1 Calculate the total surface area of the tank. (5) 7.2 Calculate the volume of the tank (5) [10] Copyright reserved Please turn over

12 Downloaded from hlayiso.com MATHEMATICS P2 (EC/NOVEMBER 2016) Give reasons for ALL statements in QUESTIONS 8, 9 and 10. QUESTION 8 In the diagram below O is the centre of the circle. Quadrilateral ABCO is drawn with A and C on the circumference. AB is a tangent to the circle at point A and BC is a tangent to circle at point C. D is the midpoint of chord AC and AD = DC. E is the point on the circumference of the circle with centre O. The length of: AC = 8 cm, OD = 3 cm, BE = x and AB = x + 4. 8.1 Write down the size of D̂1 with a reason. (2) 8.2 Calculate the length of OA. (2) 8.3 Determine the size of  and give a reason for your answer. (2) 8.4 Calculate the value of x if x  1 . (4) [10] Copyright reserved Please turn over

13 Downloaded from hlayiso.com MATHEMATICS P2 (EC/NOVEMBER 2016) QUESTION 9 In the diagram ∆LNR is drawn with points L and R on the circumference of a circle. NR cuts the circle at M. K is a point on LR with KN the bisector of LN̂R . KN cuts the chord LM at O. LK = LO. LÔK  650 and KN̂M  25 0 . 9.1 Prove that LN is a diameter of a circle passing through L, M and N. (5) 9.2 Determine the size of L̂ . (5) 9.3 Show that LN is a tangent to circle LMR. (3) 9.4 Is LR a diameter to circle LMR? Motivate your answer. (2) [15] Copyright reserved Please turn over

14 Downloaded from hlayiso.com MATHEMATICS P2 (EC/NOVEMBER 2016) QUESTION 10 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. 10.1 Name FOUR other angles each equal to x with reasons. (8) 10.2 Determine the size of Ŵ2 in terms of x with reasons. (4) 10.3 Prove that WS = WV. (5) [17] TOTAL: 150 Copyright reserved Please turn over

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