Mathematics Grade 11 P2 National Exemplar November 2013 QP

Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE GRADE 11 MATHEMATICS P2 EXEMPLAR 2013 MARKS: 150 TIME: 3 hours This question paper consists of 12 pages and 3 diagram sheets. Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 2 DBE/2013 NSC – Grade 11 Exemplar INSTRUCTIONS AND INFORMATION Read the following instructions carefully before answering the questions. 1. This question paper consists of 11 questions. 2. Answer ALL the questions. 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. You may use an approved scientific calculator (non-programmable and non- graphical), unless stated otherwise. 6. If necessary, round off answers to TWO decimal places, unless stated otherwise. 7. THREE diagram sheets for QUESTION 2.1, QUESTION 2.2, QUESTION 9.2, QUESTION 10.1, QUESTION 10.2 and QUESTION 11.2 are attached at the end of this question paper. Write your name on these diagram sheets in the spaces provided and insert them inside the back cover of your ANSWER BOOK. 8. Number the answers correctly according to the numbering system used in this question paper. 9. Write neatly and legibly. Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 3 DBE/2013 NSC – Grade 11 Exemplar QUESTION 1 The data below shows the number of people visiting a local clinic per day to be vaccinated against measles. 5 12 19 29 35 23 15 33 37 21 26 18 23 18 13 21 18 22 20 1.1 Determine the mean of the given data. (2) 1.2 Calculate the standard deviation of the data. (2) 1.3 Determine the number of people vaccinated against measles that lies within ONE standard deviation of the mean. (2) 1.4 Determine the interquartile range for the data. (3) 1.5 Draw a box and whisker diagram to represent the data. (3) 1.6 Identify any outliers in the data set. Substantiate your answer. (2) [14] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 4 DBE/2013 NSC – Grade 11 Exemplar QUESTION 2 A group of Grade 11 learners were interviewed about using a certain application to send SMS messages. The number of SMS messages, m, sent by each learner was summarised in the histogram below. Histogram showing the number of SMS messages sent by learners 50 45 40 36 35 31 29 30 Frequency 26 25 20 15 15 14 10 7 5 2 0 0 2 4 6 8 10 12 14 16 18 Number of SMS messages (m) 2.1 Complete the cumulative frequency table provided in DIAGRAM SHEET 1. (2) 2.2 Use the grid provided in DIAGRAM SHEET 2 to draw an ogive (cumulative frequency curve) to represent the data. (3) 2.3 Use the ogive to identify the median for the data. (1) 2.4 Estimate the percentage of the learners who sent more than 11 messages using this application. (2) 2.5 In which direction is the data skewed? (1) [9] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 5 DBE/2013 NSC – Grade 11 Exemplar QUESTION 3 A(1 ; 6), B(3 ; 0), C(12 ; 3) and D are the vertices of a trapezium with AD || BC. E is the midpoint of BC. The angle of inclination of the straight line BC is θ, as shown in the diagram. y D A(1 ; 6) C(12 ; 3) E θ x B(3 ; 0) 3.1 Calculate the coordinates of E. (2) 3.2 Determine the gradient of the line BC. (2) 3.3 Calculate the magnitude of θ. (2) 3.4 Prove that AD is perpendicular to AB. (3) 3.5 A straight line passing through vertex A does not pass through any of the sides of the trapezium. This line makes an angle of 45° with side AD of the trapezium. Determine the equation of this straight line. (5) [14] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 6 DBE/2013 NSC – Grade 11 Exemplar QUESTION 4 In the diagram below, P(–3 ; 17), Q, O and S are the vertices of a parallelogram. The sides OS and OQ are defined by the equations y = 6 x and y = − x respectively. QÔS = α . y P(–3 ; 17) S Q α x O 4.1 Determine the equation of QP in the form y = mx + c . (3) 4.2 Hence, determine the coordinates of Q. (4) 4.3 Calculate the length of OQ. Leave your answer in simplified surd form. (2) 4.4 Calculate the size of α. (3) 4.5 If OS = 148 units, calculate the length of QS. (3) [15] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 7 DBE/2013 NSC – Grade 11 Exemplar QUESTION 5 5.1 In the figure below, the point P(–5 ; b) is plotted on the Cartesian plane. OP = 13 units and RÔP = α . y P(–5 ; b) • α 13 x • O R Without using a calculator, determine the value of the following: 5.1.1 cos α (1) 5.1.2 tan(180° − α ) (3) sin(θ − 360°) sin(90° − θ ) tan(−θ ) 5.2 Consider: cos(90° + θ ) sin(θ − 360°) sin(90° − θ ) tan(−θ ) 5.2.1 Simplify to a single trigonometric ratio. (5) cos(90° + θ ) 5.2.2 Hence, or otherwise, without using a calculator, solve for θ if 0° ≤ θ ≤ 360° : sin(θ − 360°) sin(90° − θ ) tan(− θ ) (3) = 0,5 cos(90° + θ ) 8 4 4 5.3 5.3.1 Prove that − = . (5) sin A 1 + cos A 1 − cos A 2 5.3.2 For which value(s) of A in the interval 0° ≤ A ≤ 360° is the identity in QUESTION 5.3.1 undefined? (3) 5.4 Determine the general solution of 8 cos 2 x − 2 cos x − 1 = 0 . (6) [26] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 8 DBE/2013 NSC – Grade 11 Exemplar QUESTION 6 In the diagram below, the graphs of f ( x) = cos( x + p ) and g ( x) = q sin x are shown for the interval − 180° ≤ x ≤ 180° . yy g 1 A 0,5 f xx -180° -135° -90° -45° 0° 45° 90° 135° 180° - 0,5 B -1 6.1 Determine the values of p and q. (2) 6.2 The graphs intersect at A(–22,5° ; 0,38) and B. Determine the coordinates of B. (2) 6.3 Determine the value(s) of x in the interval − 180° ≤ x ≤ 180° for which f ( x) − g ( x) < 0 . (2) 6.4 The graph f is shifted 30° to the left to obtain a new graph h. 6.4.1 Write down the equation of h in its simplest form. (2) 6.4.2 Write down the value of x for which h has a minimum in the interval − 180° ≤ x ≤ 180° . (1) [9] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 9 DBE/2013 NSC – Grade 11 Exemplar QUESTION 7 sin A sin C 7.1 Prove that in any acute-angled ∆ABC, = . (5) a c 7.2 In ∆PQR, P̂ = 132°, PQ = 27,2 cm and QR = 73,2 cm. P 132° 27,2 cm Q R 73,2 cm 7.2.1 Calculate the size of R̂ . (3) 7.2.2 Calculate the area of ∆PQR. (3) 7.3 In the figure below, SP̂Q = a , PQ̂S = b and PQ = h. PQ and SR are perpendicular to RQ. P a S h b R Q 7.3.1 Determine the distance SQ in terms of a, b and h. (3) h sin a cos b 7.3.2 Hence show that RS = . sin( a + b) (3) [17] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 10 DBE/2013 NSC – Grade 11 Exemplar QUESTION 8 A solid metallic hemisphere has a radius of 3 cm. It is made of metal A. To reduce its weight a conical hole is drilled into the hemisphere (as shown in the diagram) and it is completely filled 8 with a lighter metal B. The conical hole has a radius of 1,5 cm and a depth of cm. 9 Calculate the ratio of the volume of metal A to the volume of metal B. [6] QUESTION 9 9.1 Complete the statement so that it is valid: The line drawn from the centre of the circle perpendicular to the chord … (1) 9.2 In the diagram, O is the centre of the circle. The diameter DE is perpendicular to the chord PQ at C. DE = 20 cm and CE = 2 cm. P E C O Q D Calculate the length of the following with reasons: 9.2.1 OC (2) 9.2.2 PQ (4) [7] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 11 DBE/2013 NSC – Grade 11 Exemplar QUESTION 10 10.1 In the diagram, O is the centre of the circle and A, B and D are points on the circle. Use Euclidean geometry methods to prove the theorem which states that AÔB = 2AD̂B . D O A B (5) 10.2 In the diagram, M is the centre of the circle. A, B, C, K and T lie on the circle. AT produced and CK produced meet in N. Also NA = NC and B̂ = 38° . B 38° C M A 1 2 3 K 4 1 2 T N 10.2.1 Calculate, with reasons, the size of the following angles: (a) KM̂A (2) (b) T̂2 (2) (c) Ĉ (2) (d) K̂ 4 (2) 10.2.2 Show that NK = NT. (2) 10.2.3 Prove that AMKN is a cyclic quadrilateral. (3) [18] Copyright reserved Please turn over

Downloaded from hlayiso.com Mathematics/P2 12 DBE/2013 NSC – Grade 11 Exemplar QUESTION 11 11.1 Complete the following statement so that it is valid: The angle between a chord and a tangent at the point of contact is … (1) 11.2 In the diagram, EA is a tangent to circle ABCD at A. AC is a tangent to circle CDFG at C. CE and AG intersect in D. B C 1 2 3 3 2 A x1 5 4 3 2 1 2 1 D G 2 1 F y 2 1 E If  1 = x and Ê 1 = y, prove the following with reasons: 11.2.1 BCG || AE (5) 11.2.2 AE is a tangent to circle FED (5) 11.2.3 AB = AC (4) [15] TOTAL: 150 Copyright reserved

Downloaded from hlayiso.com Mathematics/P2 DBE/2013 NSC – Grade 11 Exemplar NAME OF LEARNER: DIAGRAM SHEET 1 QUESTION 2.1 CUMULATIVE CLASS FREQUENCY FREQUENCY 0≤m<2 2≤m<4 4≤m<6 6≤m<8 8 ≤ m < 10 10 ≤ m < 12 12 ≤ m < 14 14 ≤ m < 16

Downloaded from hlayiso.com Mathematics/P2 DBE/2013 NSC – Grade 11 Exemplar NAME OF LEARNER: DIAGRAM SHEET 2 QUESTION 2.2 160 150 140 130 120 110 100 Cumulative Frequency 90 80 70 60 50 40 30 20 10 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Number of SMS messages Copyright reserved

Downloaded from hlayiso.com Mathematics/P2 DBE/2013 NSC – Grade 11 Exemplar NAME OF LEARNER: DIAGRAM SHEET 3 QUESTION 9.2 QUESTION 10.1 P D E C O O Q A B D QUESTION 10.2 QUESTION 11.2 B B 38° C M C 1 A 2 1 2 3 3 3 2 A K 1 5 x 4 1 4 2 T 3 2 1 2 1 D G 2 1 N y F 2 1 E Copyright reserved