NATIONAL SENIOR CERTIFICATE GRADE 12 MATHEMATICS P2 SEPTEMBER 2014 MARKS: 150 TIME: 3 hours This question paper consists of 10 pages, 2 diagram sheets and 1 information sheet. Copyright reserved Please turn over Mathematics/P2 2 NSC NW/September 2014 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 that 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. Diagrams are NOT necessarily drawn to scale. 8. TWO diagram sheets for QUESTION 1.2, QUESTION 2.1, QUESTION 7.1 and QUESTION 9.1 are attached at the end of this question paper. Write your name on these diagram sheets in the spaces provided and insert the diagram sheets inside the back cover of your ANSWER BOOK. 9. An information sheet with formulae is included at the end of this question paper. 10. Number the answers correctly according to the numbering system used in this question paper. 11. Write neatly and legibly. Copyright reserved Please turn over Mathematics/P2 3 NSC NW/September 2014 QUESTION 1 The distance travelled (in kilometres) by a group of 9 learners from Thohoyandou District to Thengwe Secondary school every morning is given as follows: 24 34 50 66 76 82 88 94 1.1 Calculate (where necessary to ONE decimal digit): 100 1.1.1 the mean (1) 1.1.2 the median (1) 1.1.3 the interquartile range of the data (3) 1.2 Draw a box and whisker diagram on DIAGRAM SHEET 1 to represent the data. (3) 1.3 Identify any outliers in the data set. Motivate your answer. (2) [10] QUESTION 2 The table below compares the number of hours spent by 7 Mathematics learners and the learners’ performance in the test. Learners Hours Marks (%) 1 1 35 2 3 55 3 5 60 4 6 65 5 8 75 6 10 70 7 11 80 2.1 Represent the data in a scatter plot on DIAGRAM SHEET 1. (3) 2.2 Determine the equation of the line of best fit for the data. (3) 2.3 Hence, draw the line of best fit on the scatter plot. (2) 2.4 Determine the correlation coefficient of the data. Hence, comment about the strength of the relationship between the two variables. Copyright reserved (3) [11] Please turn over Mathematics/P2 4 NSC NW/September 2014 QUESTION 3 In the accompanying figure, K(x; y), L(−2; −1) and M(4; 3) are the vertices of triangle KLM. The equation of the side KL is y – 5x – 9 = 0 and that of KM is 5y + x – 19 = 0. N is the midpoint of LM. y K(x ; y) M(4 ; 3) x L( 3.1 Calculate the coordinates of N. (2) 3.2 Show that the coordinates of K are (−1 ; 4). (4) 3.3 Determine the equation of the line through K and N in the form y = mx + c (3) 3.4 Determine the gradient of the line LM. Hence, prove that KN is the perpendicular bisector of LM . (3) 3.5 If L, M and the point J(7; a) are collinear, calculate the value of a. 3.6 Determine the size of the angle of inclination between KL and the positive x-axis. (2) [17] Copyright reserved (3) Please turn over Mathematics/P2 5 NSC NW/September 2014 QUESTION 4 In the figure below, the origin O is the centre of the circle. P(x ; y) and Q(3 ; −4) are two points on the circle and POQ is a straight line. R is the point (k ; 1) and RQ is a tangent to the circle. T is an x-intercept of the circle. y P(x ; y) R(k ; 1) T O x Q(3 ; −4) Determine: 4.1 the equation of the circle. (3) 4.2 the length of TQ. (Leave your answer in simplified surd form.) (3) 4.3 the equation of OQ. (3) 4.4 the coordinates of P. (2) 4.5 the equation of the circle with centre P, that passes through (0; 0) in the form x2 + y2 … (3) 4.6 the equation of QR. (4) 4.7 the value of k. (3) [21] Copyright reserved Please turn over Mathematics/P2 6 NSC NW/September 2014 QUESTION 5 5.1 5 and 0 180 , use a sketch to determine the value of the 12 following expression (WITHOUT USING A CALCULATOR): 3 sin 2 cos If tan (4) 5.2 Simplify the following expression to ONE trigonometric ratio of P: sin (360 P) cos 270 sin (90 P) tan (180 P) tan (360 P) 5.3 Calculate the value(s) of x , x [ 90 ; 270] if sin x = cos 2x – 1 (6) (6) [16] QUESTION 6 The sketch represents the graphs of the following functions for x [180; 0] : 1 f ( x) sin 2 x and g ( x ) tan x . 2 Line segment ABC is perpendicular to the x-axis at C(135 ; 0), with A on f and B on g. D( 120; 3 ) 2 is the point where f and g intersect. 6.1 Calculate the length of AB. (3) 6.2 Write down the period of f. (1) 6.3 Without calculations, use the graph to write down the values of x for which f ( x) g ( x) in the given interval. Copyright reserved (5) [9] Please turn over Mathematics/P2 7 NSC NW/September 2014 QUESTION 7 7.1 In ∆ ABC, Â is obtuse. C A B Redraw the sketch in your ANSWER BOOK and prove that 7.2 a b sin A sin B (5) From A the angle of elevation to the top of a vertical tower CD is x and from a point B, d metres closer to the tower, the angle of elevation is y . D 1 h y C d B 7.2.1 Show that the height of the tower is given by h x A d . sin x sin y sin ( y x) 7.2.2 Calculate the height of the tower if d = 85 m, x 10 and y 38 . (5) (2) [12] QUESTION 8 8.1 Prove that sin (x + y) + sin (x – y) = 2 sin x . cos y . 8.2 Hence, or otherwise, prove that: sin 3 x sin x 2 sin x . 1 cos 2 x (2) (5) [7] Copyright reserved Please turn over Mathematics/P2 8 NSC NW/September 2014 NOTE: GIVE REASONS FOR YOUR STATEMENTS AND CALCULATIONS FROM QUESTIONS 9 TO QUESTION 11. QUESTION 9 9.1 In the diagram below, O is the centre of the circle and OS is perpendicular to the chord RT. O 1 2 R T S Prove, using Euclidian geometry methods, the theorem that states that RS = ST. 9.2 (5) The line ABCD intersects two concentric circles with centre O as shown in the diagram below. OA = 25 cm and OC = 17 cm. O A 1 2 B X C D 9.2.1 Determine AC if OX = 15 cm. (5) 9.2.2 Show that AB = CD. (4) [14] Copyright reserved Please turn over Mathematics/P2 9 NSC NW/September 2014 QUESTION 10 10.1 Complete the statements of the following theorems by writing down only the missing word(s) in each case. 10.1.1 The opposite angles of a cyclic quadrilateral are . . . (1) 10.1.2 If two triangles are equiangular, the corresponding sides are . . . (1) 10.2 PQR is a tangent to circle QABCD. AB ║ QD. CB = CD. Let Q3 30 and D3 70 . B C 2 1 70º 1 2 A 1 2 3 1 2 3 D 30º R Q P 10.2.1 Calculate Q1 . 10.2.2 Prove that C 110 10.2.3 Calculate B1 (2) (4) 10.3 (4) DOE is a diameter of a circle with centre O. DF and EF are chords of the circle. GH DE. F G. 12 D 1 2 H . O E Prove that G1 E . (4) [16] Copyright reserved Please turn over Mathematics/P2 10 NSC NW/September 2014 QUESTION 11 11.1 In the accompanying diagram, AD = 15 m, DB = 10 m and AF = 12 m. A 15m 12m D F 10m E B C If DE ║ BC and DF ║ BE, calculate the length of AC. 11.2 (8) PQRS is a rectangle. A is a point on QR such that PAˆ S 90 . Q P 1 2 A 3 1 2 S R Prove that: 11.2.1 PÂQ AŜR (3) 11.2.2 ΔAPQ /// ΔSAR (3) 11.2.3 If RS = 8 units, QA = x units and RA = y units, express y in terms of x. (3) [17] TOTAL: 150 Copyright reserved Please turn over Mathematics/P2 11 NSC NW/September 2014 DIAGRAM SHEET 1 NAME: QUESTION 1.2 QUESTION 2.1 Scatter Plot 100 90 80 Marks (%) 70 60 50 40 30 20 10 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Hours Copyright reserved Please turn over Mathematics/P2 12 NSC NW/September 2014 DIAGRAM SHEET 2 NAME: QUESTION 7.1 C A B QUESTION 9.1 O R Copyright reserved 1 2 S T Please turn over Mathematics/P2 13 NSC NW/September 2014 INFORMATION SHEET: MATHEMATICS b b 2 4ac x 2a A P(1 i) n A P(1 ni ) A P(1 ni ) Tn a (n 1)d Tn ar n 1 Sn n Sn n 2a (n 1)d 2 a r n 1 r 1 x 1 i 1 F i P f ( x h) f ( x ) h h 0 x x2 y1 y 2 M 1 ; 2 2 m y 2 y1 a b c sin A sin B sin C cos 2 sin 2 cos 2 1 2 sin 2 2 cos 2 1 cos cos . cos sin .sin sin 2 2 sin . cos fx x 2 n n( A) nS 2 x x i 1 i n P(A or B) = P(A) + P(B) – P(A and B) b 1 ab. sin C 2 sin sin . cos cos.sin n Copyright reserved m tan a 2 b 2 c 2 2bc. cos A area ABC cos cos . cos sin .sin yˆ a bx x 2 x1 r2 sin sin . cos cos.sin P( A) a ; 1 r 1; 1 r f ' ( x) lim y y1 m( x x1 ) x a 2 y b 2 In ABC: S r 1 ; x[1 (1 i)n ] i d ( x 2 x1 ) 2 ( y 2 y1 ) 2 y mx c A P(1 i) n x x ( y y ) (x x) 2