MATHEMATICS: PAPER II Page 1 of 11 HILTON COLLEGE TRIAL EXAMINATION AUGUST 2013 MATHEMATICS: PAPER II GENERAL INSTRUCTIONS

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1 MATHEMATICS: PAPER II Page 1 of 11 HILTON COLLEGE TRIAL EXAMINATION AUGUST 01 Time: hours MATHEMATICS: PAPER II GENERAL INSTRUCTIONS 150 marks PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY. 1. This question paper consists of 11 pages. You are provided with a separate Answer Sheet (green) and a Formula Sheet (yellow). Please check that your paper is complete.. Read the questions carefully.. This question paper consists of 1 questions. Answer all questions. 4. Question 4 (b) and 4 (c) and must be answered on the Answer Sheet which must be handed in with your answer book. 5. Number your answers exactly as the questions are numbered. 6. You may use an approved non-programmable and non-graphical calculator, unless a specific question prohibits the use of a calculator. 7. Round off your answers to one decimal digit where necessary, unless otherwise stated. 8. All necessary working details must be shown. 9. It is in your own interest to write legibly and to present your work neatly. 10. Please note that the diagrams are NOT necessarily drawn to scale. Please do not turn over this page until you are asked to do so

2 MATHEMATICS: PAPER II Page of 11 QUESTION 1 SECTION A In the figure below A, B and C are points on the Cartesian plane. B(5;4) M A(-1;1) Ɵ C(8;-) (a) Determine the gradient of the line BC. () 4 mbc 5 8 (b) Determine Ɵ, the angle of inclination of line BC. () tanθ key angle 6.4 θ (c) Determine M the midpoint of the line joining A and B. () M ; 5 ; (d) Determine, to one decimal place, the length of AB. () ( 5 1) ( 4 1) units AB +

3 MATHEMATICS: PAPER II Page of 11 (e) Determine the equation of the line parallel to BC and passing through A () ( x ) y 1 1 y x 1 (f) Show that ABC is a right-angled triangle. () mab since mab mbc 1 AB BC so ABC is right angled at B (g) Determine the area of ABC. () 1 Area ABC b h units ( ) ( ) 17 marks

4 MATHEMATICS: PAPER II Page 4 of 11 QUESTION (a) Below is a picture of an Angry Bird as it moves from Point A to point A to point A y A A A x For this question: translations: ( x ; y) ( x a ; y b) + + may not be used. (i) Give a description in words of the transformation which maps A to A () The point has been rotated 90 o clockwise about the origin (ii) Give a rule for the transformation which maps A to A in the form: x ; y ; () ( ) ( ) ( x ; y) ( y ; x) (iii) Give a rule for the transformation which maps A to A () ( x ; y) ( x ; y)

5 MATHEMATICS: PAPER II Page 5 of 11 (b) y A A x (i) Describe, in words, the transformation which maps shape A to shape A () An enlargement through the origin by a scale factor of. (ii) If the area of Shape A is 147 units determine the area of shape A () Shape A will be 4 times bigger so will have an area of units 11 marks

6 MATHEMATICS: PAPER II Page 6 of 11 QUESTION (a) Given: sin α and 90 < α < With the aid of a sketch and without the use of a calculator, determine: 5 α (i) tanα () x 5 4 tanα 4 (ii) sin(90 0 α) () cosα 4 5 (iii) cosα () cos α

7 MATHEMATICS: PAPER II Page 7 of 11 (b) If sin 40 p, determine the following in terms of p (without the use of a calculator): (i) sin140 (1) sin 40 p (ii) cos( 40 ) () cos 40 1 sin 40 1 p (iii) sin 80 () sin 40 cos 40 p 1 p (c) Simplify the following (without the use of a calculator): sin150.tan 5 ( ) 0 0 sin 0.sin sin 0.tan 45 sin 0 sin 60 or (4) 0 (d) Determine the general solution to: tan( 10 ) + 1,077 0 tan 0 ( θ 10 ) + 1,077 0 tan ( θ 10 ) θ (4) key angle 1 tan θ k, k Z θ k

8 MATHEMATICS: PAPER II Page 8 of 11 (e) Below is a picture of the pyramid of The pyramid of king Khafra, at Giza. It is made out of a square base and 4 equilateral triangles. 0m D C A 0m F B 0m (i) Show that the height of the pyramid is 16.6m (4) AC ( Pythagoras) ( ) AF 16.6m mid pt of diagonal EF EF 16.6m ( Pythagoras) 1 (ii) Determine the volume of the pyramid if: ( ) Volume Area of base Height () 1 Volume m 8 marks

9 MATHEMATICS: PAPER II Page 9 of 11 QUESTION 4 79 teenagers have cellular phones purchased on pay-as-you-go packages. The table below reflects the amount of money, in rands, spent on airtime by these teenagers in a certain month. AMOUNT OF MONEY SPENT ON AIRTIME (IN RANDS) NUMBER OF TEENAGERS 0 x < x < x < x < x < x < 10 5 (a) Determine the estimated mean for the AMOUNT OF MONEY SPENT () mean 79 R44.4 (b) Complete the table on the ANSWER SHEET. () AIRTIME Freq (f) Cumulative frequency Points < ( 0;19 ) < 5 44 ( 40;44 ) < 1 56 ( 60;56 ) < ( 80;66 ) < 8 74 ( 100;74 ) < 5 79 ( 10;79 ) f 0 x 0 0 x x x x x 10 (c) Draw an ogive curve of the cumulative frequency vs amount of money spent on the ANSWER SHEET. (4) (d) Use your graph to estimate the Inter Quartile Range. () Q 1 and Q 67 1 so, the inter - quartile range R46 1 marks

10 MATHEMATICS: PAPER II Page 10 of 11 QUESTION 5 The graph below shows the dot plots for two different sets of 5 numbers. The sets of numbers have different means and different standard deviations. Data set A: Data set B: (a) Which of the data sets (A or B) has the smallest variance? () A (b) Which of the data sets (A or B) has the smallest mean? () A (c) Comment on the skew-ness of date set A () Negatively skewed or skewed to the left 6 marks SECTION A TOTAL 75 marks

11 MATHEMATICS: PAPER II Page 11 of 11 QUESTION 6 SECTION B (a) The mean price of 1 different pieces of jewelry is R After another piece of jewelry was added the mean price increased to R Determine the value of the added piece of jewelry? () the total value of the original items was let value of added item x x So, x x x R4060 (b) An investor has the following record of profits and losses on 10 different investments. R1 000 R6 000 R R 000 R1 000 R R9 000 R5 000 R1 000 R (i) Determine the mean profit of the investor. (1) R (ii) Determine the standard deviation of the investor. () R (iii) Comment if you think the investor is successful or not. () I would say that he is successful as he has averaged a positive amount. However, he is not consistent as the standard deviation is high. 9 marks

12 MATHEMATICS: PAPER II Page 1 of 11 QUESTION 7 (a) Determine the order of rotational symmetry for the union jack flag above. () 4 y A(5;9) x (b) Determine the angle of rotational symmetry for the shape above. () 60 (c) Determine, in surd form, the co-ordinate of A the image of A after a rotation of 0 o clock-wise. (Calculators may not be used.) (5) ( x ; y) ( xcosθ + ysin θ ; ycosθ xsin θ) ( ) 5;9 (5cos0 + 9sin0 ; 9cos0 5sin0 ) A' ; 9 marks

13 MATHEMATICS: PAPER II Page 1 of 11 QUESTION 8 (a) Prove the following identity: 1+ sin x cos x + sin x cos x cos x sin x 1+ sin x LHS cos x sin x + cos x + sin x cos x cos x sin x ( sin x + cos x) ( cos x sin x)( cos x + sin x) sin x + cos x cos x sin x RHS (5) (b) Hence, or otherwise, calculate the following without the use of a calculator, giving your answer in simplified surd form: cos15 + sin15 cos15 sin15 () 1+ sin 0 cos0 (c) Solve the following for the interval x ϵ ( 90 ;180 ) (5) cos(θ 0 ) cos(40 θ) ( ) θ 0 40 θ + 60k or θ 0 40 θ + 60k θ k or θ k θ 0 or 140 or 0 1 marks

14 MATHEMATICS: PAPER II Page 14 of 11 QUESTION 9 Petronas Twin Towers are the fifth highest building in the world, reaching a height of 45m. Two persons standing at A and B look up to the top (T) and to the connecting bridge (M) respectively. Angle of elevation from A to T is 5 o T Angle of elevation from B to M is 9 o Angle CAB 50 o Angle ACB 100 o M 45m A (a) Show that AC 969,m. () TC tan 5 AC TC 45 AC tan 5 tan 5 AC 969.m C 100 o 5 o 9 o 50 o (b) Determine the height above the ground of the connecting bridge at point M. (Round off your answer to the nearest meter. ) (6) B BC 969. sin 50 sin sin 50 BC sin 0 MC tan 9 BC MC BC tan 9 MC 5m 8 marks

15 MATHEMATICS: PAPER II Page 15 of 11 QUESTION 10 (a) Determine the equation of the following graphs: (i) () y cos x + 1 (ii) () x y tan

16 MATHEMATICS: PAPER II Page 16 of 11 (b) Given below are the graphs f ( x) sin x & g( x) cos x for the domain xϵ[-90;60] (i) For which x values will: f (x) > g (x) () 10 < x < 0 (ii) For which x values will: f ( x) g ( x ) > 0 () 90 < x < 0 or 90 < x < 180 or 70 < x < 60 9 marks

17 MATHEMATICS: PAPER II Page 17 of 11 QUESTION 11 A proposed way to cut budget is to get sheep to eat the grass on the sports fields. To prevent them from wandering on the roads they will be tied to poles resulting in them being able to eat a circle of grass. The plan is to put poles at A, B and C to cover the sports field as in the diagram below. A is the point (0 ; 0) C C is the point (80 ; 55) and circle center C has a radius of 4m A The equation of circle center B is: ( ) x 70 + y 0y b B (a) If circle center A touches the x-axis determine the equation of the circle. () ( x ) ( y )

18 MATHEMATICS: PAPER II Page 18 of 11 (b) If circle center B goes through point C. Determine, to the nearest meter, the length of the rope tied to the sheep at B (5) ( ) x 70 + y 0y b ( ) ( ) ( 70;10) x 70 + y 10 b B ( 80 70) ( 55 10) 46m rope + (c) Prove that there is no area on the field where all sheep can eat. (5) 65 ( 0 80) ( 55 0) AC + but radius of circle C + radius of cirlce A so, sheep A and C cannot graze the same area so there is no area grazed by all sheep (d) Can you list 1 problem with the proposal () Some areas are not grazed at all OR Sheep dung on the field 14 marks

19 MATHEMATICS: PAPER II Page 19 of 11 QUESTION 1 Daniel Craig staring as James Bond needs to run between two cameras for a scene in the next James Bond movie. The two cameras are positioned at A and B. And Daniel needs to run on a straight line in such a way that he is always the same distance from each camera. If A(10 ; 0) and B(5 ; -5) are the locations of the cameras, then determine the equation of the line Daniel Craig needs to run. He must run on the perpendicular bisector 45 gradient of AB 15 perpendicular line has a slope of 5 5 mid - point of AB ; so y + x 1 50 y x marks

20 MATHEMATICS: PAPER II Page 0 of 11 QUESTION 1 Mr Lombard s son asked him to cut up his toast into 5 parts with equal area. So Mr Lombard decided to cut it as in the picture below. C A E B D Note: A is a reflection of B and C is a reflection of D Mr Lombard needed to find out where to cut the bread to make the area the same. So he assumed the bread to be perfectly rectangular and put it on a Cartesian plane y 4 6 x Mr Lombard worked out that he needed to cut: From point (0 ; 0) to point (0 ; 8) From point (0 ; 0) to point (4,8 ; 8) Work out the other cuts that Mr. Lombard needs to make in order to grant his son s request.

21 MATHEMATICS: PAPER II Page 1 of 11 Area of toast cm so each piece must have an area of 8.4 Piece D is made up of triangles and a rec tan gle D + D + D ( 1. 8) + 6 h 8.4 where h is height of D h 9.6 h. so, by symmetry, the other cuts are : ( 0;0) to ( 6;.) ( 0;0) to ( 6;.) ( 0;0) to ( 4.8; 8) D 1 D D 7 marks SECTION B TOTAL 75 marks GRAND TOTAL 150 marks

22 MATHEMATICS: PAPER II Page of 11 MATHEMATICS INFORMATION SHEET x b ± b a 4ac n i 1 1 n i T a ( n 1) d n n n( n + 1) i 1 + S [ a + ( n 1) d ] n n Tn ar n 1 n a( r 1) Sn ; r 1 r 1 a S ; 1 r 1 1 r < < f ( x) lim h 0 f ( x + h) f ( x) h P( 1 ni) A P( 1 ni) A + P ( 1 i) n A P ( 1 i) n A + F x n ( 1 + i) 1 1 ( 1 + i) i P x i n x1 + x y1 + y d ( x x1 ) + ( y y1 ) M ; y m x + c y y m( x ) y 1 m m tan θ x y x 1 1 x1 ( x a) + ( y b) r

23 MATHEMATICS: PAPER II Page of 11 In ABC : a sin A b sin B c sin C a b + c b c. cos A area ABC 1 a b. sin C sin ( α + β) sin α.cosβ + cos α. sin β sin ( α β) sin α.cosβ cos α. sin β cos ( α + β) cosα.cosβ sin α. sin β cos ( α β) cos α.cosβ + sin α. sin β cos α cos α sin 1 sin α cos α 1 α sin α sin α. cosα ( x ; y) ( xcosθ + ysin θ ; ycosθ xsin θ) ( x ; y) ( xcosθ ysin θ ; ycosθ + xsinθ) x fx n σ n ( x x) i 1 i n n ( A) P ( A) P ( A or B) P ( A) + P ( B) P ( A and B) n ( S) ŷ a + bx b ( x x) ( y y) ( x x)

24 MATHEMATICS: PAPER II Page 4 of 11 STUDENT NUMBER 4. (b) 4. (c) M E M O R A N D U M AIRTIME Frequency (f) Cumulative frequency Points to plot < ( 0;19 ) < 5 44 ( 40;44 ) < 1 56 ( 60;56 ) < ( 80;66 ) < 8 74 ( 100;74 ) < 5 79 ( 10;79 ) f 0 x 0 0 x x x x x 10