SAJC H2 MATHS P2

SAINT ANDREW’S JUNIOR COLLEGE Preliminary Examination MATHEMATICS Higher 2 9758/02 Wednesday 13 September 2017 3 hours Additional materials : Answer paper List of Formulae (MF 26) Cover Sheet READ THESE INSTRUCTIONS FIRST Write your name, civics group and index number on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use a soft pencil for any diagrams or graphs. Answer all the questions. Total marks : 100 Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. You are expected to use a graphic calculator. Unsupported answers from a graphic calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphic calculator are not allowed in a question, you are required to present the mathematical steps using mathematical notations and not calculator commands. You are reminded of the need for clear presentation in your answers. The number of marks is given in brackets [ ] at the end of each question or part question. At the end of the examination, fasten all your work securely together. This document consists of 6 printed pages including this page. [Turn over

2 [Turn Over] Section A: Pure Mathematics [40 marks] 1 Without the use of a calculator, find the complex numbers z and w which satisfy the simultaneous equations i3zw  2 63 i 0zw   [6] 2 (i) Show that 3 231 11 An B nn n n n    , where A and B are constants to be found. [2] (ii) Hence find 3 2 26n r r rr   . [3] (iii) Use your answer to part (ii) to find   2 21 0 123 n r r rr r    . [3] 3 The function f is defined by 2 1f : 1x x  , ,x 1x  . (i) Find 1f( ) x and write down the domain of 1f . [3] (ii) On the same diagram, sketch the graphs of f( )yx , 1f( )y x and 1ff ( )y x stating the equations of any asymptotes and showing the relationships between the graphs clearly. [4] (iii) State the set of values of x such that 11f f () f f ()x x  . [1] 4 Referred to the origin O, the point A has position vector −5i + 2j + 2k and the point B has position vector i + 3j − 2k. The plane  has equation: (1 2 ) (3 2 ) ( 2)     ri j k where ,  (i) Find the vector equation of plane  in scalar product form. [2] (ii) Find the position vector of th e foot of perpendicular, C, from A to . [3] The line l1 passes through the points A and B. (iii) The line l2 is the reflection of the line l1 about the plane . Find a vector equation of l2. [3]

3 [Turn Over] 5 It is given that DEFG is a square with fixed side 2 a cm and it is inscribed in the isosceles triangle ABC with height AH , where AB = AC and angle BAH = . (i) Taking tant  , show that the area of the triangle ABC is given by 2 144Sa t t   . [3] (ii) Find the minimum area of S in terms of a when t varies. [4] (iii) Hence sketch the graph showing the area of the triangle ABC as  varies. [3]                                        D G F E H C B A 2a  2a

4 [Turn Over] Section B: Statistics [60 marks] 6 (a) There are three yellow balls, three red balls and three blue balls. Balls of each colour are numbered 1, 2, and 3. Find the number of ways of arranging the balls in a row such that adjacent balls do not sum up to two. [2] (b) In a restaurant, there were two round tables available, a table for five and a table for six. Find the number of ways eleven friends can be seated if two particular friends are not seated next to each other. [4] 7 For the events A and B, it is given that P( ') 0.6, P( ') 0.83 and P( | ') 0.83AB AB A B    Find, (i) P( )B [2] (ii) P( )A B [2] (iii) P( | ')B A [2] Hence determine whether A and B are independent. [1] 8 A fairground game involves trying to hit a moving target with a gunshot. A round consists of a maximum of 3 shots. Ten points are scored if a player hits the target. The round ends immediately if the player misses a shot. The probability that Linda hits the target in a single shot is 0.6. All shots taken are independent of one another. (i) Find the probability that Linda scores 30 points in a round. [1] The random variable X is the number of points Linda scores in a round. (ii) Find the probability distribution of X. [3] (iii) Find the mean and variance of X. [3] (iv) A game consists of 2 rounds. Find the proba bility that Linda scores more points in round 2 than in round 1. [3]

5 [Turn Over] 9 Six cities in a certain country are linked by rail to city O. The rail comp any provides the information about the distance of each city to city O and the rail fare from that city to city O on its website. Charles copied the table belo w from the website, but he had copied one of the rail fares wrongly. City A B C D E F Distance, x km 100 270 120 56 289 347 Rail fare, $y 11.1 17.1 6.44 7.62 17.9 18.8 (i) Give a sketch of the scatter diagram for the data as shown on your calculator. On your diagram, circle the point that Charles has copied wrongly. [2] For parts (ii), (iii) and (iv) of this question you should exclude the point for which Charles has copied the rail fare value wrongly. (ii) Find, correct to 4 decimal places, the produc t moment correlation coefficient between (a) ln x and y, (b) 2x and y. [2] (iii) Using parts (i) and (ii), explain which of the cases in part (ii) is more appropriate for modelling the data. [2] (iv) By using the equation of a suitable regression line, esti mate the rail fare when the distance is 210 km. Explain if your estimate is reliable. [3] 10 A factory manufactures round tabl es in two sizes: small and la rge. The radius of a small table, measured in cm, has distribution N 2(30, 2 ) and the radius of a large table, measured in cm, has distribution N 2(50,5 ) . (i) Find the probability that the sum of the radi us of 5 randomly chosen small tables is less than 160 cm. [2] (ii) Find the probability that the sum of the radius of  3 randomly chosen small tables is less than twice the radius of a randomly chosen large table. [2] (iii) State an assumption needed in your calculation in part (ii). [1] A shipment of 12 large tables is to be e xported. Before shipping, a check is done and the shipment will be rejected if there are at least two tables whose radius is less than 40 cm. (iv) Find the probability that the shipment is rejected. [3] The factory decides now to manufacture medi um sized tables. The radius of a medium sized table, measured in cm, has distribution N 2(, )  . It is known that 20% of the medium sized tables have radius greater than 44 cm and 30% have radius of less than 40 cm. (v) Find the values of  and . [4]

6 [Turn Over] 11 The Kola Company receives a number of complaints that the volume of cola in their cans are less than the stated amount of 500 ml. A st atistician decides to sample 50 cola cans to investigate the complaints. He measures the volume of cola, x ml, in each can and summarised the results as follows: 24730x , 2 12242631x  . (i) Find unbiased estimates of the population m ean and variance correct to 2 decimal places and carry out the test at the 1% level of significance. [6] (ii) One director in