EJC H2 2022 Prelim P2

EJC 2022 JC2 Prelim/9758/02 1 Section A: Pure Mathematics [40 marks] 1 It is given that ( )f 4 3xx=− . (a) Find the binomial expansion for ( )f x , up to and including the term in 2x . Give the coefficients as exact fractions in their simplest form, and state the range of values of x for which this expansion is valid. [4] (b) By taking 1 4x= , show that 184713 512 . [3] (c) We may also substitute 1 13x= to obtain an approximation of 13 . Without further calculation, explain why this leads to a better approximation of 13 than the value shown in (b). [1] 2 The graph of ( )fyx= is given below. It has one vertical asymptote at 1x= and two horizontal asymptotes 0y= and 2y= . The graph passes through the origin O and has a turning point at ( )1, 2−− . (a) On separate diagrams, sketch the following graphs, indicating the coordinates of the points where the graphs cross the axes, the turning points, and the equations of any asymptotes if possible. (i) ( ) 1 fy x= [3] (ii) ( )f'yx= [3] (iii) ( )f1yx=− [2] (b) State the range of values of a such that f ( 1)xa−= has only positive root(s). [1] y x O

EJC 2022 JC2 Prelim/9758/02 2 3 The sum nS of the first n terms of a sequence u1, u2, u3, … is given by 2 eln 3 n n nS =   . (a) Show that 1 (1 2 )ln3nun= + − . [2] (b) Hence, show that the sequence is an arithmetric progression. [2] (c) Find the sum of the first ten odd-numbered terms. [2] (d) A geometric sequence 312e , e , e , ...uuu has a common ratio, r. Find the value of r. [2] (e) Find the least value of n for which the sum of the first n terms of this geometric sequence is within 810− of its sum to infinity. [3] 4 (a) One of the roots of the equation 4 3 2 5 26 0z pz z qz+ + + − = , where p and q are real, is 1 2 3i− . Find the other roots of the equation, and the values of p and q. [5] Do not use a calculator in answering part (b). (b) The complex number w is given by ( ) 3 4 i 3i w= −+ . (i) Find the exact value of the modulus and argument of w . [4] (ii) Find the smallest positive integer n such that i * nw w is purely imaginary. [3] Section B: Probability and Statistics [60 marks] 5 In this question you should state the parameters of any normal distribution you use. The masses in grams of oranges have the distribution N(150, 14 2) and the masses in grams of kiwis have the distribution N(70, 82). (a) Find the probability that the mass of a randomly chosen orange is less than 180 grams. [1] 6 oranges and 4 kiwis are randomly selected and packed into a randomly chosen empty basket to make a fruit basket. The masses of the empty baskets in grams have the distribution N(750, 168). (b) Find the probability that a randomly chosen fruit basket is within 25 grams of its mean. [3] (c) Sketch the distribution for masses of fruit baskets between 1710 grams and 2150 grams. [2] (d) Three fruit baskets were randomly chosen. Find the probability that exactly one fruit basket weighs more than 2000 grams and exactly one fruit basket weighs less than 1900 grams. [2]

EJC 2022 JC2 Prelim/9758/02 3 6 A company manufactures screen protectors for handphones. On average, 1 7% of the screen protectors are cracked. The screen protectors are packaged into boxes of 20. It should be assumed that the number of cracked screen protectors in a box of 20 screen protectors follows a binomial distribution. (a) Show that the probability of having no more than 3 cracked screen protectors in a randomly chosen box is 0.55041, correct to 5 significant figures. [1] (b) Find the most probable number of cracked screen protectors in a randomly chosen box. [1] Every month, the company exports a large shipment of n cartons of screen protectors. In each carton, there are 8 boxes of screen protectors. Every carton is assumed to be filled completely. (c) Using a suitable approximation, find the least value of n such that in a randomly chosen month, there is at least a 99% chance that the average number of boxes per carton having no more than 3 cracked screen protectors is at most 5. [4] To determine whether cartons are ready for local or overseas sales , cartons are randomly selected for inspection. In a carton, • if there are at least 4 boxes that contain no more than 3 cracked screen protectors each , the carton is ready for local sales; • if there are at least 6 boxes that contain no more than 3 cracked screen protectors each , the carton is ready for overseas sales. (d) A carton that is not ready for overseas sales is randomly selected. Find the probability that the carton is ready for local sales. [3] 7 In a carnival lucky dip game, a game master places n consolation tickets, m blank tickets, and one golden ticket into a box. A contestant taking part in the game would pay $1 to draw two tickets from the box. They would then be awarded $1 for each consolation ticket drawn, and $10 for the golden ticket if it is drawn. Nothing is awarded for the blank tickets drawn. Let $W represent the total amount awarded to a contestant after one game. (a) Show that 2P( 1) ( 1)( ) nmW n m n m== + + + , and determine the probability distribution of W. [4] (b) By considering E( )W , show that if the game master expects to make a profit from the lucky dip game, then 19mn− . [3] The game master then decides to run the game with 10 consolation tickets and 40 blank tickets. He is also issued with carnival lucky draw tickets to give away, and gives each contestant Y lucky draw tickets after the game, where 4YW=− . (c) Find ( )E Y and ( )Var Y . [2]

EJC 2022 JC2 Prelim/9758/02 4 8 In an art installation, three points on the floor, X, Y and Z, were connected using a rope to form a triangle XYZ. 9 bulbs, labelled A, B, C, D, E, F, G, H, I, are then to be placed along the rope to improve the aesthetic appearance. (a) The 9 bulbs are first distributed into 3 groups, namely Group XY, Group YZ and Group ZX. Find the number of ways the bulbs can be distributed if each group consists of at least 2 bulbs. [2] In one particular distribution from part (a), the bulbs from Group XY, Group YZ and Group ZX are placed on the sides XY, YZ and ZX of the triangle XYZ respectively. The bulbs are arranged as shown in the diagram below. (b) By connecting the bulbs in this arrangement using additional rope, we can form a smaller triangle within the triangle XYZ. For example, the bulbs B, I, G form a smaller triangle when connected. How many different smaller triangles can be formed? [2] (c) Each bulb can be programmed to produce either red, blue or green light when it is switched on. Find the number of ways the colours can be programmed so that all the 3 colours are used. [3] In another art installation, 9 bulbs, labelled A, A, B, B, C, C, D, E, F, are laid on the floor in a circular arrangement. (d) Find the number of possible arrangements if the 2 A’s are separated and the 2 B’s are separated. [3] X Y Z A G F E I H D C B

EJC 2022 JC2 Prelim/9758/02 5 9 Orange is a company that produces laptops and tablets. The company has a patented design for their ultra-slim laptop cooling fans, which operate at an optimal speed of 3100 RPM, such that they maximise the cooling effectiveness while minimising the noise level. In a routine quality check by the company’s internal surveyors, a random sample of 13 laptops were tested and their cooling fan speeds are recorded in the table below. 3202 3033 3103 3013 3023 3032 3145 3054 3154 3083 3099 3101 3102 (a) Calculate the unbiased estimates of the population mean and population variance of the fan speeds. [2] Based on historical data, it is known that the fan speeds have a standard deviation of 30 RPM. (b) Stating any necessary assumptions, test at the 10% lev el of significance whether the population mean fan speed is different from the optimal speed of 3100 RPM. [6] (c) Sta