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 exact