2024 NYJC Prelim Exam H2MATH Paper 2

2024 Nanyang Junior College Preliminary Examination H2 Mathematics Paper 2 Section A: Pure Mathematics [40 marks] 1 The area bounded by the loop of the curve ( ) ( ) 22 13y x x= − + is given by ( )2 1 3 d x x x   −+ . (a) State the values of α and . [1] (b) Use the substitution 3ux=+ to find the exact area bounded by this loop. [4] 2 With reference to the origin O, the points A, B and R have position vectors a, b and r respectively. The points A and B are fixed and R varies. (a) Interpret geometrically the vector equation =r a where 0 1 . [1] (b) Interpret geometrically the vector equation ( ) − =r a b 0 . [2] (c) Interpret geometrically if ( ) 0−=r r a . [2] (d) Given that ( ) ()−  − =r a r b 0 . Find r in terms of a and b, showing your working clearly. [2] 3 Do not use a calculator in answering the question. Three complex numbers are 1 1 3iz =− + , 2 1iz =− − and 3 112 cos isin12 12z =−  . (a) Represent 1z , 2z and 3z on an Argand diagram. [2] (b) Find ( ) 1 2 23 z zz in the form ier  , where 0r and   −   . [4] A fourth complex number, 4z , is such that ( ) 14 2 23 zz zz is purely real and ( ) 14 2 23 1zz zz = . (c) Find the possible values of 4z in the form ( )cos isinr + , where 0r and   −   . [3] 4 (a) Write 2 1 4 8 3rr −+ in partial fractions. [2] (b) Find an expression in terms of n for 3 2 2 1 . 4 8 3 n r rr=   −+ [3] (c) Find 2 2 1 . 4 8 3r rr  =   −+ [1] (d) Using the result in part (b), find 3 2 1 1 . 4 8 3 n rn rr=+   −+ Write your answer as a simplified fraction in terms of n. [3]

2 5 In this question units are in centimetres (cm). The diagram shows part of the circle 22 256xy+= and the line y = q, where q < 0. The circle and the line cross the negative y-axis at points P and Q respectively. (a) The shaded region between the circle and the line is rotated about the y-axis t o form a wok of negligible thickness. The wok has depth PQ cm and a capacity of 3.3 litres. By finding the capacity of the wok in terms of q, calculate the depth of the wok giving your answer to the nearest integer. [1 litre = 1000 cm3] [4] (b) A flat frying pan of negligible thickness with a capacity of 1.464 litres is formed by rotating the part of the circle between the lines y = 0 and y = r, q < r < 0, about the y-axis. Find a cubic equation satisfied by r and hence find the value of r. [3] (c) Water is poured from a container to the wok in part (a) at a rate of 3 55t litres per second, where t is the time in seconds from when pouring begins. Find the time taken to fill this wok to its full capacity of 3.3 litres. [3] Section B: Probability and Statistics [60 marks] 6 A carpark has a row of 10 parking lots. On one day the carpark has 9 cars of different makes and 1 empty lot. Of the 9 cars, 2 are blue, 2 are red and the remaining 5 are of other distinct c