TPJC_H2_MATHS_P1

1 H2 2017 Preliminary Exam Paper 1 Question Answer all questions [100 marks]. 1 Without using a calculator, solve the inequality 2 3 7 1 2 1 3 x x x x + + < − + . [4] 2 The function p is defined by 2 2 1 p : , 1 x x x x − ∈ + a

. (i) Find algebraically the range of p, showing your working clearly. [3] (ii) Show that p( ) p( ) for all . x x x = − ∈

[1] It is given that 1 q( ) p 4 , 2 x x x   = − ∈    

. (iii) State a sequence of transformations that will transform the graph of p on to the graph of q. Hence state the line of symmetry for the graph of q. [3] 3 The function f is defined by ( ) 2 f : , where 5 x x k x k k − < > a . (i) Find 1 f ( ) x − and state the domain of 1 f − . [3] ( ) g y x = The diagram above shows the curve with equation ( ) g y x = , where 2 2. x − ≤ ≤ The curve crosses the x –axis at 2, 1, 1 and 2, x x x x = − = − = = and has turning points at ( ) 1.5, 1 , (0,4) − − and ( ) 1.5, 1 . − (ii) Explain why the composite function fg exists. [2] (iii) Find in terms of k , (a)the value of fg ( 1 − ) [1] (b)the range of fg. [2 ] 4 It is given that 1 i 3 z = − − √ . (i) Given that ( ) 2 i n z z is purely imaginary, find the smallest positive integer n . [4] The complex number w is such that 2 * 5 π 4 and arg . 6 w wz z   = = −     (ii) Find the value of w and the exact value of arg( ) w in terms of π . [3] | | | | | | 2 − 2 1 − 4 1 − 1 1.5 1.5 − x y O

On an Argand diagram, points A and B represent the complex numbers and w z respectively. (iii) Referred to the origin O , find the exact value of the angle OAB in terms of π . Hence, or otherwise, find the exact value of arg( ) z w − in terms of π . [2] 5 A metal cylinder of radius r cm and height h cm is inscribed in a circular cone paperweight of base radius 4 cm and height 6 cm (see diagram). It is determined that the volume of the cylinder, 3 cm V , should be as large as possible to provide weight to the paperweight. Show that ( ) 2 3 4 π 36 12 9 V h h h = − + . [2] Hence find the exact maximum value of V . [5] The metal cylinder is known to expand under heat. An experiment shows that the height of the cylinder is increasing at a rate of 1 0.04 cm s − at an instant when 1.