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 1.5 h = . Find the rate of change of V at this instant. [2] 6 Timber cladding is the application of timber planks over timber planks to provide the layer intended to control the infiltration of weather elements. (a) Using method A , 20 rectangular planks are used and the lengths of the planks form an arithmetic progression with common difference d cm. The shortest plank has length 65 cm and the longest plank has length 350 cm. (i) Find the value of d . [2] (ii) Find the total length of all the planks. [2] (b) Using method B , a long plank of 2000 cm is sawn off by a machine into n smaller rectangular planks. The length of the first plank is cm a and each successive plank is 8 9 as long as the preceding plank. (i) Show that the total length of the planks sawn off can never be greater than k times the length of the first plank, where k is an integer to be determined. [2]

3 (ii) Given that 423, a = find the greatest possible integral value of n and the corresponding length of the shortest plank. [4] 7 (i) Express 2 1 1 r − in partial fractions, and deduce that 2 1 1 1 1 ( 1) 2 ( 1) ( 1) r r r r r r   = −   − − +   . [2] (ii) Hence, find the sum, n S , of the first n terms of the series 1 1 1 ... 2 3 3 8 4 15 + + + × × × . [4] (iii) Explain why the series converges, and write down the value of the sum to infinity. [2] (iv) Find the smallest value of n for which n S is smaller than the sum to infinity by less than 0.0025. [3] 8 A drug is administered by an intravenous drip. The drug concentration, x , in the blood is measured as a fraction of its maximum level. The drug concentration after t hours is modelled by the differential equation ( ) 2 d 1 2 , d x k x x t = + − where 0 1, x ≤ < and k is a positive constant. Initially, 0. x = (i) Find an expression for x in terms of and . k t [5] After one hour, the drug concentration reaches 75% of its maximum level. (ii) Show that the exact value of k is 1 ln10 3 , and find the time taken for the drug concentration to reach 90% of its maximum level. [3] A second model is proposed with the following differential equation 2 d 1 sin , d 2 x t t   =     where x is the drug concentration, measured as a f raction of its maximum level, in the blood after t hours. Initially, 0. x = (iii) Find an expression for x in terms of . t [3] (iv) Explain, with the aid of a sketch, why this proposed second model is inappropriate. [2]

9 The figure above shows a cross-section of a searchlight whose inner reflective surface is modelled, in suitable units, by the curve 2 2 , 4 , 2 2. x t y t t = = − √ ≤ ≤ √ The inner reflective surface of the searchlight has the shape produced by rotating the curve about the x -axis. (i) Show that the curve has cartesian equation 2 8 y x = , and find the volume of revolution of the curve, giving your answer as a multiple of π . [3] ( ) 2 2 ,4 P t t is a point on the curve with parameter t . TS is the tangent to the curve at P , and PR is the line through P parallel to the x -axis. Q is the point ( ) 2, 0 . The angles that PS and QP make with the positive x -direction are θ and φ respectively. (ii) By considering the gradient of the tangent TS , show that cot t θ = . [2] (iii) Find the gradient of the line QP in terms of t . Hence show that 2 φ θ = , and show that angle TPQ is equal to θ . [5] A lamp bulb is placed at Q . (iv) Use your answer to part (iii) to describe the direction of the reflected light from the bulb. [1] (v) Find a cartesian equation of the locus of the mid-point M on PQ as t varies. [2] 10 Federal Aviation Administration data shows that there were an increase in aviation incidents caused by laser illuminations reported by pilots in 2015 and 2016. A simplified laboratory model is set up to investigate the effects of a laser beam on plexiglass, a common material used to make cockpit windscreen.

5 The piece of plexiglass is represented by a plane 1 p with equation 2 3 0 x y z + − = . Referred to the origin, a laser be