JPJC 9758 2023 Prelim P2

Jurong Pioneer Junior College H2 Mathematics JC2-2023 Prelim Paper 2 Question Paper Section A : Pure Mathematics [40 Marks] 1 It is given that f(x) = ( )ln 1 e x−+ . (i) Using standard series from the List of Formulae (MF26) or otherwise, show that the first three non- zero terms in the Maclaurin series for f(x) is 2ln a bx cx++ , where a, b and c are constants to be determined. [4] (ii) Hence find an expansion for 1 1e x+ , up to and including the term in x. [2] 2 In military exercises, parachutists jump from stationary helicopters and their motion is tracked by sensors tagged to their bodies. One parachutist jumps vertically from a helicopter. At time t seconds, the parachutist is x metres below the helicopter with velocity 1 msv − . It experiences a downward gravitational acceleration of 9.8 2ms− and an upward acceleration R due to air resistance, where 9.8R< . R is directly proportional to v , with a constant of proportionality 0k > . The rate of change of v for the parachutist is modelled as the difference between the downward gravitational acceleration and R . Write down a differential equation relating v , t and k, and solve this differential equation to find an expression for v in terms of t and k. [6] The terminal velocity of a falling object is the value of its velocity after a long time. State the terminal velocity of the parachutist in terms of k. [1] 3 (a) Describe a sequence of two transformations which maps the graph of exy= onto the graph of e2xy= −+ . [2] (b) The line with equation 6yx= + undergoes the following transformations, in succession: A: Scaling by a scale factor of 1 3 parallel to the x-axis, B: Translation of 2 units in the positive x-direction, C: Scaling by a scale factor of 1 3 parallel to the y-axis. Find the equation of the resulting line. [3]

2 (c) The curve ( )fyx= cuts the axes at ( ),0a and ( )0,b . xc= and yd= are asymptotes of ( )f x . It is given that ( ) 1f x− exists. State the equations of the asymptotes and the coordinates of the axial intercepts of the following curves. (i) ( ) 1 ,fy x= [2] (ii) ( ) 1f.yx −= [2] 4 The diagram shows a sketch of the curve ( )fyx= . The region under the curve between 2x= and 6x= , shown shaded in the diagram, is A. This region is split into 6 vertical strips of equal width, h. (i) State the value of h and show, using a sketch, that ( ) 6 1 f (2 ) n nh h = +∑ is greater than the area of A. [3] (ii) Find a similar expression that is less than the area of A.