2024 Y6 Timed Prac Rev Paper 2 (Qns)

RAFFLES INSTITUTION H2 Mathematics (9758) 2024 Year 6   2024 Year 6 Timed Practice Revision Practice Paper 2 (Source: 2020 Year 6 Term 3 Timed Practice) The solution will be released in Ivy on 3 June (Mon) Total Marks: 100 Duration: 3h **** Note that Qn 1 (Complex Numbers) will NOT be examined in the coming Timed Practice. So, you should complete this paper within 2 hrs 51 mins. Section A: Pure Mathematics (60 marks) 1 The complex number z is given by ixy , where x and y are real numbers. (i) Express y in terms of x if  2arg 2z  . [2] (ii) State the values of x and y if  Re 0z  and 2z  . [1] Using these values of x and y, find the smallest positive integer n for which * n z z is a negative real number. [2] 2 (a) By writing  2 2 1rr  in partial fractions, find an expression for  2 2 2 . 1 n r rr  [ 3 ] (b) A geometric series has first term a and common ratio r, where a and r are non- zero and 1r . The 3rd and 9th terms of the series are 448 and 7 respectively. Given also that the sum of the first n terms is 1197, find the values of a, r and n. [ 4 ]   3 The functions f and g are defined by      f: 1 6 4 , , 4xx x x  αϒ ,      2g: , xx x αϒ . (i) Sketch on the same diagram, the graphs of f and 1f , giving the coordinates of all points of intersection. [4] (ii) Explain why the composite function fg does not exist. [1] (iii) Find gf in similar form and state its range. [2]

2     4 A curve C has equation  41, , 1 1yk x x x x      , where 02 k is a constant. (i) Sketch C, labelling clearly the axial intercept(s), the coordinates of turning points and equations of the asymptotes. [4] The graph of C is transformed by a reflection in the x-axis, followed by a translation of 1 unit in the positive x-direction, followed by a stretch with scale factor 1 2 parallel to the y-axis. (ii) Find the equation of the resulting curve in the form f.yx [3] 5 The curve C has equation  21 2 e. x y   (i) Sketch C, labelling clearly the coordinates of the axial intercept(s) and turning point(s), if any. [2] (ii) Show that the equation of the tangent to C at the point where x p  can be expressed as 21 22(2 1) e 2 1 p p x y pp     . Hence find the equations of the tangents to C which passes through the origin. [4] (iii) The straight line y = mx intersects C at two distinct points. State the range of values of m. [2]

3     6 A frigate is stationed at position 1, 2, 0F . Two submarines 1S and 2S are under the sea surface. Submarine 1S is at position  2, 1, 1A   and travelling in a path parallel to vector –3 i + 2 j – k. An enemy submarine 2S is detected at position 3, 2, 2B  travelling in a path parallel to vector –2i – 3j + k.