2024 Y6 Timed Prac Rev Paper 1 (Qns)

RAFFLES INSTITUTION H2 Mathematics (9758) 2024 Year 6 2024 Year 6 Timed Practice Revision Practice Paper 1 (Source: 2019 Year 6 Term 3 Common Test) The solution will be released in Ivy on 27 May (Monday) Total Marks: 100 Duration: 3h **** Note that Qn 2 (Complex Numbers) will NOT be examined in the coming Timed Practice. So, you should complete this paper within 2 hrs 47 mins. Section A: Pure Mathematics [59 marks] 1 The curve 1C with equation 22 22 14 xy aa is transformed to curve 2C by a translation of a units in the positive x-direction, followed by a stretch with scale factor 2 parallel to the x-axis, and followed by a reflection in the y-axis, where a is a positive constant. (i) Find the equation of 2.C [3] (ii) Describe the shape of 2C geometrically. [2] 2 Do not use a calculator in answering this question. The equation 2 23 4 0zz  has two complex roots 1z and 2 ,z where 10a r g ( ) 2z  . (i) Find 1z and 2z in polar form. [3] (ii) Show that 4 1 2 2 4z z  . [1] (iii) Find the set of possible values of n, ,n for which 1 13 i nz  is a real number. [3]

2 2022 Year 6 Common Test Revision Practice Paper 3 3 The function f is defined as follows. 2f: 3 ,x xx for x a . (i) State the largest value of a for which the function 1f exists. Hence find  1f x and state the domain of 1f for this value of a. [3] In the rest of the question, use 1a . The function g is defined on an interval A as follows. g: l n 3 2 ,xx  for .x A (ii) A student suggests 2 possible intervals for A as follows. (a) 21, 33  (b) 1 , 02  Determine which of the above intervals will result in the existence of the composite function fg, justifying your answ er. In the case(s) that fg exists, find the range of fg. [4] 4 (a) A geometric series has first term 22 s i n and second term . (i) Show that the series is convergent for 2    . [2] (ii) It is given that 3 4   and nS denotes the sum of the first n terms of the series. Find nS and hence determine the exact value of .S [3] (b) (i) Show that 1 1 1 nn nn    , where n   . [1] (ii) Hence find the least possible value of N such that 4 1 100. 1 N n nn   [2] sin 2

3 2022 Year 6 Common Test Revision Practice Paper 3 5 The parametric equations of a curve are 2xt , 21y t t  , where t  , 1 .2t  (i) Sketch the curve, stating the equations of any asymptotes and the coordinates of any points where the curve crosses the axes. [3] (ii) The tangent to the curve at the point 2, 21 pp p    intersects the x-axis at point A and the y-axis at point B. Find, in terms of p, an expression for the area of the triangle OAB.