2024 Y6 Timed Prac Rev Paper 1 (Soln)

RAFFLES INSTITUTION H2 Mathematics (9758) 2024 Year 6   2024 Year 6 Timed Practice Revision Practice Paper 1 Solutions Source: 2019 Year 6 Term 3 Common Test 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]   1(i) [3] 22 22 22 22 2 2 22 a translation of units in the po sitive -direction i.e replace with a stretch with scale factor 2 parallel to 14 () 14 1 2 the -axis i.e replace 4 with 2 xy aa xa y aa xy a a axx x a xxx a          22 22 22 22 2 22 a reflection in the -axis i.e r (2 )11 44 (2 ) 1( 2 ) ( epla 2)44 ce with xa y aa xa y xa y aa yx x a           It is good practice to simplify as you go along. Otherwise, you may end up with a very complicated expression which makes simplification much harder. (ii) [2] C 2 is a circle with the centre (2, 0 )a , with a radius of 2a units.

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(i) [3] 2 23 4 0 2 3 12 4(4) 2 23 4 2 3i zz z           2 2 12 31 2zz    1 1 1arg( ) tan 3 z    πi 6 1 3i2 e z    and πi 6 2 3i 2 ez     Avoid the tedious method of replacing z by ix y in the equation and comparing real and imaginary parts. (ii) [1]   4πi 6 4 1 22 πi2 6 4 42i π i π66 2 2e 2e 2 e4 e 4 ( S h o w n )2 z z             (iii) [3] 1 1 ππarg arg( ) arg(1 3 i) 6313 i nzn nz      For 1 13 i nz  to be a real number, ππ π,w h e r e 63 n kk   Setting ππ π 62 , 63 n kn k k       Therefore the set of possible values is  |6 2 , nn k k  .

3    3 The function f is defined as follows. 2f: 3 ,xx x  for xa . (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 .xA (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 answer. In the case(s) that fg exists, find the range of fg. [4]   3(i) [3] From