2024 Y6 Timed Prac Rev Paper 4 (Soln)

RAFFLES INSTITUTION 2022 Year 6 H2 Mathematics Common Test Questions and Solutions with comments     Page 1 of 19   1 The curve C has equation 2 24 , xxyx b xb   . (i) Given that y cannot take values between 2 and 6, show that b = 2 using an algebraic method. [3] (ii) Describe a sequence of 2 transforma tions that transform the graph of C onto the graph of 2 54 , 0xxyx x  . [3] (i) [3] Method 1 Since y cannot take values between 2 and 6, then the graph of this rational function will have 2 stationary points, with minimum point at 6y and maximum point at 2y . Consider 2 2 24 2 42 0 xx xb xb     and 2 2 24 6 84 60 xx xb xx b     Since 2y and 6y are tangents to 2 24 ,xxy xb   then the discriminant of the above 2 quadratic equations is equal to 0. 4(4 2 ) 0 2 b b   and 64 4(4 6 ) 0 2 b b   Thus, b = 2. Method 2 2 2 2 24 24 (2 ) ( 4 ) 0 xxy x b yx by x x xx y b y       2 2 Discriminant ( 2 ) 4(1)(4 ) (4 4 ) 12 yb y yy b       Since y cannot take values between 2 and 6, then 26 y  is the solution for 2 (4 4 ) 12 0yy b  . We can rewrite 26 y  as   2 26 0 41 2 0 yy yy    Comparing coefficients of y, b = 2. 6y 2y 2024 Y6 H2 Math Timed Practice Revision Paper 4 Solution

Page 2 of 19 (ii) [3] We can rewrite the 2 equations as 2 24 4 22 xxyx x x      and 2 54 4 5xxyx x x        Replace by 2 Replace by 3 44 22 432 45 xx yy yx yx xx yx x yx x                The sequence of transformation is 1) Translate C in the negative x-direction by 2 units to get 42yx x , 2) followed by a translation of 3 units in the positive y- direction. Note that for such question, you need to describe (in words) the transformation used. Stating “Replace x by x+2” is not a description, but is part of your working. Need to use the proper wordings for describing transformations. In this question, “translate by k units positive/negative x/y-direction”. It is mentioned by the Cambridge report that “along the x-axis”, “in the x-axis” or “on the x-axis” are not accepted.

Page 3 of 19 2 The function g is defined by g: s i n ,x xx for ,x  .x  (i) Show that g0 x  for .x  Hence, or otherwise, show that g has an inverse. [2] (ii) Sketch on the same diagram the graphs of g( )yx  and 1g( )yx  . [3] (iii) Show that the composite function 2g exists. [1] (iv) Find the exact solutions of the equation   2g. x x [2] (i) g( ) sinx xx g( ) c o s 1x x  For ,x  1c o s 1 0c o s 12 0g ( )2 x x x      Thus, g0 x  for .x  Since g is a strictly increasing function on x