RI 2023 MATH CT SOLN

RAFFLES INSTITUTION H2 Mathematics (9758) 2023 Year 6   ________________________________________________ 2023 Y6 H2 Math Common Test: Solutions with comments Page 1 of 23     2023 Year 6 H2 Mathematics Common Test: Solutions with Comments 1 The points A, B and C have position vectors ,a b and c respectively. The points A and B are fixed while C varies. (a) Given that ,a b and c are non-zero vectors such that () ()ca b b ca , find the relationship between ()ca  and b . [2] (b) Hence describe geometrically the set of all possible positions of the point C. [2] Solutions Comments (a) [2] Since  () ()bc a c a b  , () () ca b b ca   () ()ca b ca b  2( )  ca b 0   ()ca b 0 Hence, () ca and b are parallel or ca . ()ca b 0 implies sin 0ca b  , where  is the angle between ()ca and b. Thus either 0ca or sin 0  (Note b0 ) (b) [2] From (a), since ca or () ca and b are parallel. () , , kk kk     ca b ca b   Point C lies on the line passing through the point A and the line is parallel to the vector b. Note that ca when 0k .     A

Raffles Institution H2 Mathematics 2023 Year 6 _____________________________________________________________________________________________   ________________________________________________ 2023 Y6 H2 Math Common Test: Solutions with comments Page 2 of 23       2 Complete the square for 229 5 .xx    [1] The function f is defined by 2f : 2 9 5 , for .xx x x   (a) Sketch the graph of f( ),yx stating the coordinates of any turning points and points of intersection with the axes. [2] (b) If the domain of f is further restricted to 1, xk state with a reason the largest value of k for which the function 1f exists. [2] In the rest of the question, the domain of f is ,1 2 .xx  (c) Find the solution set of 1f( ) f ( )xx  . [1] The function g is defined by 1g: x x , for x  , 05 x .  (d) State whether the composite function 1fg exists, justifying your answer. [2] (e) If the domain of g is further restricted to pxq and the composite function gg exists, find an equation involving p and q. [2] Solutions Comments [1] 2 2 94 129 5 2 48xx x     (a) [2] Take note of details when drawing a graph. 5.125 > 5, so the turning point should be slightly higher than the y- intercept (0,5) Graph should also be symmetrical about 2.25x (0.649, 0) (3.85, 0) or 94 1,48  

Raffles Institution H2 Mathematics 2023 Year 6 _____________________________________________________________________________________________   ________________________________________________ 2023 Y6 H2 Math Common Test: Solutions with comments Page 3 of 23     (b) [2] Largest value of k is 9 4 . Since the line 41,2 8ym m  , cuts the graph of f( )yx at exactly one point,