RI Post+Prelim+Revision+P2 Qns

RAFFLES INSTITUTION H2 Mathematics 9758 2025 Year 6 Term 4 Post-Prelim Revision Paper 2 (Source: Other JCs’ Prelim Qns) Total number of marks: 100 3 hour Please attempt this paper in one sitting under exam conditions within 3 hours before the lesson on 24 October 2025. Section A: Pure Mathematics [40 Marks] 1 (a) The diagram shows the curve h( ).yx The curve has maximum points at  6, 4 and the origin, and crosses the x-axis at  5, 0 . The lines y = 0, x = 4 and y = x + 3 are the horizontal, vertical and oblique asymptotes to the curve respectively. (i) On the diagram given above, sketch the graph of   22 269 ,xy r  where r is a positive constant. Find the range of values of r for the equation     22 26h 9xx r  to have at least one real root. [3] (ii) On a separate diagram, sketch the graph of  1 .hy x [3] (b) The graph of 10 1yx  undergoes a sequence of transformations which transform its equation into 1.yx Describe and write down the transformations. [3] y = h(x) x = 4 y = x + 3 (5, 0) (6, 4) y = 0 y O

Raffles Institution H2 Mathematics 2025 Year 6 _______________________________________________________________________________________________ ______________________________________ Y6 H2 Math Term 4 Post Prelim Revision: Paper 2 Page 2 of 7 2 Referred to the origin O, the points A, B and X are such that OA a  , OB  b  and 3OX ab  . It is given that 2a , 5b and 73OX   . (a) By considering the scalar product of OX  with itself, or otherwise, find the value of .ab . [3] (b) Find the area of triangle OAX. [5] (c) The variable point V, not necessarily coplanar with A, B and X, has position vector v. Given that ,va vb describe geometrically the set of all possible positions of the point V. [2] 3 (a) It is given that sin , for 0 3 π,2f( ) 4, for 3π 4ππ x x x x x        and that f( ) f( 4 π)xx for all real values of x. (i) Sketch the graph of f( )yx for π 6π.2 x [3] (ii) Find 6π π 2 f( ) d ,x x  leaving your answer in exact form. [2] (b) From the diagram above, the region A is bounded by the curve ln(2 ),yx the line ,yh ,h   the x-axis and the y-axis while the region B is bounded by the curve ln(2 )y x and the lines e 2x and .yh Given that the volumes of the solids generated when A and B are rotated completely about the y-axis are equal, find the exact value of h. [5]

Raffles Institution H2 Mathematics 2025 Year 6 _______________________________________________________________________________________________ ______________________________________ Y6 H2 Math Term 4 Post Prelim Revision: Paper 2 Page 3 of 7 4 [The volume of a right square-based pyramid is 1 base area height.3 ]