2025 RI Math Prelim P2_solutions with comments

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 6 Page 1 of 30 H2 Math Year 6 Preliminary Examination Paper 2: Solutions with comments 1 (a) Without using a calculator, solve the inequality 2 2 5 6 2 3 .4 2 x x x x x     [4] (b) (i) Sketch on the same diagram the graphs of lny x and 5,y x  giving the equations of any asymptotes and the x-coordinates of the points of intersection between the two graphs. [2] (ii) Hence solve the inequality ln 5.x x  [2] Solution Comments (a) [4] 2 2 5 6 2 3 4 2 x x x x x     , 2x ( 2)( 3) 2 3 ( 2)( 2) 2 3 2 3 2 2 x x x x x x x x x x          3 2 3 0 2 x x x     0 2 ( 2) 0 0 or 2 and 2 OR 2 or 0 2 or 2 x x x x x x x x x x            A number of students did not simplify the LHS to 3 2 x x   and so arrive at a more complicated inequality 2( 2) ( 2) 0x x x   . Most of them managed to arrive at the second form of the solution from here. Many students who gave the first form of the solution omitted to exclude 2 from the answer. A few students cross multiplied and did not get any mark for this part.

2025 H2 Math Year 6 Preliminary Examination Paper 2: Solutions with Comments _________________________________________________________________________________ Page 2 of 30 (bi) [2] From GC, x-coordinates of intersection are 6.94 (3sf) or 0.00678 (3sf)x x  The graph of lny x was not well drawn with some students copying directly from the GC to have a hanging graph that does not extend to negative infinity. Some students even put in a horizontal asymptote of ey ! Students are reminded to round off the answers to 3 sf. (bii) [2] For ln 5x x  , 6.94 or 0.00678 0 or 0 0.00678 or 6.94 x x x x        OR 6.94 or 0.00678 0.00678 or 6.94 and 0x x x x      Alternative Method For ln 5,x x  the solution is 0 0.00678 or 6.94x x   . To solve ln 5x x  , we replace x byx . The solution is then 6.94 or 0.00678 0 or 0 0.00678 or 6.94 x x x x        As this part carries 2 marks, one mark is for some explanation of the method used and one mark is for the completely correct answer. Hence students who put down an answer with no explanation will not get any mark unless all the intervals are correct. Students should remember that disjoint intervals should be separated by the conjunction “or” and not a comma. lny x 0 x y 0 lny x x y 1 5

2025 H2 Math Year 6 Preliminary Examination Paper 2: Solutions with Comments _________________________________________________________________________________ Page 3 of 30 2 (a) In a triangle ABC, 2,AB angle CAB x radians and angle π 6CBA radians. (i) Show that 3 2 .ic nos sx xAC  [2] (