RI Post+Prelim+Revision+P1+%28LT5%29_solutions

RAFFLES INSTITUTION H2 Mathematics 9758 2025 Year 6 Term 4 Post-Prelim Revision Paper 1 (Source: A level questions) Total number of marks: 100 3 hour 1 9758/2024/01/Q1 The graph of 2 1xy ax bx c += ++ , where ,ab and c are non-zero constants, has an asymptote at 1 2x = − . The graph also has a turning point at 12, 9 −− . Find the values of ,ab and c . [4] Suggested Solution Remarks for Student [4] 2 1xy ax bx c += ++ Αsymptote at 2 11 1 022 2x a bc  = − ⇒ − + − +=   2 4 0 (1)abc⇒− + =  Graph passes through 11 12, 9 9 42 a bc − − ⇒− =− −+ 4 2 9 (2)a bc⇒ − +=  ( )( ) ( ) ( ) 2 22 2 22 12d d 2 ax bx c x ax by x ax bx c ax ax c b ax bx c + +− + += ++ − − +−= ++ 1Turning point at 2, 4 4 09 0 (3) a acb bc − − ⇒− + + − = ⇒− + =  Solving: 2, 1ab= = − and 1c = −

Raffles Institution H2 Mathematics 2025 Year 6 _______________________________________________________________________________________________ _____________________________________ Y6 H2 Math Term 4 Post Prelim Revision: Paper 1 Page 2 of 20 2 9758/2023/01/Q1 Find the exact equation of the tangent to the curve ( ) 2 ln 11 5yx= − at the point where 2.x = [5] No. Suggested Solution Remarks for Student ( ) 2 ln 11 5yx= − Differentiate with respect to x, ( ) ( ) ( ) 1d 2 11 5 ( 5) 10 11 5d d 10 11 5d y xxyx y yxx = − −= − − = −− When 2x = , ( ) 2 ln 11 5(2) 1 eyy= −= ⇒= ( )d 10 11 5 10(e)(11 10) 10ed y yxx = − − = − −= − The equation of tangent is e 10e( 2) 10e 21e yx yx −= − − = −+ 3 9758/2024/01/Q4 (a) Without using a calculator, solve the inequality 43 .2 x xx −≥+ [4] (b) Hence, solve the inequality 34 .2 x xx −≥+ [2] No. Suggested Solution Remarks for Student (a) [4] 43 2 x xx −≥+ ( )( ) ( ) 4 23 02 xx x xx −+ −⇒≥ + ( )( ) ( ) 4 23 02 xx x xx ++ − + − ≤⇒ ( ) ( ) 22 5 6 0 and 0 and 2xx x x x x⇒ + − − ≤ ≠ ≠− ( )( )( )2 6 1 0 and 0 and 2xx x x x x⇒ + − + ≤ ≠ ≠− 2 1 or 0 6xx∴ −<≤ − <≤ No calculator is allowed so key algebraic steps need to be included in your working Multiplying by “−1” to convert to “ +x 2 ” term results in an easier algebraic manipulation that is less prone to careless mistakes −2 −1 0 6

Raffles Institution H2 Mathematics 2025 Year 6 _______________________________________________________________________________________________ _____________________________________ Y6 H2 Math Term 4 Post Prelim Revision: Paper 1 Page 3 of 20 (b) [2] To solve 43 2 x xx −≥+ , we replace x by x in part (a): 2 1 or 0 6xx− <≤ − <≤ 0 2 1 has no solutionxx≥ ⇒ − < ≤− To solve 0 6, 0 and 6 0 and 6 6 60 o r 06 x xx xx xx <≤ < ≤ ⇒≠ −≤≤ ∴ − ≤< <≤