RI Post+Prelim+Revision+P1%28LT1%29_Solutions

RAFFLES INSTITUTION H2 Mathematics 9758 2025 Year 6 Term 4 Post-Prelim Revision Paper 1 (Source: Other JCs’ Prelim Qns) Source of Question: MI Prelim 9758/2024/01/Q1 1 (i) A quadratic curve passes through the point ( )1, 4−− and has its turning point at ( )2,5 . Find the equation of the curve. [4] (ii) Given instead that a cubic curve passes through the same point ( )1, 4−− and has the same turning point as stated in part (i). Explain whether it is possible to obtain a unique equation of the curve based on given information. [1] Solution: 1(i) METHOD 1 METHOD 2 - Preferred ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 Let . For point 1, 4 : 114 4 ---- (1) For point 2,5 : 225 4 2 5---- (2) d 2.d dAt turning point, when 2, 0. d 22 0 4 0 ---- (3) y ax bx c a bc abc a bc a bc y ax bx yx x ab ab = ++ −− − + −+= − −+= − + += + += = + = = += += ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 Let . Using turning point 2,5 : Clearly 2, 5 25 For point 1, 4 : 12 5 4 1 2 5 41 y ax h k hk y ax a a y x xx =−+ = = =−+ −− −− + = − =− = − − += + + From GC, 1, 4, 1a bc= −= = . Hence, equation of the curve is 2 4 1.yx x= −++ (ii) Not possible. For a general cubic equation 32y ax bx cx d= + ++ , there are 4 unknowns to solve for. But we can only form 3 equations from the given information. Therefore, we will obtain infinitely many solutions.

Raffles Institution H2 Mathematics 2025 Year 6 _______________________________________________________________________________________________ ______________________________________ Y6 H2 Math Term 4 Post Prelim Revision: Paper 1 Page 2 of 23 Source of Question: ACJC Prelim 9758/2024/02/Q1 2 The diagram shows part of the graph cos2 ,xyx= for 05 x≤≤ , which represents the path of a roller coaster. The horizontal distance travelled by the roller coaster is denoted by x units and its vertical distance travelled is denoted by y units. (a) Show that ( )cos 2d cos 2 2sin 2 lnd xyx x xxxx = − . [2] (b) At the point on the graph where x π= , find the rate at which the roller coaster is moving vertically when it is moving horizontally at a rate of 8 units per hour. [2] (c) Find the acute angle that the tangent to the graph where x π= makes with the horizontal. [1] Solution: 2(a) cos2 xyx= ( )ln cos 2 lny xx= ( )1 d cos 2 2sin 2 lnd yx xxyx x = +− ( )cos 2d cos 2 2sin 2 lnd xyx x xxxx = − (shown) (b) dd d dd d yy x tx t= × ( )cos 2 cos 2 2sin 2 ln 8x xx xxx =−×  ( )cos 2 cos 2 2sin 2 ln 8π ππ π ππ = −×  1 08 8 π π = −× = (c) Since d 1d x y x π= = , gradient of tangent is 1. Thus the angle that the tangent makes with the horizontal is or 454 π °.

Raffles Institution H2 Mathematics 2025 Year 6 ______________________________________________________________