RI Post+Prelim+Revision+P1 Qns

RAFFLES INSTITUTION H2 Mathematics 9758 2025 Year 6 Term 4 Post-Prelim Revision Paper 1 (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 17 Oct 2025 (Friday). 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] 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 2dc o s 2 2sin2 lnd xyx xx xxx   . [2] (b) At the point on the graph where x  , 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  makes with the horizontal. [1] 3 Do not use a calculator to solve this question. (i) Solve the inequality 2 6 145 x xx   . [3] (ii) Hence solve the inequality 2 2 6 145 xx xx   . [2]

Raffles Institution H2 Mathematics 2025 Year 6 _______________________________________________________________________________________________ ______________________________________ Y6 H2 Math Term 4 Post Prelim Revision: Paper 1 Page 2 of 6 4 A sequence is defined by the recurrence relation 1 1 1 n n n uu u    , for 1n . (a) State what happens to the sequence when 1 0u  . [1] It is now given that 1 2u  . (b) Find 2 345 6, , , and uu u u u . [2] (c) By observing the pattern in part (b), find 4 1 n r r u   in terms of n. [2] 5 Elly started planking as an exercise and she c ontinues the exercise every day to build her core muscles. If she meets her target duration, she increases the target duration of the exercise by an additional 4 seconds on the next day. On any day, she will stop her exercise once she meets her target duration for the day. However, Elly does not always meet her target. Each day when Elly misses her target, she decreases her target duration by 5% on the following day. On Day 1, Elly carries out 20 seconds of planking, and she hopes to reach her target of 2 minutes by the end of 30 days. (a) Assume that Elly met her targets for the fi rst 11 days but missed her target duration from Day 12 to Day 15. Determine whether Elly will be able to reach her target of 2 minutes by the end of 30 days, if she met all her targets from Day 16 onwards. [3] Due to the difficulty level, Elly decides to rest art the programme by increasing the target duration of the exercise by a% each day, regardless of whether she meets her target. (b) Find in terms of a, the total target duration Elly has completed by the end of 30 days if she carries out 20 seconds of planking on Day 1. [2] [You may assume that on any day, she will st op her exercise once she meets her target duration for that day.] (c) If the total target duration she has completed by the end of 30 days is at least 30 minutes, find, to the nearest integer, the least value of a. [1]

Raffles Institution H2 Mathematics 2025 Year 6 _______________________________________________________________________________________________ ______________________________________ Y6 H2 Math Term 4 Post Prelim Revision: Paper 1 Page 3 of 6 6 (a) By using the substitution secx  , where π0 2 , show that  2 1 2 22 1 dg d 1 x x      , where 1 and 2 are exact constants to be stated, and g is a single trigonometric function to be determined. [4] (b) Hence find the exact value of 2 22 1 d 1 x x  . [3] 7 The points A, B and C represent the complex numbers a, b and c respectively, such that 0,a 3b  and 2i .c   The three complex numbers are roots to the equation f( ) 0z  where f(z) is a quartic polynomial with real coefficients and z is a complex variable. (a) Express f( z) as a product of two quadratic factors with real coefficients. [3] (b) Sketch an Argand diagram showing the roots of the equation f( ) 0z  . [2] (c) The point W represents the complex number w, such that i.cw Find the value of ACA W     and the area of the triangle APW where P represents the complex number .cb [4] 8 The curve C is defined by the parametric equations 11xa t   and 2 1ya t t   , where a is a positive constant and 0t  . (a) Show that 3d2 d yt x t    . [3] (b) Find the coordinates of the turning point on C, and explain why it is a maximum. [4] (c) Sketch C. [3]

Raffles Institution H2 Mathematics 2025 Year 6 _______________________________________________________________________________________________ ______________________________________ Y6 H2 Math Term 4 Post Prelim Revision: Paper 1 Page 4 of 6 9 The functions f and g are defined by 2f: 4 2x xx , , 3 . 5xx , g: 4 e axx  , , 1xx  , where a > 0. (a) Find 1f( ) x and state its domain. [3] (b) Find the value of x for which 1f( )f ( )x x  . [2] (c) Show that the composite function fg exists and express the exact range of fg in the form of 2eeaaAB C  , where A, B and C are real constants. [4] (d) Without the use of a graphing calculator, solve the inequality 2 g( ) 0 22 x xx   . Leave your answer in exact form. [3] 10 Game developers closely monitor the number of people playing thei r game. Understanding player numbers and behaviour can not only help in optimising in-game purchases, advertisement placements, and other revenue-generating aspects, it can also help the company manage server loads and ensure the game runs smoothly without performance issues. Two game developers are interested in the number of players playing the mobile game “Mobile Saga”. They attempt to model the number of players x, in hundred thousands, at time t months after the launch of the game using a differentia l equation. On the day of the launch, there were 55 000 players. (a) One game developer suggests that x and t are related by the di fferential equation 2d3 d5 x x ktt  , where k is a positive constant. (i) By substituting 3 5e t x u , show that the differential equation can be written as 3 2 5d ed tu ktt   . [2] (ii) Hence show that 3 2 55 50 250 11 250 e3 9 27 20 27 tkkk kxt t    . [4] (iii) Company A intends to place an advertisemen t in the game only if there are more than 76 000 players playing the game. Given that 1 10k , find the length of time for which Company A will place an advertisem ent in “Mobile Saga”, giving your answer correct to the nearest month. [2] (b) The other game developer suggests that x and t are related by the differential equation  2 32 d1 0 d 1 x t t   . Given further that there were 180 000 players playing “Mobile Saga” after 1 month, find x in terms of t. [4]

Raffles Institution H2 Mathematics 2025 Year 6 _______________________________________________________________________________________________ ______________________________________ Y6 H2 Math Term 4 Post Prelim Revision: Paper 1 Page 5 of 6 11 (a) The diagram shows part of the graph of 2 2yx from 1x to 2x . The area under the curve in this interval may be approximated by the total area of n rectangles, A, as shown.