2026 RVHS JC2 H2MA Prelim P1 Qn

©RIVER VALLEY HIGH SCHOOL 9758/01/2026 [Turn Over RIVER VALLEY HIGH SCHOOL 2026 JC2 Preliminary Examination Higher 2 NAME CLASS 2 5 J INDEX NUMBER MATHEMATICS Paper 1 Additional Materials: Printed Answer Booklet List of Formulae (MF27) 9758/01 16 September 2026 3 hours READ THESE INSTRUCTIONS FIRST This document consists of 6 printed pages and 2 blank pages. Write your class, index number and name on all the work you hand in. Write in dark blue or black pen. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. Answer all the questions. Write your answers on the Printed Answer Booklet. Follow the instructions on the front cover of the answer booklet. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. You are expected to use an approved graphing calculator. Unsupported answers from a graphing calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphing calculator are not allowed in a question, you must present the mathematical steps using mathematical notations and not calculator commands. You must show all necessary working clearly. The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 100.

2 ©RIVER VALLEY HIGH SCHOOL 9758/01/2026 1 It is given that ( ) 1tany px −= , where p is a constant and ( ) 1ππ tan22 px−−   . (a) Show that ( ) 22 d1 d yp x p x+= . [2] (b) By further differentiation of the result in part (a), find the first two non-zero terms in the Maclaurin series for y. [4] 2 (a) Solve the inequality 2 1e2 1 x x − − using a graphical approach. [3] (b) Without the use of a graphing calculator, solve the inequality 22 13 x xx + −− . [3] 3 A curve C has equation 2 3 3e 0 yyx −− + = , for 0x . The normal to C at a point P has positive gradient and makes an angle of o30 with the x-axis. Find the exact equation of this normal. [6] 4 With reference to the origin O, the points A and B are such that OA= a  and OB= b  , where a and b are both non-zero and non-parallel. The point X lies on OA such that : 2 :1OX XA = . The point Y lies on OB such that : 3: 2OY YB = . The lines AY and BX intersect at the point P. (a) Express OP  in terms of a and b. [5] (b) Point Z lies on AB such that 43AZ BZ= . Express OZ  in terms of a and b. [1] (c) Show that O, P and Z are collinear. [2] 5 (a) By writing cos3x as ( )cos 2xx+ and applying addition formula, show that ( ) 2cos3 cos 4cos 3x x x=− . [2] (b) By using integration by parts, or otherwise, find tan sin d3 xx x . [5]

3 ©RIVER VALLEY HIGH SCHOOL 9758/01/2026 [Turn Over 6 A colony of bacteria is cultivated in a tank under experimental conditions. The amount of bacteria is y (in grams), at time t (in days) after the start of the experiment . The total growth rate of the bacteria is determined by the following two factors. • The bacteria reproduce naturally and grow a rate directly proportional to the amount of bacteria present. • An automated feeder continuously supplies a liquid nutrient solution to the tank. This nutrient solution stimulates an additional growth rate which is directly proportional to the time taken from the start of the experiment. Initially, there is 3 grams of bacteria, and the total growth rate of the bacteria is 6 grams per day. After 1 day, the additional growth rate, due to the liquid nutrient, is 3 grams per day. (a) Show that d 23d y ytt =+ . [2] (b) By substituting 23u y t=+ , show that the differential equation can be written as d 23d u ut =+ . [2] (c) Find u in terms of t and hence y in terms of t. [5] 7 Let a and b be fixed positive real numbers. A variable l ine C passing through ( ),ab with varying gradient m has equation ( )y b m x a− = − , where 0m . The line C cuts the x-axis and y-axis at the points P ( ),0p and Q ( )0,q respectively. (a) (i) Show that A, the area of triangle OPQ, where O is the origin, is given by 2 21 22 bA ab a m m = − − units2. [2] (ii) Using differentiation, find the value of m that gives the minimum A. [3] (b) As m varies, the line C rotates about point ( ),ab causing the points P and Q to move along the x- and y- axes, respectively. At the instant when 1m=− , point P is moving to the right, where the rate of change of p is 1 unit per second and point Q is moving downwards. At this instant, find the rate of change of q, in units per second, leaving your answer in terms of a and b. [4]

4 ©RIVER VALLEY HIGH SCHOOL 9758/01/2026 8 (a) A sequence 1 2 3, , ,...u u u is such that 1 5u = and 1 32nnu u n+ =+ . It is known that the nth term of this sequence is given by (3 )n nu a bn c= + + , where a, b and c are constants. Find the constants a, b and c. [4] (b) (i) State the value of k such that 2 11 1 1 1 22 nn rr r r k r== =+  + . [1] (ii) Hence, show that 2 11 1 1 1 2 1 2 nn r n r r r r= + = =−  − . [2] (iii) Write down the value of 2 1 1lim n n rn r→ =+  . [1] 9 (a) The function f is defined by f : 4 e xx −+  , where 04 x . (i) Show that 1f− exists. [1] (ii) Find the ( )1f x− and the domain of 1f− . [3] (b) The function g is defined by g : f ( )xx  , for 04 x , and that g( ) g( 4)xx=+ for all real values of x. (i) Explain why the composite function g2 exists. [1] (ii) Find the exact value of ( ) 2g2 . [2] (iii) State the range of the composite function g2. [2]

5 ©RIVER VALLEY HIGH SCHOOL 9758/01/2026 [Turn Over 10 (a) The diagram shows the graph of f ( )yx= . The graph intersects the ax es at (2,0) and (0,3). It has a turning point at ( –5, 6) and asymptotes with the equation s 4x= , 0y= and 3y= . On separate diagrams, sketch each of the following graphs. For each graph, state the equation of any asymptotes, the coordinates of any points where the curve crosses the axes and the coordinates of any turning points. (i) f (| |)yx= , [2] (ii) 1 f ( )y x= . [3] (b) Express 95 32 x x + + in the form 32 ba x− + where a and b are positive constants. Hence, describe a sequence of transformations which transform the graph of 95 32 xy x += + to the graph of 1y x= . [5] y x 4 (2,0) (0, 3) O

6 ©RIVER VALLEY HIGH SCHOOL 9758/01/2026 11 Do not use a calculator in answering this question. (a) The function f is defined by ( ) 2f ( ) 2 i 10 20 10iz z z= − + + + . (i) Find the roots of f ( ) 0z = . [3] The function p is defined by 4 3 2p( )z z az bz cz d= + + + + , where a, b, c and d are real numbers. It is given that f ( )z is a factor of p( )z . (ii) Without any further calculations, state all the roots of p( ) 0z = , explaining your answer clearly. [2] (b) Solve the simultaneous equations, giving z and w in the form iab+ , where a and b are real numbers. * 8 20i, 2 10. zw wz =− −= [5] 12 Underwater sonar buoys are deployed, in the ocean, to detect and monitor threats in the ocean and on the ocean surface . Two underwater sonar buoys, buoy A and buoy B, are located at points A and B respectively. The underwater sonar buoys are in the water body. Points are defined relative to an origin ( )0,0,0 on the ocean bed, with units in metres. The ocean surface is modelled as the plane with equation 3 4 240yz+= . A