YI 2026 H2 Math JC2 Prelim Paper 1 (Qns)

This document consists of 5 printed pages and 1 blank page. [Turn Over YISHUN INNOVA JUNIOR COLLEGE JC 2 PRELIMINARY EXAMINATION Higher 2 CANDIDATE NAME CG MATHEMATICS Paper 1 Additional Materials: Printed Answer Booklet List of Formulae and Results (MF27) 9758/01 20 August 2026 3 hours READ THESE INSTRUCTIONS FIRST Write your CG, index number and name on the work you hand in. Answer all 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 ©YIJC 9758/01/JC2PE/26 1 A function f is defined by ( ) 32f. x ax bx cx d= + + + The graph of f( )yx= has turning points at ( )1,4− and 1 11,24 − . Find the values of a, b, c, and d. [4] 2 A geometric series has first term a and common ratio r. The sum to infinity of the series is 16 and its fourth term is 3 16 . (a) Show that r satisfies the equation 43256 256 3 0rr − + = . Given that 0.5r  , solve this equation and hence show that 12a= . [5] (b) Find the exact value of the sum to infinity of the even -numbered terms of the geometric series. [2] 3 It is given that 2 for 1 1,f () for 1 3, 10 94 2 x xx x  −  =  −  and that ( ) ( )f f 4xx=+ for all real values of x. (a) Evaluate f( 49)− [1] (b) Sketch the graph of ( )fyx= for 27 x−   . [3] (c) Find the exact value of ( ) 6 4 fd xx . [3] 4 (a) Find the exact solutions of the equation 2 6 2 7xx− = + . [4] (b) Sketch, on the same axes, the graphs of 2 6yx=− and 27yx=+ . Hence solve exactly the inequality 2 6 2 7xx−  + . [3]

3 ©YIJC 9758/01/JC2PE/26 [Turn Over 5 The sum of the first n terms of a sequence 1u , 2u , 3u , … is given by 1 2 ( 2)! n r r aua n= =− + , where a is a positive constant. (a) Find nu in terms of a, expressing your answer as a single algebraic fraction. [2] (b) Explain why the series 1 r r u  =  converges. [1] (c) Find 7 r r u  =  in terms of a. [2] (d) Find the least value of n for which the sum of the first n terms of the sequence is within 810a − of its sum to infinity. [3] 6 (a) Describe a sequence of transformations that transform the graph of lnyx= onto the graph of ( )lny ax b=+ , where a and b are positive constants. [2] (b) The curve f( )yx= cuts the axes at (0, 3)− and ( 1, 0).− It is given that the equation of the asymptote is 2x= . State, if it is possible to do so, the equations of any asymptotes for the following curves and the coordinates of any points where the curves cut the axes. (i) ( )f,yx= [2] (ii) ( ) 1 fy x= . [2] 7 A curve has parametric equations cos 2 , ln(sin 2 ), 0 2xy   = =   . (a) Sketch the curve, stating the equations of the two vertical asymptotes. [2] (b) State the equation of the line of symmetry of the curve. [1] (c) Given that the gradient of the curve at a point P is between 1.5− and 1.5 , find the range of possible values of the x-coordinate of P. [4] 8 A curve has equation 11 3xy−= , where 1 3x . (a) Show that 2 2 d d yy x x= . [2] (b) Using calculus, find the equation of the tangent to the curve at the point where 1x=− , giving your answer in the form y mx c=+ . [3] (c) Given that y increases at a constant rate of 5 units per second, find the rate of change of x when 1y= . [3]

4 ©YIJC 9758/01/JC2PE/26 9 (a) Without using a calculator, solve the equation 2 9 40iz =− , leaving your answer s in the form iab+ , where ,ab   . [5] (b) It is given that the Maclaurin series for ex can be generalised to expressions of the form ewx for all complex numbers w and real numbers x. (i) Find the first six non-zero terms of the Maclaurin series for ie x . [2] (ii) By considering the Maclaurin series for cos x and sin x , verify that ie cos isinx xx=+ . [1] 10 It is given that ( )1sin 3e, xy −= where 11 33 x−   . (a) Show that ( ) 22 2 dd1 9 9 9 0 dd yyx x y xx− − − = . [3] (b) By differentiating the above result, find the Maclaurin series for y, up to and including the term in 3x . [4] (c) Using your result in part (b) and the standard series in the List of Formulae (MF27), obtain the Maclaurin series for 1sin (3 ),x− up to and including the term in 3x . [3] 11 The plane 1 has equation 2 2 14.x y z+ + = The line 1l passes through the point A with coordinates (1, 0, 2) and is parallel to +ij . (a) Find the acute angle between 1 and 1l . [2] (b) Find the coordinates of the point of intersection of 1l and 1 . [3] (c) Find the position vector of the foot of perpendicular from A to 1 . [3] The line 2l is the reflection of 1l in 1 . (d) Show that 2l is parallel to 4.− + +i j k [2] (e) The plane 2 contains 1l and 2.l Find a cartesian equation of 2 . [3] (f) Explain whether 1 and 2 are perpendicular. [1]

5 ©YIJC 9758/01/JC2PE/26 [Turn Over 12 Scientists are investigating the toxin levels in marine fish living in a particular habitat. The rate of change of the toxin level in a marine fish, u milligrams (mg), with respect to time, t days, is proportional to the difference between the toxin level in the habitat and the toxin level in the fish. It may be assumed that the toxin level in the habitat, a mg, is constant. Two models describe the fluctuation of th e toxin level over time. Model A is a continuous model using differential equations. (a) Write down a differential equation for this model. [1] (b) By s olving the differential equation in part (a), show that e ktu a A −=− , where A and k are constants. [3] The scientists narrow their investigation to a particular species of marine fish for which 23a= . Based on published data, when a fish of this species has an initial toxin level of 13 mg, its toxin level increases to 19.3 mg after 15 days. (c) Find the toxin level in the fish after 30 days. [4] (d) Sketch the graph of u against t . [2] Model B is a discrete model using the recurrence relation 1 0.96nnu b u+ =+ , where nu is the toxin level in the fish on the nth day and b is a constant. The starting term is 0u . This recurrence relation has a general solution ( )0.96 nnu C D=+ for some constants C and D . (e) Use the earlier published data on the toxin levels to find the values of C and D , giving your answers correct to 3 decimal places. [2] (f) Describe how the toxin level changes over time based on this model. [1] The fish will die if the toxin level in its body reaches 25 mg. A study reveals that many of the fish have died from the toxin in recent years. (g) Explain which of the models, A or B, should be used in view of this study. [1]

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