2025 YIJC Prelim H2 Maths Papers 1&2 (QP & Soln)

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 29 August 2025 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/25 1 Show that the differential equation 24 d 2d e xx yxyy x §· ¨¸©¹ can be reduced by the substitution 24e xzy to 2 d2 d ex zx x . Hence, find the general solution in the form 2 fyx . [4] 2 Ronald is writing a novel. He began his new novel on 1 March 2025, writing 23 pages on the first day. On each subsequent day, he writes 90% of the number of pages he has written the previous day. (a) Find the total number of pages Ronald will have written by 31 March 2025. [1] Sam is also writing a novel. He began writing his novel on 8 March 2025, writing 2 pages on the first day. On each subsequent day, he writes 1 more page than the day before. (b) Find the first date on which Sam writes more pages in a day than Ronald. [2] (c) Find the first date on which Sam’s total num ber of pages written exceeds Ronald’s. [2] 3 (a) Find the series expansion of 1 15 x , up to and including the term in x2. State the range of values of x for which the expansion is valid. [3] (b) Hence, use the substitution 1 20x to obtain an approximation of 5, expressing your answer as a fraction. [2] (c) Without any further calculation, explain whether using the substitution 4 25x gives a better approximation of 5 than the substitution used in part (b). [1] 4 A curve has parametric equations 34 3 txt , 23yt t . (a) For the part of the curve where 0,x! find the equation of the tangent to the curve which is parallel to the y-axis. [3] (b) Find the coordinates of the point where the tangent meets the curve again. [2] 5 (a) Without using a calculator, solve the inequality 2 98 2 23 x xx t

. [4] (b) Hence solve the inequality 2 9ln 8 2 ln 2ln 3 x xx t

. [2]

3 ©YIJC 9758/01/JC2PE/25 [Turn Over 6 The diagram below shows the graph of f( )yx . The graph has asymptotes ya b x and 0x , intersects the x-axis at the point (, 0 )c and has a turning point (,)de . Sketch the following graphs on separate diagrams, lab elling the equations of any asymptotes and the coordinates of any points where the graphs cross the axes and of any turning points. (a) 3fyx c ,[ 2 ] (b)

1 fy x , [3] (c) fyx c . [2] 7 (a) On the same diagram, sketch the graphs 21yx and 2 42yx x . [2] (b) Hence find the set of values of x for which 221 42xx x . [2] (c) Without using a calculator, find the range of values of k for which 221 42xx x k has only positive real roots. [4] 8 (a) Find the exact value of 1 4 π 0 ec o s 2dx xx³ . [4] (b) (i) Show that d ln(tan )ds i n 2 kxxx , where k is a constant to be determined. [2] (ii) Hence find >@cosec2 ln(tan ) cos 2 dxx x x ³ . [2] y O O (d, e) c x

4 ©YIJC 9758/01/JC2PE/25 9 Do not use a calculator in answering this question. (a) The complex numbers z and w satisfy the following equations. i3 iwz 23 1 2 iwz Find z and w , giving your answers in the form ixy where x and y are real numbers. [4] (b) It is given that 43 2f( ) 9 3 5xa xxb x x , where a and b are real numbers, and f( ) 0x has no repeated roots.The graph of f( )yx intersects the x-axis exactly once. (i) Explain why 0a . [2] One of the roots of f( ) 0x is 12 i . (ii) Find the other roots and the value of b. [3] 10 Planes p and q are perpendicular. Plane p has equation (2 ) 6 .˜ ri jk Plane q contains the line l with equation (4 10 )12 O jjk i kr where Ois a parameter. The point A has coordinates (0,1, 12). (a) Find a cartesian equation of q.[ 2 ] (b) Find a vector equation of the line m, where p and q meet. [2] l passes through p at point B. Point C is the foot of the perpendicular of A to p. (c) Find the position vector of B. [2] (d) Find the position vector of C.[ 2 ] (e) Find the exact area of triangle ABC.[ 2 ] 11 The function f is defined by 2 1f: 2 x xx 2 1 2 xx 2

, x , 1, 2xx z . (a) Show that f has an inverse. [1] (b) Define 1f in similar form. [4] (c) The function g is such that 2fg( ) 1xx . Find g(x). [2] The function h is defined by 2 1h: x x 2 1 x , x , 0xz . (d) Only one of the composite functions fh and hf exists. Give a definition (including the domain) of the composite that exists, and explain why the other composite does not exist. [3]

5 ©YIJC 9758/01/JC2PE/25 [Turn Over 12 (a) Use the substitution sinx T toshow that 1 2 0 π1d 4xx ³ . [4] The diagram shows the region R bounded by the ellipse with equation 224 ( 2) 4xy and the line 2.yx The ellipse and the line intersect at the origin and the point (1, 2). (b) Find the area of R, giving your answer in terms of π. [4] (c) Find the volume of solid generated when R is rotated 2π radians about the x-axis. [2] 13 Around November 2019, there was an outbreak of COVID-19 in Wuhan, China, caused by the coronavirus SARS-CoV-2 virus, which subsequently spread to the rest of the world. On 23 January 2020, there was 1 confirmed case in Singapore. 70 days later, there were 1 049 confirmed cases. A student is interested to study the spread of the virus in Singapore and uses the modeld d N rNt , where r is a positive constant, N denotes the total number of confirmed cases, and t is the number of days after 23 January 2020. (a) Solve the differential equation to express N in terms of t. [4] (b) Use the model to estimate the number of days it will take to reach 1 000 000 cases. [1] To contain the spread of the virus, Singapore implemented strict circuit breaker lockdown measures starting on 7 April 2020, when the number of confirmed cases stood at 1,481. By the time the measures were lifted 55 days later, the number of confirmed cases had surged to 35 292. To study the spread of the virus during this period, the student uses a second model d d N sNu , where s is a positive constant, and u is the number of days after 7 April 2020. (c) Find the value of s. [2] (d) By comparing the values of r and s, comment on whether the lockdown measures were effective in containing the spread of the virus. [1] (e) Give a reason why neither model can estimate the actual number of confirmed cases accurately. [1] On 30 December 2020, Singapore became the first count ry in Asia to start its COVID-19 vaccination campaign. The student now uses a third model d 1d NN aNvb §· ¨¸©¹ , where a and b are positive constants, and v is the number of days after 30 December 2020. (f) Find the maximum rate of change of N in terms of a and b. [3] (1, 2) O x y

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1 Show that the differential equation 24 d 2d e xx yxyy x §· ¨¸©¹ can be reduced by the substitution 24e xzy to 2 d2 d ex zx x . Hence, find the general solution in the form 2 fyx . [4] 1 24 42 4 4 e dd 2e 4 edd d 2 e 2 d x xx x zy zy yyxx