ASRJC 2026 Prelim P1 QP

[Turn Over ANDERSON SERANGOON JUNIOR COLLEGE H2 MATHEMATICS JC2 Prelim Paper 1 (100 marks) 9758/01 25 Aug 2026 3 hours Additional Materials: Printed Answer Booklet List of Formulae (MF 27) READ THESE INSTRUCTIONS FIRST Write your name and class in the boxes on the Printed Answer Booklet. Please use a dark blue or black pen and write clearly. An HB pencil may be used for graphs and diagrams only. Do not use staples, paper clips, glue, or correction fluid. Answer all the questions. Write your answers on the Printed 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 p resent the mathematical steps using mathematical notations and not calculator commands. You must show all necessary workings clearly. All work must be handed in at the end of the examination. If you have used any additional paper, please insert them inside the Printed Answer Booklet. The number of marks is given in brackets [ ] at the end of each question or part question. This document consists of 7 printed pages and 1 blank page

2 1 A graph with equation f( )yx= undergoes, in succession, the following transformations: I: A translation of 3 units in the negative y -direction; II: A translation of 1 unit in the negative x -direction; III: A scale parallel to the x-axis by a scale factor of 12 ; IV: A reflection in the y-axis. The equation of the resulting curve is ln(2 4 ) 2yx= −− . Determine the equation of function f, showing your workings clearly. [4] 2 (i) Using an algebraic method, solve the inequality 2 2 12 17 373 10xx xx++ ≤+− exactly. [3] (ii) Hence, solve 2 2 12 17 373 10xx xx+ + ≤− + . [2] 3 A curve C has parametric equations sin , cos 2x ty t= = , for 322 tππ<< , t π≠ . (a) Find the equation of the normal at the point P(sin p, cos 2p). [2] (b) The normal at P passes through another distinct point Q (sin q, cos 2q). Show that ( )8sin sin sin 1pp q += − . [3] 4 A curve C has equation 2 3 3 20x x xy y+ − + −= . (i) For real values x, use a non -graphical method to determine that y cannot lie between a and b, where a and b are real constants to be determined. [3] (ii) Sketch the graph of C, stating clearly the equations of any asymptotes and the coordinates of any turning points and axial intercepts. [3]

3 [Turn Over 5 (a) Find the exact value of 2 1 e 1 dx xx− − −−∫ . Show your workings clearly. [3] (b)(i) Show that ( )d ln sec tan secd xx xx += for 0 2x π<< . [1] (ii) Hence, using integration, show that ( )( ) 3 1sec d sec tan ln sec tan2xx x x x x C= + ++∫ for 0 2x π<< . [3] 6 Relative to the origin O, the position vectors of A, B and C are a, b and c respectively where 3= −c ba . Given that b is a unit vector, 3=a , 7−=ba and the angle between and is 3π , (i) find the exact shortest distance from C to the line AB. [4] (ii) Find the angle between b and AC . [3] 7 Given that 1e xy −= , show that 22 2 d 2d d2ln ddd yy yy yx xx += − . [2] (i) By further differentiation of this result, or otherwise, find the Maclaurin’s series of y up to and including the term in 3x . [4] (ii) Deduce the series expansion of 1 1 e x y x − = − up to and including the term in 2x . [2] a b

4 8 For this question, all answers for distance are to be expressed in kilometres. Madilyn and Nancy are participating in a 60 -day charity running challenge. The amount donated by their sponsors depends on the total distance each participant runs during the challenge. Madilyn runs on alternate days, starting on Day 1. On Day 1, she runs 2 km. On each subsequent running day, she runs 100 m more than she did on her previous run. Her running distances are therefore 2 km on Day 1, 2.1 km on Day 3, 2.2 km on Day 5, and so on. Nancy runs every day. On Day 1, she runs 2 km. On Day 2, she runs 200 m more than she did on Day 1. On each subsequent day, the additional distance she runs is 90% of the additional distance she ran on the previous day. Her daily running distances are therefore 2 km, 2.2 km, 2.38 km, and so on. At the end of the 60-day challenge, (a) find the total distance run by Madilyn. [1] (b) Show that the daily distance, d, run by Nancy on Day n is in the form 14 (0.9) nk −− , where k is a real constant. Hence find the total distance covered by Nancy in 60 days. Leave your answer correct to 2 decimal places. [4] (c) Show that Nancy's daily running distance converges to a limiting value. Hence, determine the limit of her daily running distance. [2] (d) Determine the day on which Madilyn will first run more daily distance than Nancy. [1]

5 [Turn Over 9 The functions f is defined by 2f : 4 2 2, , 2 2. x xx x x− −+ ∈ −≤< (a) Sketch the graph of ( )fyx= , stating clearly the coordinates of the axial intercepts and end points. [2] (b) Explain why 1f− does not exist. [1] The function g is defined by 2 1g : 4 2 2, , 2 . 4x xx x x− − + ∈ − ≤ <− (c) Find 1g− in a similar form. [4] The function h and k are defined by 2 h : ln( 1), , 1, k: 4 2 2, , 1 0 . x xx x x xx x x − + ∈ >− + − ∈ −≤ <   (d) Show that hk exists and find the exact range of the composite function hk. [2] 10 (a) Given that k is a real constant, find ( ) 30 12 5 22 r r kr− =  −−∑ , leaving your answer in the form ( )1 910A B Ck−++ , where A, B and C are exact real constants. [3] (b) Another series is such that ( )( )1 1 11 1 1 2 2 2 ( 1)( 2) N r rr r N N= = −++ ++ ∑ . (i) Find ( )( )1 1 .12r rr r∞= ++∑ Explain your answer clearly. [1] (ii) Find ( )( )13 1 12 N r rr r= −−∑ in terms of N. [3] (c) A sequence of positive numbers 123, , , ...xxx satisfies the recurrence relation 1 10 3 ,nnxx+ = − for 1, 2, 3,...n= . (i) Given that 4 1,x = find the exact value of 1x . [2] It is also known that the sequence converges to l . (ii) Find the value of l . [2] (iii) Prove that ( ) ( )2 21 3nnx l lx+ −= − . [1] (iv) Use the result in part (iii) to show that if nxl < , then 1nxl+ > . [2]

6 11 (a) Find the complex number z in the form ixy+ , where x, y ∈ ℝ such that 10i 1 7i1 2izz∗ = +−+ . [3] (b) The cubic equation 32 20az bz+ −= has real coefficients a and b, 0a≠ . The roots of the equation are 1z , 2z and 3z with 1z being a real root, 2z and 3z are non-real roots. If 23 3zz−= and 2 1z cz++ , where 0c> , is the quadratic factor of the cubic function in z, find the roots of the cubic equation. [5] (c) The point A represents a fixed complex number a such that arg24 aππ− < <− . The complex numbers i a and iaa+ are represented by the points B and C respectively. On a clearly labelled Argand diagram, show the points A, B and C. Deduce the shape of the quadrilateral OACB. [4] The set of points on the quadrilateral OACB is represented by z. If 1 3ia= − , find the exact maximum z . [2]

7 [Turn Over 12 An ocean-monitoring sensor stores electrical energy generated from wave motion. Let E joules be the amount of energy stored t hours after the sensor is activated. At the time of activation, • wave motion supplies energy at a rate of 121t+ joules per hour; • energy i