ASRJC 2026 Prelim P2 QP

[Turn Over ANDERSON SERANGOON JUNIOR COLLEGE H2 MATHEMATICS JC2 Prelim Paper 2 (100 marks) 9758/02 31 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. Section A: Pure Mathematics [40 marks]

2 1 Two complex numbers z and w are such that 25zw+= . State, with a reason, an inequality satisfied by z. Given also that ( )arg 2.411865w = and 1z=− find w. [5] 2 (a) The diagram below shows the part of curve C with equation ( ) 2 4cosec 3 cot cosec 2cot 4cosec 7 x xxy xx −= −+ for 62 xππ≤≤ . One wing of a toy plane is modelled by the region bounded by the curve C, the x-axis and the lines 6x π= and 2x π= . Using the substitution cosec ux= , find the exact area of the wing. [7] (b) The diagram below shows the part of curve S with equation 22 219 14 25 5 yyx += −   for 1 and 0 54xy≤− ≤ ≤ . The region bounded by the curve S, the axes and the line 5y= is rotated through 2π radians about the y-axis to form the body of the toy plane. Find the volume of the plane’s body, giving your answer correct to 2 decimal places. [2] x y y x O O

3 [Turn Over 3 A fantasy role-playing game is set in an enchanted kingdom where all positions and movements are represented using a 3D coordinate system with origin O. The player controls a wizard and starts the game by traveling along a straight path 1l given by the equation 5 1, = 32x zy−=+ . (a) Write down a vector equation form of this path. [1] After a fierce battle, the injured wizard enters a Cursed Forest along another path 2l given by 2 41 : 1 0 , where , 0 04 l µ µµ −     = + ∈>      r  . This Cursed Forest is bounded by two cursed planes p1 and p2, given by the equations 223xyz++= and 2 25xy z+− = respectively. To recover from his injury, the wizard must collect a healing potion at point B on 2l , which is the closest point to the crystal shrine A(6, 3, 9). (b) Find the coordinates of B. [4] Planes p1 and p2 intersect along a dangerous fault line that the wizard must avoid. (c) Find the vector equation of the fault line. [1] The wizard then discovers an ancient scroll reveal ing an escape spell that can only be cast from points on l2 that are equidistant from planes p₁ and p₂. (d) Find the position vectors of these points. [4] From either of these points , the wizard casts the escape spell along l2 toward p2. If the spell strikes p 2 at an angle of less than 50° to the plane, it is absorbed by the cursed boundary, trapping the wizard in the Cursed Forest. Otherwise, the spell will pass through p2 allowing the wizard to escape safely. (e) Determine whether the wizard escapes the Cursed Forest or remains trapped. [3]

4 4 Paul, a lifeguard standing at point A on a straight stretch of beach, sees a swimmer in difficulty at point B in the sea. Point P is the point on the beach nearest to 𝐵𝐵, where 1 kmBP= and 2 kmAP= , as shown in the diagram. To reach the swimmer, Paul first runs from A to a point Q and then swims in a straight line from Q to B. Paul takes 4 minutes to run 1 km and 10 minutes to swim 1 km. (i) Assume that the swimmer remains stationary. Given that kmPQ w= , where 02 w≤≤ , show that the time T minutes taken by Paul to reach the swimmer is given by 28 4 10 1Tw w= −+ + . Hence, find the exact value of w for which T is a minimum. [5] (ii) Sketch the graph of T against w for 02 w≤≤ . [2] (iii) Find the minimum time taken by Paul to reach the swimmer, giving your answer to the nearest minute. [1] (iv) Suppose instead that, as Paul runs from A towards P, the swimmer is carried by a current directly away from the beach, perpendicular to the beach, at a constant speed of 0.01 km/min. At time t minutes after they start moving, Paul and the swimmer are at points R and S respectively, where R lies between A and P. Let kmPR x= , kmPS y= and PRSθ =  . Show that 2 2d 25sec d 100xytxθθ += . Hence find the rate of change of θ when 1x= . [5] B w km 1 km 2 km P A Q

5 [Turn Over Section B: Probability and Statistics [60 marks] 5 To celebrate the opening of a Fun Festival, the organizers set up a "Lucky Card Flip" prize booth. At the booth are two separate decks of cards, Deck A and Deck B. Deck A consists of 9 cards: 2 cards showing “$10”, 3 cards showing “$20”, and 4 cards showing “$50”. Deck B consists of 9 cards: 6 cards showing “x1” and 3 cards showing “x2”. A player randomly selects one card from each deck. The amount shown on the card selected from Deck A is the base prize, while the number shown on the card from Deck B is the multiplier. The player’s cash prize is the product of the base prize and the multiplier. For example, if the player selects a “$20” card from Deck A and a “x2” card from Deck B, the player receives $40. Let X denote the cash prize, in dollars, received by a player. (a) Write down the probability distribution of X. [2] (b) Find E(X) and Var(X). [3] (c) Find P( 57 30)X −> . [2] 6 A study reported that the population mean daily screen time of teenagers is 8.5 hours. A random sample of 200 teenagers was selected, and its standard deviation was found to be 1.5 hours. (i) Find the unbiased estimate of the population variance. [1] (ii) Estimate the probability that the total daily screen time of the 200 teenagers is between 1640 hours and 1730 hours. [3] The Ministry of Education suspects that the mean daily screen time of teenagers exceeds 8.5 hours. An independent organisation conducts a further investigation by selecting another random sample of 200 teenagers. This sample has a mean daily screen time of 8.665 hours. It is known that the daily screen time of teenagers is normally distributed with population standard deviation of 1.2452 hours. (iii) Using the new sample, test at the 5% significance level whether there is sufficient evidence to support the Ministry of Education’s suspicion. State clearly the hypotheses and define any parameters used. [4] (iv) If instead a random sample of 10 teenagers is selected, find the range of values of the sample mean for which there is sufficient evidence, at the 1% significance level, to conclude that the population mean daily screen time of teenagers differs from 8.5 hours. [2]

6 7 The continuous random variable X has distribution ( )2N 593, 127 . (a) A random sample of ten observations of X is taken.