XJC H2 Math - Set 1 - P2 (ANS)

X Junior College Preparatory Examinations 9758 Mathematics Suggested Solutions and Post-mortem © X Junior College Set I – Paper 2 1 MATHEMATICS Topic identification and short answers Qn Topic(s) Part Answers Section A: Pure Mathematics [40 marks] 1 Vectors (two dimensions); Graphs and transformations Required sequence of transformation: 1st:Scaling by a scale factor of 𝑘2 parallel to the 𝑥–axis 2nd:Translation of 𝑘2−1−1𝑘+𝑘 units in the positive 𝑦–direction 2 Graphs (sketching using GC); Inequalities • If 0<𝑎<1, then −1<𝑥≤−𝑎 or 𝑎≤𝑥<2. • If 𝑎=1, then 1≤𝑥<2. • If 1<𝑎<2, then −𝑎≤𝑥<−1 or 𝑎≤𝑥<2. • If 𝑎=2, then −2≤𝑥<−1. • If 𝑎>2, then −𝑎≤𝑥<−1 or 2<𝑥≤𝑎. 3 Differentiation (maxima and minima) ℎ=√3𝑟 gives minimum volume. Minimum volume=4√3𝑟3 units3 4 Complex numbers (cartesian form, conjugate) (a) [shown] (b) 𝑦=1+2𝑥+(2−𝑎22)𝑥2+(43−𝑎2)𝑥3+⋯ (c) e2𝑥sin𝑎𝑥=𝑎𝑥+2𝑎𝑥2+⋯ 5 Sequences and series (geometric series) (a) End of July 2026 (b) [shown] Amount=$9,477.56 (c) August 2029 Paper 9758/02 Set I – Paper 2

X Junior College Preparatory Examinations 9758 Mathematics Suggested Solutions and Post-mortem © X Junior College Set I – Paper 2 2 Section B: Probability and Statistics [60 marks] 6 Normal distribution (a) 𝜇=6.667𝑎; 𝜎=0.667𝑎 (b) 7 Binomial distribution (a) [shown] (b) 𝑝≈0.968 or 0.252 (c) 12<𝑝<1 8 Discrete random variables (a) [shown] (b) E(𝑋)=4𝑁2+3𝑁−16𝑁 (c) [shown] 𝑚=71 9 Probability (a) P(𝐵)=25; P(𝐴∩𝐵)=730 (b) [shown] (c) P(𝐴∩𝐵∩𝐶)=115 (d) 720≤P(𝐴′∩𝐵′∩𝐶′)≤1330 10 Sampling; Hypothesis testing (a) • The sample is taken from the population of 1 000 chips of the newest model. • The sample size (or number of chips) 𝑛 is greater than 30. • The 𝑛 chips are obtained randomly (or independently of one another with equal probability). (b) [shown] Var(𝑥)=98.75 (c) 𝐻1:𝜇<40; 6.66≤𝛼<13.3

X Junior College Preparatory Examinations 9758 Mathematics Suggested Solutions and Post-mortem © X Junior College Set I – Paper 2 3 11 Linear regression (a) (b) [shown] The square of the residual horizontal distances between the relevant data points and the best-fit line is minimised. (c) Required value =8.2 degree Celsius (allow marginal error) This value is unreliable. (d) 𝑡=417.40−259𝑇 𝑟 =−0.9561828875≈ 0.956 (e) 𝑘=3 (allow marginal error) The data is selected in the range of 𝑡 where 𝑇 is strictly increasing and such that the product moment correlation coefficient is as close to 1 as possible.

X Junior College Preparatory Examinations 9758 Mathematics Suggested Solutions and Post-mortem © X Junior College Set I – Paper 2 4 Suggested solutions and post-mortem Qn Suggested Solutions Comments Section A: Pure Mathematics [40 marks] 1 [4] Finding the cartesian equation of 𝐿1, 𝐫=(𝑘1)+𝜆(1𝑘) →𝑥−𝑘=𝑦−1𝑘 →𝑦=𝑘𝑥−𝑘2+1 Finding the cartesian equation of 𝐿2, 𝐫=(1𝑘)+𝜇(𝑘1) →𝑥−1𝑘=𝑦−𝑘 →𝑦=1𝑘𝑥−1𝑘+𝑘 Required sequence of transformation: 1st:Scaling by a scale factor of 𝑘2 parallel to the 𝑥–axis 2nd:Translation of 𝑘2−