XJC H2 Math - Set 3 - P2 (ANS)

X Junior College Preparatory Examinations 9758 Mathematics Suggested Solutions and Post-mortem © X Junior College Set III – Paper 2 1 MATHEMATICS Topic identification and short answers Qn Topic(s) Part Answers Section A: Pure Mathematics [40 marks] 1 Complex numbers (cartesian form, conjugate); System of linear equations 𝑢=3+2i 𝑤=5−i 𝑧=−4+3i 2 Differentiation (connected rates of change) 𝑠=14√𝑥2+42 (or equivalent) 𝑇=256 3 Integration techniques (by substitution); Definite integrals (volume of revolution) (a) [shown] (b) 23𝜋2−√34𝜋 units3 4 Vectors (two dimensions, collinearity) (a) [shown] (b) 𝑂𝑌⃖⃖⃖⃖⃑=(𝑎𝑎−𝑐)𝐫−(𝑐𝑎−𝑐)𝐩 𝑂𝑍⃖⃖⃖⃖⃖⃑=(𝑏𝑏−𝑐)𝐫−(𝑐𝑏−𝑐)𝐪 [shown] (c) 𝑐=𝑎𝑏2𝑎−𝑏 5 Maclaurin series; Differential equations (a) [shown] (b) 𝑦=1−2𝑘2𝑥2+23𝑘2(𝑘2−1)𝑥4+⋯ (c) 𝑦=cos(2𝑚sin−1𝑥) (d) 𝑘=𝑚 Paper 9758/02 Set III – Paper 2

X Junior College Preparatory Examinations 9758 Mathematics Suggested Solutions and Post-mortem © X Junior College Set III – Paper 2 2 Section B: Probability and Statistics [60 marks] 6 Probability (permutations and combinations) Number of ways =1016 7 Hypothesis testing; Sampling (a) The probability of obtaining a sample which gives a test statistic that is as extreme or more extreme than the one being observed, assuming that the null hypothesis is true. (b) 𝐻0∶𝜇=30 and 𝐻1∶𝜇≠30 𝑛=100 𝛼=5 Reject the null hypothesis and conclude that there is sufficient evidence at 5% significance level that the average duration of uploaded videos differs from 30.0 seconds. (c) The second sample is large and yields 𝑝-value >0.05 8 Discrete random variables (expectations); Probability (a) [shown] (b) 𝑎=7 (c) Least integer 𝑦=2 9 Linear regression; Normal distribution (normal curve) (a) (b) In the sketch, f(𝑥) increases from 𝑥=59 to 𝑥=61, then decreases from 𝑥=61 to 𝑥=64. This implies that f(𝑥) is maximum at either 𝑥=61 or somewhere else in the range 59<𝑥<64. (c) 𝜇=62 𝑎≈0.0399 𝑏≈−0.00499 (d) 𝑥≈78.6 or 45.4 Both estimates are equally reliable (e) (71.8,0.0247) or (52.2,0.0247)

X Junior College Preparatory Examinations 9758 Mathematics Suggested Solutions and Post-mortem © X Junior College Set III – Paper 2 3 10 Binomial distribution (a) The event that the stock price rises from one period to the next is independent of any other event where the price rises from any other period to the next. (b) [shown] (c) P(sell at period 2𝑘)=(12)2𝑘[(2𝑘𝑘+1)+(2𝑘𝑘+2)] P(sell at period 2𝑘+1)=(12)2𝑘+1[(2𝑘+1𝑘+2)+(2𝑘+1𝑘+3)] (d) 𝑡odd=13 or 15 𝑡even= 6 or 8 Sell at 𝑡=13 or 15 to get 𝑟3𝑆0 or 𝑟5𝑆0 with probability 0.2444 Sell at 𝑡=6 or 8 to get 𝑟2𝑆0 or 𝑟4𝑆0 with probability 0.3281 (Any suggestion acceptable if justified using values above.) 11 Normal distribution (a) (b) Required probability ≈0.158 (c) [shown] Required least integer 𝑚=94 A fruit of either kind (apple or pear) will almost absolutely exceed 94 grams. To have a higher chance of distinguishing the fruit correctly, the vendor may use a larger value of 𝑚.