XJC H2 Math - Set 2 - P2 (ANS)

X Junior College Preparatory Examinations 9758 Mathematics Suggested Solutions and Post-mortem © X Junior College Set II – Paper 2 1 MATHEMATICS Topic identification and short answers Qn Topic(s) Part Answers Section A: Pure Mathematics [40 marks] 1 Graph (sketching, transformation); Sequences and series (geometric series) (a) (b) (𝜋4+12)𝑘2 units2 (c) −(𝜋+2)𝑘2 2 Connected rates of change (a) [shown] (b) [shown] 68 seconds 3 Differentiation (maxima and minima) 𝜃=109.5° gives maximum volume. Maximum volume=6481𝑟3 units3 4 Complex numbers (cartesian form, conjugate) (a) [shown] (b) 𝛾=±25𝑞i 5 Vectors (two dimensions, modulus); (a) [shown] (b) |𝐛−𝐚|2=|𝐚|2+|𝐛|2−2|𝐚||𝐛|cos𝜃 𝜃=72° (c) Regular pentagon −2𝑘 2𝑘 4𝑘 𝑂 𝑦 𝑥 (3𝑘,𝑘2) (−𝑘,2𝑘) (𝑘,𝑘) 𝑦=𝑓(𝑥) Paper 9758/02 Set II – Paper 2

X Junior College Preparatory Examinations 9758 Mathematics Suggested Solutions and Post-mortem © X Junior College Set II – Paper 2 2 Section B: Probability and Statistics [60 marks] 6 Probability (permutations and combinations) Probability=1.83×10−37 7 Discrete random variables (a) [shown] (b) 𝑥 𝑃(𝑋=𝑥) 3 1−3𝑝+3𝑝2 4 3𝑝−9𝑝2+12𝑝3−6𝑝4 5 6𝑝2−12𝑝3+6𝑝4 (c) 0.354≤𝑝≤0.646 8 Normal distribution (a) [shown] (b) 𝜎≈50.01044≈50.0 (c) 𝑘=12 𝑚=325 Probability=1.53×10−6 9 Binomial distribution (a) [shown] (b) [shown] Most probable value of 𝑋=100 (c) Required probability ≈0.0781 (d) 𝑘𝑚𝑎𝑥=10 10 Sampling; Hypothesis testing (a) [shown] (b) E(𝑥)=95.02 Var(𝑥)=2.5796 (c) Since 𝑛 is a multiple of 50, 𝑛>30, and thus the sample size is large enough so that, by Central Limit Theorem, the sample mean approximately follows a normal distribution. (d) 𝑛𝑚𝑖𝑛=17450

X Junior College Preparatory Examinations 9758 Mathematics Suggested Solutions and Post-mortem © X Junior College Set II – Paper 2 3 11 Linear regression (a) 𝐴 is Quick Sort. 𝐵 is Merge Sort. 𝐶 is Selection Sort. (b) [shown] (c) Algorithm 𝐴, Quick Sort; Best-case 𝑇=−0.248+(2.46×10−4)𝑛log2𝑛 𝑛≈1277 bytes. (d) [shown] (e) Suggestion: 𝑆=0.368+0.540ln𝑇 (Any suitable suggestion acceptable.) 1.2 7.5 0.3 1.3 𝑂 𝑆 / bytes 𝑇 / ms

X Junior College Preparatory Examinations 9758 Mathematics Suggested Solutions and Post-mortem © X Junior College Set II – Paper 2 4 Suggested solutions and post-mortem Qn Suggested Solutions Comments Section A: Pure Mathematics [40 marks] 1 (a) [2] To begin, it must first be understood that the given piecewise function gives rise to vertically scaled sketches of the same graph in different periods. Be mindful of the domain of the sketch and indicate inclusion of the point (−2𝑘,0) by means of a solid point, and exclusion of the point (4𝑘,0) by means of a hollow point. 1 (b) [1] Required area = Area of a quarter circle + Area of a right triangle =14𝜋𝑘2+12𝑘2 =(𝜋4+12)𝑘2 units2 There may be a tendency to resort to integration to find the required area due to keywords such as finding “the exact area bounded”. Considering the mark