XJC H2 Mathematics 9758 – Set I: Paper 2 (Answers)

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 Complex numbers (cartesian form) 𝐴=−1 𝐵=1 2 Complex numbers (modulus, real and imaginary, argument); Graph (sketching, conics section, transformation) (i) [shown] 𝑦=±√3(𝑥−1) (ii) (iii) −𝜋3<arg(𝑧−1)<𝜋3 3 Differentiation (maxima and minima) ℎ=√3𝑟 gives minimum volume. Minimum volume=4√3𝑟3 units3 4 Complex numbers (cartesian form, conjugate) (i) [shown] (ii) 𝑦=1+2𝑥+(2−𝑎22)𝑥2+(43−𝑎2)𝑥3+⋯ (iii) e2𝑥sin𝑎𝑥=𝑎𝑥+2𝑎𝑥2+⋯ 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 5 Vectors (three dimensions, vector product, magnitude, angle); (i) [shown] 𝑏=9.5 (ii) 𝑝=78; 𝑞=78; 𝑟=932 Area≈132.98 units2≈133 units2 (iii) Distance ≈14.60407≈14.6 units Section B: Probability and Statistics [60 marks] 6 Normal distribution (i) 𝜇=6.667𝑎; 𝜎=0.667𝑎 (ii) 7 Binomial distribution (i) [shown] (ii) 𝑝≈0.968 or 0.252 (iii) 12<𝑝<1 8 Discrete random variables (i) [shown] (ii) E(𝑋)=4𝑁2+3𝑁−16𝑁 (iii) [shown] 𝑚=71 9 Probability (i) P(𝐵)=25; P(𝐴∩𝐵)=730 (ii) [shown] (iii) P(𝐴∩𝐵∩𝐶)=115 (iv) 720≤P(𝐴′∩𝐵′∩𝐶′)≤1330 10 Sampling; Hypothesis testing (i) • 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). (ii) [shown] Var(𝑥)=98.75 (iii) 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 (i) (ii) [shown] The square of the residual horizontal distances between the relevant data points and the best-fit line is minimised. (iii) Required value =8.2 degree Celsius (allow marginal error) This value is unreliable. (iv) 𝑡=417.40−259𝑇 𝑟 =−0.9561828875≈ 0.956 (v) 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 [3] By remainder theorem, 𝑃(𝑧)=(𝑧2+1)𝑔(𝑧)+𝐴𝑧+𝐵 Divided by (𝑧+i), remainder is 1+i → 𝑃(−i)=−𝐴i+𝐵=1+i Divided by (𝑧−i), remainder is 1−i→𝑃(i)=𝐴i+𝐵=1−i Adding the two results to eliminate 𝐴i, −𝐴i+𝐵+𝐴i+𝐵=1+i+1−i 2𝐵=2→𝐵=1 Substituting 𝐵=1 into previous result, −𝐴i+1=1+i→𝐴=−1 While this question mainly tests on complex numbers expressed in cartesian form, it also demands familiarity with facto