XJC H2 Mathematics 9758 – Set II: Paper 1 (Answers)

X Junior College Preparatory Examinations 9758 Mathematics Suggested Solutions and Post-mortem © X Junior College Set II – Paper 1 1 MATHEMATICS Topic identification and short answers Qn Topic(s) Part Answers 1 Graphs and transformations Required sequence of transformations: 1st: Translation of 12𝜋 units to the positive 𝑥–direction 2nd: Scaling of scale factor 12 parallel to the 𝑥–axis 3rd: Translation of 1 unit to the positive 𝑦–direction 4th: Scaling of scale factor 12 parallel to the 𝑦–axis (Other possible sequences acceptable.) 2 Maclaurin series (i) cos(𝑥+sin𝑥)≈1−2𝑥2+𝑥4 (ii) 𝜋3−𝜋3324+𝜋538880 or 2sin12 3 Sequences and series (arithmetic series) 𝑢𝑛=5+(𝑛−1)3 4 Differentiation (concept); Maclaurin series (small angle approximation) (i) [shown] (ii) [shown] 5 Inequalities (i) [shown] (ii) 𝜋4<𝑥≤𝜋3 or 2𝜋3≤𝑥<3𝜋4 6 Sequences and series (geometric series); Complex numbers (polar form, modulus); Differentiation (i) [shown] (ii) [shown] (iii) [shown] 7 Integration techniques (by parts); Definite integrals (volume of revolution) (i) 𝐼1=tan−1𝑐 (ii) [shown] (iii) Volume=3748𝜋−564𝜋2 units3 Paper 9758/01 Set II – Paper 1

X Junior College Preparatory Examinations 9758 Mathematics Suggested Solutions and Post-mortem © X Junior College Set II – Paper 1 2 8 Functions (inverse, composite, domain restriction); Inequalities (a) (i) [shown] (a) (ii) Rh=(−∞,𝑞2−6] [shown] (a) (iii) gh(𝑥)=𝑥2−2𝑞𝑥+13|𝑥2−2𝑞𝑥+16|=𝑥2−2𝑞𝑥+13𝑥2−2𝑞𝑥+16 Rgh=[13−𝑞2|16−𝑞2|,1)=[13−𝑞216−𝑞2,1) (Other equivalent forms acceptable.) (b) (i) XY(2)=8 YX−1(5)=7 X−1X−1Y(4)=2 (b) (ii) Both Y−1 and YX does not exist. 9 Graph (parametric); Differentiation; Definite integrals (i) (ii) 3𝛼𝑦2=𝑥(𝑥−𝛼)2 (Other equivalent forms acceptable.) (iii) Surface area=𝜋3𝛼2 units2 𝑦 𝛼 0 𝑥

X Junior College Preparatory Examinations 9758 Mathematics Suggested Solutions and Post-mortem © X Junior College Set II – Paper 1 3 10 Differential equations; Differentiation (connected rates of change); Graph (i) [shown] (ii) 𝑘=[𝑠𝐴𝜆+(𝑘01−𝑎−𝑠𝐴𝜆)𝑒−𝜆(1−𝑎)𝑡]11−𝑎 (iii) (iv) (Any of) increase 𝑠 / 𝐴 / 𝑎 or decrease 𝜆 Assumption: other constants remain unchanged 11 Vectors (tree dimensions, angle between two lines, cartesian equations, distances) (i) [shown] (ii) [shown] 𝛿=𝜋6 (iii) The direction vector of 𝑚 is not parallel to (10−1). (iv) 𝑃∶(2cos𝜃+2√3,2√2sin𝜃,2cos𝜃−2√3) (v) 𝐶∶(2√3,0,−2√3) |𝐶𝑃⃖⃖⃖⃖⃑|=2√2 units As 𝜃 varies, 𝑃 is a circle with radius 2√2 units centred at 𝐶. (vi) 2√6 units 𝑘 425 𝑂 𝑡 𝑘=649 𝑘=(83−3415𝑒−34𝑡)2

X Junior College Preparatory Examinations 9758 Mathematics Suggested Solutions and Post-mortem © X Junior College Set II – Paper 1 4 Suggested solutions and post-mortem Qn Suggested Solutions Post-mortem 1 [4] Using trigonometric identities in MF26, sin2𝑥=12(1−cos2𝑥)=12(1+sin(2𝑥−𝜋2)) Transforming from 𝑦=sin𝑥, 1 ←←←→𝑦=sin(𝑥−𝜋2) 2 ←←←→𝑦=sin(2𝑥−𝜋2) 3 ←←←→𝑦=1+sin(2𝑥−𝜋2) 4 ←←←→𝑦=12(1+sin(2𝑥−𝜋2))=sin2𝑥 Requ