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

X Junior College Preparatory Examinations 9758 Mathematics Suggested Solutions and Post-mortem © X Junior College Set I – Paper 1 1 MATHEMATICS Topic identification and short answers Qn Topic(s) Part Answers 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 Sequences and series (arithmetic series) (i) 𝑎=𝑚2+(𝑛−1)(𝑚+𝑛)𝑚𝑛 𝑑=−2(𝑚+𝑛)𝑚𝑛 (ii) [shown] 4 Differentiation (implicit functions, equations of tangents) (i) d𝑦d𝑥=2𝑥+3𝑦+42𝑦−3𝑥 (ii) [shown] (iii) 3𝑥+5𝑦+2=0 11𝑥+28𝑦+46=0 5 Complex numbers (polar form, modulus and argument, Argand diagram) (i) 𝑢−𝑣𝑢+𝑣=itan(𝛼−𝛽2) (ii) [shown] Right angle at 𝑊𝑂𝑍 (or 𝑍𝑂𝑊) (iii) 𝜃=𝛼−𝛽2 |𝑢−𝑣|=2𝑟sin𝜃 |𝑢+𝑣|=2𝑟cos𝜃 6 Vectors (ratio theorem, angle between two vectors, scalar products) (i) 𝜆=|𝐛||𝐚|+|𝐛| (ii) [shown] Paper 9758/01 Set I – Paper 1

X Junior College Preparatory Examinations 9758 Mathematics Suggested Solutions and Post-mortem © X Junior College Set I – Paper 1 2 7 Functions (inverse, composite); System of linear equations (a) (i) f−1(𝑥)=𝑥+12𝑥−1, 𝑥∈ℝ, 𝑥≠12 (a) (ii) g(𝑥)=4𝑥−3𝑥−1 (b) (i) h(2)=0.5 h(0.5)=−1 h(−1)=2. (b) (ii) h(𝑥)=1−1𝑥 8 Integration techniques (by substitution, by parts); Definite integrals (area of a region) (i) 𝐼0=𝜋𝑎28 (ii) [shown] (iii) 5𝜋16 units2 9 Differentiation (parametric functions, stationary points, tangents); Graphs (parametric equation, transformations); Definite integrals (volume of revolution) (i) d𝑦d𝑥=𝑡2−12𝑡 (ii) 𝑡1=e−1 𝑡2=e (iii) Angle=49.6° (iv) [shown] 𝑘=𝑎2(e+e−1) (v) Volume=𝜋𝑎34(8−𝑒2+5𝑒−2) units3 10 Differential equations; Differentiation (local maxima and minima, connected rates of change) (i) (a) d𝑅d𝑡=0.6𝑅−0.4𝑅𝑊 d𝑊d𝑡=−0.8𝑊+0.2𝑅𝑊 (i) (b) [shown] (ii) (a) Rabbit: smallest = 900, largest = 10,700. Wolf: smallest = 300, largest = 4,500. (ii) (b) 𝐵2 11 Sequences and series (geometric series) (i) End of July 2026 (ii) [shown] Amount=$9,477.56 (iii) August 2029

X Junior College Preparatory Examinations 9758 Mathematics Suggested Solutions and Post-mortem © X Junior College Set I – Paper 1 3 Suggested solutions and post-mortem Qn Suggested Solutions Post-mortem 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 𝑘2−1−1𝑘+𝑘 units in the positive 𝑦–direction This question mainly assesses on converting vector equations into cartesian equations. Admittedly, despite being within the expectations of the syllabus, two-dimensional vector equations r