XJC H2 Math - Set 1 - P1 (ANS)

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 Complex numbers (cartesian form) 𝐴=−1 𝐵=1 2 Sequences and series (arithmetic series) (a) 𝑎=𝑚2+(𝑛−1)(𝑚+𝑛)𝑚𝑛 𝑑=−2(𝑚+𝑛)𝑚𝑛 (b) [shown] 3 Complex numbers (modulus, real and imaginary, argument); Graph (sketching, conics section, transformation) (a) [shown] 𝑦=±√3(𝑥−1) (b) (c) −𝜋3<arg(𝑧−1)<𝜋3 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 4 Differentiation (implicit functions, equations of tangents) (a) d𝑦d𝑥=2𝑥+3𝑦+42𝑦−3𝑥 (b) [shown] (c) 3𝑥+5𝑦+2=0 11𝑥+28𝑦+46=0 5 Complex numbers (polar form, modulus and argument, Argand diagram) (a) 𝑢−𝑟𝑢+𝑟=itan12𝜃 (b) [shown] Right angle at 𝑊𝑂𝑍 (or 𝑍𝑂𝑊) (c) ∠𝑂𝑊𝑍=12𝜃 |𝑢−𝑟|=2𝑟sin12𝜃 |𝑢+𝑟|=2𝑟cos12𝜃 6 Vectors (ratio theorem, angle between two vectors, scalar products) (a) 𝜆=|𝐛||𝐚|+|𝐛| (b) [shown] 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) (a) 𝐼0=𝜋𝑎28 (b) [shown] (c) 5𝜋16 units2 9 Vectors (three dimensions, vector product, magnitude, angle); (a) [shown] 𝑏=9.5 (b) 𝑝=78; 𝑞=78; 𝑟=932 Area≈132.98 units2≈133 units2 (c) Distance ≈14.60407≈14.6 units

X Junior College Preparatory Examinations 9758 Mathematics Suggested Solutions and Post-mortem © X Junior College Set I – Paper 1 3 10 Differentiation (parametric functions, stationary points, tangents); Graphs (parametric equation, transformations); Definite integrals (volume of revolution) (a) d𝑦d𝑥=𝑡2−12𝑡 (b) 𝑡1=e−1 𝑡2=e (c) Angle=49.6° (d) [shown] 𝑘=𝑎2(e+e−1) (e) Volume=14𝜋𝑎3(8−𝑒2+5𝑒−2) units3 11 Differential equations; Differentiation (local maxima and minima, connected rates of change) (a) (i) d𝑅d𝑡=0.6𝑅−0.4𝑅𝑊 d𝑊d𝑡=−0.8𝑊+0.2𝑅𝑊 (a) (ii) [shown] (b) (i) Rabbit: smallest = 900, largest = 10,700. Wolf: smallest = 300, largest = 4,500. (b) (ii) 𝐵2

X Junior College Preparatory Examinations 9758 Mathematics Suggested Solutions and Post-mortem © X Junior College Set I – Paper 1 4 Suggested solutions and post-mortem Qn Suggested Solutions Post-mortem 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 This question mainly assesses on complex numbers expressed in cartesian form. Candidates are expected to recall the factor-remainder theorem from O–Level Additional Mathematics, which will lead to a system of two complex equations relating 𝐴 and