XJC H2 Math - Set 3 - P1 (ANS)

X Junior College Preparatory Examinations 9758 Mathematics Suggested Solutions and Post-mortem © X Junior College Set III – Paper 1 1 MATHEMATICS Topic identification and short answers Qn Topic(s) Part Answers 1 Differentiation; Sequences and series (Geometric sequence) 𝑟=−e𝜋 2 Sequences and series (Arithmetic sequence) (a) [shown] (b) 𝑑=2 3 Graphs (sketching); Inequalities (a) (b) 𝑥=0 or 3−√52≤𝑥≤1 or 𝑥≥3+√52 4 Differentiation (maxima and minima) (a) [shown] (b) 𝑥=−1 and 𝑥=2 5 Integration techniques (by parts); Definite integrals (a) ∫{f(𝑥)+g(𝑥)}d𝑥=𝑥ln(ln𝑥)+𝐶 (b) e2ln2−e22+e Paper 9758/01 Set III – Paper 1

X Junior College Preparatory Examinations 9758 Mathematics Suggested Solutions and Post-mortem © X Junior College Set III – Paper 1 2 6 Differentiation (implicit); Integrals; Graphs (sketching, conics) (a) 𝑘=1 f(𝑥)=𝑥+2𝑥−2 (b) 7 Recurrence relations; Complex numbers (four operations) (a) [shown] Base cases: 𝑢1=1, 𝑢2=5 (b) 𝑟2−2𝑟+5=0 (c) [shown] 𝑚=4 𝛼4+𝛽4=−6 8 Complex numbers (conjugate, argument, modulus, Argand diagram, geometrical effects) (a) [shown] (b) 𝑧3=1 [shown] (c) |𝑧4−𝑧3|=√(5−√52) (d) Required area =58√(10+2√5) units2 (e) Im Re 3 3 3 3 3

X Junior College Preparatory Examinations 9758 Mathematics Suggested Solutions and Post-mortem © X Junior College Set III – Paper 1 3 9 Differentiation (parametric equations, tangents); Graphs (parametric, conics) (a) d𝑦d𝑥=cos𝜃−1sin𝜃 (b) [shown] (c) 𝑞=𝑝+𝜋 [shown] 𝑦=sin2𝜃 (d) Circle with radius 1 unit and centre (0,0) Clockwise direction 10 Differential equation; Differentiation; Graphs (sketching) (a) d2𝑥d𝑡2+7d𝑥d𝑡+6𝑥=0 (b) 𝑥=4e−𝑡+e−6𝑡 𝑦=2e−𝑡−2e−6𝑡 (c) 11 Functions (inverse, composite) (a) (i) [shown] (a) (ii) Rh−1=[0,1] Df=[0,1] [shown] (a) (iii) Dh−1=[−2,2] Rfh−1=[0,1] [shown] (b) [shown] (c) Q(𝑥)=1−|1−|4𝑥−2||, 0≤𝑥≤1

X Junior College Preparatory Examinations 9758 Mathematics Suggested Solutions and Post-mortem © X Junior College Set III – Paper 1 4 12 Vectors (three dimensions, normal, angle between two vectors, distances, relationships); Inequalities; Differentiation (implicit, maxima-minima) (a) (√30−1) (b) (001) The orbits are not coplanar. (c) [shown] (d) 14𝜋≤𝑡≤12𝜋 or 54𝜋≤𝑡≤32𝜋 (e) 𝑡=38𝜋,118𝜋 Distance≈2.31 units (f) The perfect condition is not feasible. The statement claims that the distance between the satellite and the starspot is the shortest possible when they are collinear with the centre of the star. However, mathematically the three points cannot be collinear.

X Junior College Preparatory Examinations 9758 Mathematics Suggested Solutions and Post-mortem © X Junior College Set III – Paper 1 5 Suggested solutions and post-mortem Qn Suggested Solutions Post-mortem 1 [3] For stationary points, →d𝑦d𝑥=e𝑥cos𝑥−e𝑥sin𝑥=e𝑥(cos𝑥−sin𝑥)=0 →cos𝑥=sin𝑥 (∵e𝑥>0) →tan𝑥=1 ∴𝑥=14𝜋+𝑛𝜋, where 𝑛∈ℤ Substituting this result into the curve equation, →𝑦=e(14𝜋+𝑛𝜋)cos(14𝜋+𝑛𝜋) →𝑦=e(14𝜋+𝑛𝜋)[c