JC and Polytechnic Mathematics Material Compilation - Pure Mathematics

Title Junior College (H2) and Polytechnic Mathematics Material Compilation – Pure Mathematics Author AprilDolphin Date 16/2/2025 Topic Page Graphs and Functions – Conic Sections 2 Inequalities – Involving Quadratic, Cubic and Rational Functions 5 Inequalities – Involving Modulus Functions 11 Complex Numbers – Basic Operations 17 Complex Numbers – Argument & Modulus 21 Calculus – Differentiation – Involving Inverse Trigonometric Functions 27 Calculus – Differentiation – Implicit Differentiation 29 Calculus – Differentiation – Parametric Differentiation 33 Calculus – Integration – Integration by Substitution 36 Calculus – Integration – Integration by Parts 39 Calculus – Differentiation Equations – Basic Concepts & Separation of Variables 42

Title Graphs and Functions – Conic Sections – Hyperbola and Ellipse Author AprilDolphin Date 21/12/2024 Note This article assume you already understand the properties of circle and parabola as taught in secondary school Additional Mathematics and Elementary Mathematics respectively. In general, all conic sections can be represented using the equation below, 𝐴𝑥2 + 𝐵𝑥 + 𝐶𝑦2 + 𝐷𝑦 + 𝐸 = 0 Different types of conic sections appears when we begin changing the value of 𝐴, 𝐵, 𝐶, 𝐷 and 𝐸 If the conic section is an Ellipse, it has the following equation required in general: (Make sure the right-hand side is exactly 1 before proceeding!) (𝑥 − ℎ)2 𝑎2 + (𝑦 − 𝑘)2 𝑏2 = 1 For an ellipse to look vertically stretched, 𝑎 < 𝑏 For an ellipse to look horizontally stretched, 𝑎 > 𝑏

For an ellipse to have an exactly circular shape, 𝑎 = 𝑏, where 𝑎 represents the radius of a circle. (Circle is a special case of an ellipse.) Properties of an ellipse. The point where the centre lies is 𝐶(ℎ, 𝑘) The coordinates of vertices from up, down, left and right can be represented as the following Upper Vertex (Top of Ellipse) 𝑈(ℎ, 𝑘 + 𝑏) Lower Vertex (Bottom of Ellipse) 𝐷(ℎ, 𝑘 − 𝑏) Left Vertex (Left side of Ellipse) 𝐿(ℎ − 𝑎, 𝑘) Right Vertex (Right side of Ellipse) 𝑅(ℎ + 𝑎, 𝑘) The lines of symmetry of an ellipse are represented by 𝑥 = ℎ, 𝑦 = 𝑘 If the conic section is a Hyperbola, it has the following equation in general: (Make sure the right-hand side is exactly 1 before proceeding!) Hyperbola can be left and right opening as seen with the below diagram Properties of left-and-right-opening hyperbola as follows: Can be reduced to the following equation, (𝑥−ℎ)2 𝑎2 − (𝑦−𝑘)2 𝑏2 = 1, where RHS is exactly 1

Hyperbola can also be up-and-down-opening as seen with the below diagram Properties of up-and-down-opening hyperbola as follows: Can be reduced to the following equation, (𝑦−𝑘)2 𝑏2 − (𝑥−ℎ)2 𝑎2 = 1, where RHS is exactly 1 Further Properties of Hyperbola Left-and-Right-Opening Up-and-Down-Opening Centre 𝐶(ℎ, 𝑘) 𝐶(ℎ, 𝑘) Vertices Left Vertex, 𝐿(ℎ − 𝑎, 𝑘) Right Vertex, 𝑅(ℎ + 𝑎, 𝑘) Upper Vertex, 𝑈(ℎ, 𝑘 + 𝑏) Bottom Vertex,𝐵(ℎ, 𝑘 − 𝑏) Lin