Secondary School Additional Mathematics Notes Compilation-15

Title Secondary School Additional Mathematics Materials Version 15 Author AprilDolphin – Lim Wang Sheng Date 12/6/2026 Page Number Topic 2 Surds Manipulation 7 Quadratic Functions and Quadratic Equations 10 Polynomial and Partial Fractions Decomposition 26 Exponents and Logarithms 32 Binomial Expansion and Binomial Theorem 38 Coordinate Geometry of Circles 43 Trigonometry 55 Differentiation of Algebraic Functions 66 Differentiation of Exponential & Logarithmic Functions 70 Differentiation of Trigonometric Functions 73 Applications of Differentiation – Tangent & Normal Lines 77 Applications of Differentiation – Rates of Change 84 Applications of Differentiation – First & Second Derivative Application to Stationary Points 95 Integration of Algebraic Functions 102 Integration of Exponential Functions and Integration Leading to Logarithmic Functions 104 Integration of Trigonometric Functions 105 Definite Integrals – Concept of Area between Curve and 𝑥-axis

Title Surds Manipulation Author - Date 31/12/2022 Basic Surds Rules to Understand before Proceeding Given the following expression can be written in the following form 𝑔√𝑎𝑏2 It can be rewritten as the following 𝑔𝑏√𝑎 Example 1.1 √8 can be decomposed into the following √2 × 4 Since 4 = 22, we can rewrite in the following manner: 2√2 Law of Surds (Only involving square roots) √𝑎 × √𝑏 = √𝑎𝑏 𝑎√𝑏 × 𝑐√𝑑 = 𝑎𝑐(√𝑏𝑑) √𝑎2 = 𝑎 √𝑎 √𝑏 = √𝑎 𝑏 𝑚√𝑎 + 𝑛√𝑎 = √𝑎(𝑚 + 𝑛) 𝑚√𝑎 − 𝑛√𝑎 = √𝑎 (𝑚 − 𝑛) Rules of rationalizing the denominator in surds calculation and manipulation Rule 1 If expression is in the following form 𝑎 𝑔√𝑏 Multiply by the denominator to both numerator and denominator to get the following 𝑎 𝑔√𝑏 × 𝑔√𝑏 𝑔√𝑏

Rule 2. If the expression is in the following form or show some near resemblance to the following form 𝑎 + 𝑔√𝑏 𝑐 − 𝑑√𝑝 Find the conjugate value of the denominator whereby the sign in between 𝑐 − 𝑑√𝑝 is flipped to positive and multiply conjugate value to both numerator and denominator to get the following 𝑎 + 𝑔√𝑏 𝑐 − 𝑑√𝑝 × 𝑐 + 𝑑√𝑝 𝑐 + 𝑑√𝑝 Rule 3. If the expression is in the following form or show some near resemblance to the following form 𝑎 + 𝑔√𝑏 𝑐 + 𝑑√𝑝 Find the conjugate value of the denominator whereby the sign in between 𝑐 + 𝑑√𝑝 is flipped to negative and multiply conjugate value to both numerator and denominator to get the following 𝑎 + 𝑔√𝑏 𝑐 + 𝑑√𝑝 × 𝑐 − 𝑑√𝑝 𝑐 − 𝑑√𝑝 Question 1. 1.1 Simplify the following expression (a) 11√7 + 6√28 − 5√63 (b) (4√3 − √2)(√3 − 5√2)

(a) 11√7 + 6√28 − 5√63 Rewrite as 11√7 + 6√7 × 4 − 5√9 × 7 Once again can be rewritten as 11√7 + 6(2)√7 − 5(3)√7 We then proceed to simplify the expression as (11 + 12 − 15)√7 = 8√7 (b) (4√3 − √2)(√3 − 5√2) Expand the expression into the following 4√3(√3) − √2 (√3) − 5√(2)(4√3) + 5√2(√2) 4(3) − √6 − 20√6 + 5(2) = 12 + 10 − √6 − 20√6 = 22 − 21√6 1.2 Rationalize the denominator of the follo