Secondary School Additional Mathematics Notes Compilation-12

Title Secondary School Additional Mathematics Materials Version 12 Author AprilDolphin – Lim Wang Sheng Date 6/3/2026 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 37 Coordinate Geometry of Circles 42 Trigonometry 54 Differentiation of Algebraic Functions 65 Differentiation of Exponential & Logarithmic Functions 69 Differentiation of Trigonometric Functions 72 Applications of Differentiation – Tangent & Normal Lines 76 Integration of Algebraic Functions 83 Integration Leading to Logarithmic Functions and Integration of Exponential Functions 85 Integration of Trigonometric Functions 86 Definite Integrals – Area Between Function 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 following (a) 12 √3 (b) 2−√7 3+4√7 (c) 1 3−√5 Solutions (a) 12 √3 × √3 √3 = 12√3 3 = 4√3 (b) 2−√7 3+4√7 × 3−4√7 3−4√7 = (2−√7)(3−4√7) (3+4√7)(