RI C1 Basics Lect Notes

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 5 ________________________________ Chapter 1: Assumed Knowledge Page 1 of 37 Chapter 1: Basics CONTENT FROM O LEVEL ADDITIONAL MATHEMATICS The following content is stated as assumed knowledge in your 9758 H2 Mathematics syllabus document: ALGEBRA A1, A2 Quadratic functions; equations and inequalities • Finding the maximum or minimum value of a quadratic function using the method of completing the square • conditions for a quadratic equation to have: – two real roots, two equal roots, no real roots • conditions for ax2 + bx + c to be always positive (or always negative) • solving simultaneous equations in two variables by substitution, with one of the equations being a linear equation A3 Surds • four operations on surds, including rationalising the denominator • solving equations involving surds A4 Polynomials and partial fractions • multiplication and division of polynomials • use of remainder and factor theorems • partial fractions with cases where the denominator is not more complicated than: – (ax + b)(cx + d) – (ax + b)(cx + d)2 – (ax + b)(x2 + c2) A6 Exponential and Logarithmic functions • Exponential and logarithmic functions ax, ex, loga x, ln x and their graphs, including – laws of logarithms – equivalence of y = ax and x = loga y – change of base of logarithms • Simplifying expressions and solving simple equations involving exponential and logarithmic functions

Raffles Institution H2 Mathematics 2025 Year 5 ______________________________________________________________________________________________ ________________ Chapter 1: Basics Page 2 of 37 GEOMETRY AND TRIGONOMETRY G1 Trigonometric functions, identities and equations • six trigonometric functions, and principal values of the inverses of sine, cosine and tangent (which will be discussed in the chapter on Functions) • trigonometric equations and identities (see Formulae List) • expression of in the forms and . G2 Coordinate geometry in two dimensions • coordinate geometry of the circle with the equation in the form CALCULUS (We will revisit this section in Chapters 5 and 8) C1 Differentiation and Integration • derivative of f(x) as the gradient of the tangent to the graph of y = f(x) at a point • derivative as rate of change • derivatives of xn for any rational n, sin x, cos x, tan x, ex and ln x, together with constant multiples, sums and differences • use of Chain Rule • derivatives of products and quotients of functions • increasing and decreasing functions • stationary points (maximum and minimum turning points and points of inflexion) • use of second derivative test to discriminate between maxima and minima • connected rates of change • maxima and minima problems • integration as the reverse of differentiation • integration of xn for any rational n, ex, sin x, cos x, sec2 x and their constant multiples, sums and differences • integration of (ax + b)n for any rational n, sin(ax + b), cos(ax + b) and eax + b The objective of this chapter is to familiarize you with content above (although you should already be familiar) that is often required for solving exercises and problems in H2 Mathematics.

Raffles Institution H2 Mathematics 2025 Year 5 ______________________________________________________________________________________________ ________________ Chapter 1: Basics Page 3 of 37 CONTENT 1 Set Language and Notation 1.1 Sets and Elements 1.2 Subsets, union and intersection of sets 1.3 Interval Notation 2 Algebra 2.1 Quadratic Expressions 2.2 Polynomials and Partial F ractions 2.3 Power, Exponential and Logarithmic Functions 3 Geometry and Trigonometry 3.1 Sine Rule and Cosine Rule 3.2 Trigonome tric Functions, Identities and Equations 4 2D Vectors 4.1 Scalars and Vectors 4.2 Notation and Geometrical Representation 4.3 Position & Free (Displacement) Vectors 4.4 Vectors in Two Dimensions (2D Vectors) 4.5 Negative Vectors 4.6 The Zero Vector 4.7 Vector Addition and Subtraction 4.8 Scalar Multiplication and Parallel Vectors 4.9 Laws of Vector Algebra 4.10 Equal Vectors 5 Introduction to Modulus 5.1 Basic Definition 5.2 Graphs of modulus functions

Raffles Institution H2 Mathematics 2025 Year 5 ______________________________________________________________________________________________ ________________ Chapter 1: Basics Page 4 of 37 1 Set Language and Notation 1.1 Sets and Elements A set is a collection of objects such as numbers, letters, etc. Each object in a set is known as an element (or member) of the set. We usually use capital letters such as A, B, C, etc to denote a set, small letters such as a, b, c, etc to denote the elements of a set. We write a∈A to mean that the element a belongs to the set A, or a is an element of A. We write b∉A to mean that b is not an element of A. Example: Suppose the set A contains all positive even integers 2, 4, 6, 8,  . There are two ways in general to represent the set A, namely: 1) V enn diagram 2) S et-builder notation (i) by listing the elements of the set within curly brackets, i.e. A = {2, 4, 6, 8,  }. (ii) by using a mathematical relationship to describe the elements of the set, i.e. A = {x=2k : k is a positive integer}. Some examples of commonly occurring sets: Notation Description ∅ or { } The empty set or null set. This set has no elements. Eg, The set of all integers who are odd and divisible by 2 = ∅  the set of natural numbers = {1, 2, 3, …}.  the set of integers = {…, −3, −2, −1, 0, 1, 2, 3, …} + the set of positive integers = {1, 2, 3, …} =  − the set of negative integers = {…, −3, −2, −1} 0 + the set of nonnegative integers = {0, 1, 2, 3, …} 0 − the set of nonpositive integers = {…, −3, −2, −1, 0} 2, 4, 6, 8, A

Raffles Institution H2 Mathematics 2025 Year 5 ______________________________________________________________________________________________ ________________ Chapter 1: Basics Page 5 of 37  the set of rational numbers of the form , where , and 0a ab bb ∈≠ , e.g. 1 2 , 4 3 , − 6 5 , 7, −0.89898989, … , etc  the set of real numbers You should be familiar with + , − , 0 + , 0 − . 1.2 Subsets, union and intersection of sets a) A set A is a subset of a set B, written A ⊆ B, if every element of A is an element of B. Mathematically, A ⊆ B if for all a∈A, then a∈B. This can be shown on a Venn diagram as depicted in Fig. 1. Also, we write A ⊆ B if the set A is not a subset of the set B. Below are some properties involving subsets. 1) ∅ ⊆ A, i.e. the empty set is a subset of any set. 2) A ⊆ A, i.e. any set is a subset of itself. 3) If A ⊆ B and B ⊆ C, then A ⊆ C. If A ⊆ B and B ⊆ A, then the two sets A and B are said to be equal, written A = B. If A ⊆ B and A ≠ B, then the set A is a proper subset of the set B, written A ⊂ B. Note: ⊂⊂⊂ . b) The union of two sets A and B, denoted by A ∪ B, is the set of elements which belongs to the set A or the set B or both. Mathematically, A ∪ B = {x : x∈A or x∈B}. In Fig. 2, the shaded region represents A ∪ B in the Venn diagram. c) The intersection of two sets A and B, denoted by A ∩ B, is the set of elements which belongs to both the set A and the set B. Mathematically, A ∩ B = {x : x∈A and x∈B} In Fig. 3, the shaded region represents A ∩ B in the Venn diagram. A B Fig. 1 A B Fig.2 A B Fig.3

Raffles Institution H2 Mathematics 2025 Year 5 _____________________________________________________________________________________