O Level Math Syllabus | Best Math Tuition Singapore

SUBJECT CONTENT

ALGEBRA

A1. Equations and inequalities
A2. Indices and surds
A3. Polynomials and Partial Fractions
A4. Binomial expansions
A5. Power, Exponential, Logarithmic, and Modulus functions

GEOMETRY AND TRIGONOMETRY

G1. Trigonometric functions, identities and equations
G2. Coordinate geometry in two dimensions
G3. Proofs in plane geometry

CALCULUS

C1. Differentiation and integration

Knowledge of the content of O Level Mathematics syllabus is assumed in the syllabus below and will not be tested directly, but it may be required indirectly in response to questions on other topics.

	Topic/Sub-topics	Content
ALGEBRA
A1	Equations and inequalities	Conditions for a quadratic equation to have: (i) two real roots (ii) two equal roots (iii) no real roots and related conditions for a given line to: (i) intersect a given curve (ii) be a tangent to a given curve (iii) not intersect a given curve Conditions for $a x^{2} + b x + c$ to be always positive (or always negative) Solving simultaneous equations in two variables with at least one linear equation, by substitution Relationships between the roots and coefficients of a quadratic equation Solving quadratic inequalities, and representing the solution on the number line
A2	Indices and surds	Four operations on indices and surds, including rationalising the denominator Solving equations involving indices and surds
A3	Polynomials and Partial Fractions	Multiplication and division of polynomials Use of remainder and factor theorems Factorisation of polynomials Use of: - $a^{3} + b^{3} = (a + b)$ $(a^{2} - a b + b^{2})$ - $a^{3} - b^{3} = (a - b)$ $(a^{2} + a b + b^{2})$ Solving cubic equations Partial fractions with cases where the denominator is no more complicated than: - $(a x + b) (c x + d)$ - $(a x + b) (c x + d)^{2}$ - $(a x + b) (x^{2} + c^{2})$
A4	Binomial expansions	Use of the Binomial Theorem for positive integer n Use of the notations n! and $(\begin{matrix} n \\ r \end{matrix})$ Use of the general term $(\begin{matrix} n \\ r \end{matrix}) a^{n - r} b^{r},$ $0 < r \leq n$ (knowledge of the greatest term and properties of the coefficients is not required)
A5	Power, Exponential, Logarithmic, and Modulus functions	Power functions $y = a x^{n}$ where n is a simple rational number, and their graphs Exponential and logarithmic functions $a^{x}$ x, ln x and their graphs, including: – laws of logarithms – equivalence of $y = a^{x}$ and $x = l o g_{a} y$ – change of base of logarithms Modulus functions \|x\| and \|f(x)\| where f(x) is linear, quadratic or trigonometric, and their graphs Solving simple equations involving exponential, logarithmic and modulus functions
GEOMETRY AND TRIGONOMETRY
G1	Trigonometric functions, identities and equations	Six trigonometric functions for angles of any magnitude (in degrees or radians) Principal values of $s i n^{- 1} x$ , $c o s^{- 1} x$ , $t a n^{- 1} x$ Exact values of the trigonometric functions for special angles (30°, 45°, 60°) or $(\frac{π}{6}, \frac{π}{4}$ , $\frac{π}{3})$ Amplitude, periodicity and symmetries related to the sine and cosine functions Graphs of $y = a s i n (b x) + c$ , $y = a s i n (\frac{x}{b}) + c$ , $y = a c o s (b x) + c$ , $y = a c o s (\frac{x}{b}) + c$ and $y = a t a n (b x)$ , where a is real, b is a positive integer and c is an integer. Use of the following - $\frac{s i n A}{c o s A} = t a n A$ , $\frac{c o s A}{s i n A} = c o t A$ , $s i n^{2} A + c o s^{2} A = 1$ , $s e c^{2} = 1 + t a n^{2} A$ , $c o s e c^{2} A = 1 + c o t^{2} A$ - the expansions of sin(A $ \plusmn $ B), cos(A $ \plusmn $ B) and tan(A $ \plusmn $ B) - the formulae for sin 2A, cos 2A and tan 2A - the expression for a cos $\theta$ + b sin $\theta$ in the form Rcos(u $\theta$ α) or R sin (u $\theta$ α) Simplification of trigonometric expressions Solution of simple trigonometric equations in a given interval (excluding general solution) Proofs of simple trigonometric identities
G2	Coordinate geometry in two dimensions	Condition for two lines to be parallel or perpendicular Midpoint of line segment Area of rectilinear figure Graphs of parabolas with equations in the form $y^{2}$ = kx Coordinate geometry of circles in the form: - $(x - a)_{2}$ + $(y - b)_{2} = r_{2}$ - $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ (excluding problems involving 2 circles) Transformation of given relationships, including y = $a x^{n}$ and y = $k b^{x}$ , to linear form to determine the unknown constants from a straight line graph
G3	Proofs in plane geometry	Use Of: - properties of parallel lines cut by a transversal, perpendicular and angle bisectors, triangles, special quadrilaterals and circles♦ – congruent and similar triangles♦ – midpoint theorem – tangent-chord theorem (alternate segment theorem)
CALCULUS
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 Use of standard notations $f^{^{'}} (x), f^{^{″}} (x), \frac{d y}{d x}, \frac{d^{2} y}{d x^{2}} [\begin{matrix} = \frac{d}{d x} (\frac{d y}{d x}) \end{matrix}]$ Derivatives of $x^{n}$ , for any rational n, sin x, cos x, tan x, $e^{x}$ , and ln x, together with constant multiples, sums and differences Derivatives of products and quotients of functions Derivatives of composite functions Increasing and decreasing functions Stationary points (maximum and minimum turning points and stationary points of inflexion) Use of second derivative test to discriminate between maxima and minima Applying differentiation to gradients, tangents and normals, connected rates of change and maxima and minima problems Integration as the reverse of differentiation Integration of $x^{n}$ , for any rational n, sin x, cos x, $s e x^{2} x$ and $e^{x}$ , together with constant multiples, sums and differences Integration of $(a x + b)^{n}$ , for any rational n, sin(ax + b), cos(ax + b), and $e^{a x + b}$ Definite integral as area under a curve Evaluation of definite integrals Finding the area of a region bounded by a curve and line(s) (excluding area of region between two curves) Application of differentiation and integration to problems involving displacement, velocity and acceleration of a particle moving in a straight line Finding areas of regions below the x-axis

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