Additional Mathematics
SPM · 58 topics available
Form 4
Chapter 1 · Functions
Functions
A function maps each input to exactly one output. Learn notation, domain, range, images and objects.
OpenComposite Functions
Combine two functions by feeding the output of one into the other: fg(x) = f(g(x)).
OpenInverse Functions
The inverse function reverses a function, sending each image back to its object: f⁻¹(f(x)) = x.
OpenChapter 2 · Quadratic Functions
Quadratic Equations and Inequalities
Solve ax² + bx + c = 0 by factorisation or formula, and solve quadratic inequalities using the roots.
OpenTypes of Roots of Quadratic Equations
The discriminant b² − 4ac tells whether a quadratic equation has two, one or no real roots.
OpenSimultaneous Equations involving One Linear and One Non-Linear Equation
Solve a linear and a non-linear equation together by substitution, giving up to two solution pairs.
OpenQuadratic Functions
A quadratic function f(x) = ax² + bx + c graphs as a parabola with a vertex, axis of symmetry and max or min.
OpenChapter 3 · Systems of Linear Equations in Three Variables
Chapter 4 · Indices, Surds and Logarithms
Laws of Indices
Master the laws of indices to multiply, divide and raise powers, and to handle zero, negative and fractional exponents with confidence.
OpenLaws of Surds
Simplify, multiply and add surds, and rationalise denominators, using the surd laws to keep answers in exact form.
OpenLaws of Logarithms
Use the laws of logarithms to expand and combine log expressions, evaluate logs and solve simple logarithmic equations.
OpenChapter 5 · Progressions
Arithmetic Progressions
Work with arithmetic progressions: find the common difference, the nth term and the sum of terms using the standard formulas.
OpenGeometric Progressions
Explore geometric progressions: find the common ratio, the nth term, the sum of terms and the sum to infinity when |r| < 1.
OpenChapter 6 · Linear Law
Linear and Non-Linear Relations
How to tell a linear relation from a non-linear one, and why converting to a straight line Y = mX + c makes analysis easy.
OpenLinear Law and Non-Linear Relations
Reduce equations like y = ax^n and y = ab^x to Y = mX + c, then find the constants from the straight-line graph.
OpenApplication of Linear Law
Use experimental data and a line of best fit to find constants in a non-linear law, then predict values.
OpenChapter 7 · Coordinate Geometry
Divisor of a Line Segment
Find the midpoint and the point dividing a segment in a given ratio using the section formula.
OpenParallel and Perpendicular Lines
Parallel lines share the same gradient; perpendicular lines have gradients whose product is −1.
OpenAreas of Polygons
Find the area of a triangle or polygon from its vertices using the shoelace (determinant) formula.
OpenEquations of Loci
A locus is the path of a point that moves under a given rule; find its equation using the distance formula.
OpenChapter 8 · Vectors
Vectors
A vector has both magnitude and direction, unlike a scalar. Learn notation, equal and negative vectors, and scalar multiplication.
OpenAddition and Subtraction of Vectors
Combine vectors using the triangle and parallelogram laws. Learn to find a resultant and to subtract by adding the negative.
OpenVectors in a Cartesian Plane
Write vectors in component form using i and j, compute magnitude with √(x²+y²), and find unit vectors.
OpenChapter 9 · Solution of Triangles
Sine Rule
The sine rule relates each side of a triangle to the sine of its opposite angle. Use it to find sides or angles in non-right triangles.
OpenCosine Rule
The cosine rule finds a side from two sides and the included angle, or an angle from all three sides.
OpenArea of Triangles
Find the area of any triangle using ½ab sin C when two sides and the included angle are known, or Heron's formula from three sides.
OpenApplication of Sine Rule, Cosine Rule and Area of a Triangle
Choose the right rule for real problems — bearings, heights and three-dimensional shapes — and combine them to solve triangles fully.
OpenChapter 10 · Index Numbers
Form 5
Chapter 1 · Circular Measure
Radian
Learn what a radian is and how to convert between degrees and radians using π rad = 180°.
OpenArc Length of a Circle
Use s = rθ to find the arc length of a circle when the angle is measured in radians.
OpenArea of Sector of a Circle
Use A = ½r²θ to find the area of a sector, and combine it to find segment areas.
OpenApplication of Circular Measures
Combine arc length, sector area and segment area to solve problems on composite figures.
OpenChapter 2 · Differentiation
Limit and Its Relation to Differentiation
Understand limits and how the gradient of a chord leads to the derivative dy/dx.
OpenThe First Derivative
Differentiate using the power rule, and apply the product, quotient and chain rules.
OpenThe Second Derivative
Find the second derivative and use it to test whether a turning point is a maximum or minimum.
OpenApplication of Differentiation
Apply differentiation to tangents, normals, rates of change and maximum/minimum problems.
OpenChapter 3 · Integration
Integration as the Inverse of Differentiation
Integration reverses differentiation: from a derivative we recover the original function, plus an arbitrary constant c.
OpenIndefinite Integral
The indefinite integral gives the general antiderivative of a function, always with an arbitrary constant c.
OpenDefinite Integral
A definite integral has limits and gives a numerical value using F(b) − F(a). No constant c is needed.
OpenApplication of Integration
Definite integrals give areas under curves, areas between curves, and volumes of revolution.
OpenChapter 4 · Permutation and Combination
Chapter 5 · Probability Distribution
Random Variable
A random variable assigns a number to each outcome. For a discrete variable, all probabilities sum to 1.
OpenBinomial Distribution
For n independent trials each with success probability p, X ~ B(n, p) and P(X=r) = ⁿCr pʳ(1−p)ⁿ⁻ʳ.
OpenNormal Distribution
A continuous X ~ N(μ, σ²) is standardised by Z = (X − μ)/σ to use the standard normal table.
OpenChapter 6 · Trigonometric Functions
Positive and Negative Angles
Measure angles from the positive x-axis: anticlockwise is positive, clockwise is negative. Convert between degrees and radians.
OpenTrigonometric Ratios of Any Angle
Use the CAST rule and the reference angle to find sine, cosine and tangent of any angle, keeping track of the correct sign in each quadrant.
OpenGraphs of Sine, Cosine and Tangent Functions
Read amplitude, period, maximum and minimum from y = a sin bx + c and similar forms, and count cycles or solutions in a given range.
OpenBasic Identities
The Pythagorean identities and the quotient identity let you simplify expressions and find one ratio from another.
OpenAddition Formulae and Double Angle Formulae
The addition formulae expand sin(A ± B), cos(A ± B) and tan(A ± B); the double angle formulae follow by setting B = A.
OpenApplication of Trigonometric Functions
Solve trigonometric equations within a given range and interpret models such as height that varies with a sine or cosine term.
OpenChapter 7 · Linear Programming
Linear Programming Model
Translate a word problem into decision variables, linear constraints (inequalities) and an objective function to be maximised or minimised.
OpenApplication of Linear Programming
Use the graphical method: shade the feasible region, then test the vertices to find where the objective function is greatest or least.
OpenChapter 8 · Kinematics of Linear Motion
Displacement, Velocity and Acceleration as a Function of Time
For motion along a straight line, displacement, velocity and acceleration are functions of time. Signs show direction; the particle is at rest when v = 0.
OpenDifferentiation in Kinematics of Linear Motion
Velocity is the derivative of displacement and acceleration is the derivative of velocity. Use v = 0 for maximum displacement and a = 0 for maximum velocity.
OpenIntegration in Kinematics of Linear Motion
Integration reverses differentiation: integrate acceleration to get velocity and velocity to get displacement, using initial conditions to fix the constant.
OpenApplication of Kinematics of Linear Motion
Combine differentiation and integration to solve motion problems: when a particle is at rest, its maximum speed or height, and total distance when direction changes.
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