Additional Mathematics
Form 5 · 29 topics available
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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