Loading...
Quick-reference guides and worked examples.
Quick tests to check whether a number is divisible by 2 through 12, with worked examples.
A quick-reference table of squares from 1 to 30 and cubes from 1 to 20.
Common percentage calculation shortcuts and mental math tricks with worked examples.
Practice questions on adding fractions and mixed numbers, with worked solutions.
What percentage means, how to compute a combined percentage across subjects, with a worked example.
The Pythagorean theorem explained, with a worked real-world distance example.
What a quadratic equation is, solving by completing the square, the quadratic formula, and a real-life example.
Sine, cosine, tangent and their reciprocals, standard angle values, and the core trigonometric identities.
The chain rule for differentiating composite functions, with two worked derivative examples.
What a definite integral is and how to evaluate one, with a worked trigonometric example.
How matrix addition and multiplication work, with a worked multiplication example.
Finding HCF using Euclid's division algorithm and prime factorization, plus verifying LCM x HCF = product of two numbers, with worked examples.
How the zeroes of a quadratic polynomial relate to its coefficients, plus building a polynomial from a given sum and product of zeroes.
Worked examples for checking whether an equation is quadratic, solving by factorization or completing the square, and using the discriminant.
Worked examples finding trigonometric ratios, evaluating standard-angle expressions, proving identities, and using co-function identities.
Proving a number is irrational by contradiction, and telling whether a fraction has a terminating or repeating decimal expansion, with worked examples.
Simplifying and rationalizing expressions with square roots, with worked examples including a telescoping sum.
Dividing one polynomial by another to find a quotient and remainder, and using that to find missing zeroes, with worked examples.
Using the Remainder Theorem to find a remainder without dividing, and factoring with the sum/difference-of-cubes identities, with worked examples.
The power rule for differentiating x to a fixed exponent, with a worked example on a square root.
The product rule for differentiating a product of two functions, with worked examples including logarithmic differentiation.
The quotient rule for differentiating one function divided by another, with worked examples.
Using derivatives as rates of change, with worked examples on a growing circle's area and a particle's velocity and acceleration.
How to differentiate an equation that isn't already solved for y, with worked examples including an inverse trig identity.
How the sign of a function's derivative determines where it's increasing or decreasing, with worked examples including a cubic.
Finding local maxima and minima and solving optimization problems using derivatives, with worked examples.
What Rolle's theorem states and how to verify it for a function on a closed interval, with a worked example.
How the derivative gives the tangent's slope and the normal's slope is its negative reciprocal, with a worked example.
Finding the slope of a tangent line to a curve at a given point, with worked examples on a cubic and a rational function.
Splitting a rational function into simpler fractions before integrating, with worked examples.
Standard integration formulas for functions like 1/(x^2-a^2) and 1/(a^2-x^2), with worked examples.
Rewriting a trigonometric integrand with an identity before integrating, with worked examples on cos^2 x, sin^2 x, and a secant identity.
A shortcut formula for integrating e^x times a function plus its own derivative, with a worked example.
Using definite integrals to find area under a curve, including a full period of cosine and a classic circle-area derivation.
What makes a sequence an arithmetic progression, and how to tell whether a real-world situation forms one, with worked examples.
What probability measures, independent vs dependent events, and conditional probability, with worked card, dice, and survey examples.
What a logarithm is, common vs natural vs binary logarithms, and the core logarithm rules, with worked examples.
What complex numbers are, the modulus-argument form, conjugates, and basic arithmetic on complex numbers, with worked examples.
How to count arrangements with permutations, with a worked dictionary-order example finding a specific word among all arrangements of a word's letters.
What a ratio is and how to simplify one to its lowest terms, with a worked real-world example.
How to check whether four numbers are in proportion, and how to solve for a missing term, with worked examples.
How to calculate profit and loss percent, and work backwards from a selling price or discount to find the cost price, with worked examples.
How to solve a one-variable linear equation, including fractional coefficients, with worked examples translating word problems into equations.
How to form a differential equation from a family of curves, and how to solve one by separating variables, with worked examples.
The integration by parts formula for integrating a product of two functions, with a worked example.
Left-hand and right-hand derivatives, spotting a point that's continuous but not differentiable, and finding a function's domain, with worked examples.
Diagonal, scalar, and identity matrices explained, with examples showing how each one is a more specific case of the last.
What symmetric and skew-symmetric matrices are, and how to split any square matrix into a sum of both, with a worked example.
Key facts about square matrices: the determinant scaling rule, idempotent matrices, and when a matrix equation has no solution.
How to solve a system of linear equations using the matrix method, with a fully worked three-variable example.
Solving for unknown entries in a matrix using a stated property or a matching determinant, with worked examples.
The negative of a vector, the scalar triple product identity, and using the dot product on unit vectors, with worked examples.
How to find the unit vector in a given direction, between two points, or scaled to a target magnitude, with worked examples.
What the projection of a vector onto a directed line means, including what happens at special angles, with a worked example.
The section formula for a point dividing a segment internally or externally, and finding a midpoint, with worked examples.
The differences between a line segment, a ray, and a line, and what makes points or lines collinear or concurrent.
What direction cosines are and how to find them for a coordinate axis, a line through two points, or a line at given angles.
Writing a line's equation in vector and Cartesian form, given two points or a point and a parallel vector, with worked examples.
The equation of a coordinate plane, and finding a plane through the intersection of two others and a given point, with a worked example.
How to read a plane's intercepts on each axis straight from its equation, with a worked example.
How to find the angle between a line and a plane using their direction vector and normal vector, with a worked example.
The formula for the shortest distance between two skew lines in vector form, with two fully worked examples.
The determinant formula for the area of a triangle from its vertex coordinates, with a worked example.
Using the cross product to find the area of a triangle or parallelogram from vertex or side vectors, with worked examples.
Reflexive, symmetric, transitive, and equivalence relations defined and checked against worked examples.
One-one, onto, bijective, and invertible functions explained, with worked examples on checking each property and finding an inverse.
What a binary operation is, and how to check whether one has an identity element, is commutative, or is associative, with worked examples.
How to compute the composition of two functions in either order, with a worked example showing gof and fog can come out different.
Using differentials to approximate square roots and estimate small changes, with worked examples including a percentage-change setup.
Feasible regions, corner points, and finding the condition on an objective function's coefficients for it to be maximized at two corners at once.
Principal values, domains of inverse trig functions, and the complementary identity between tan⁻¹ and cot⁻¹, with worked examples.
Turning a word problem into a pair of linear equations, and checking whether two lines intersect, are parallel, or coincide by comparing coefficient ratios.
Solving a pair of linear equations by substitution and by elimination, including recognizing when a system has infinitely many solutions.
The cross-multiplication formula for solving a pair of linear equations, and reducing an equation that isn't linear yet into one that is, with worked examples.
Translating age and money word problems into a pair of linear equations and solving them, including a classic problem with two valid cases.
The simple interest and compound interest formulas, why compound interest grows faster, and worked examples for both.
The core rules for combining and simplifying exponents, including negative exponents, with worked examples.
Five worked Math Olympiad-style problems for grade 7: a Fibonacci-like sequence, an exponent equation, percentages with overlap, consecutive integers, and clock arithmetic.
The formula for counting how many divisors a number has from its prime factorization, with worked examples.
Union, intersection, and the inclusion-exclusion principle for counting overlapping groups, with a worked example.
Area and perimeter formulas for rectangles, squares, circles, sectors, triangles, and regular pentagons, including Heron's formula, with a worked example.
Surface area and volume formulas for cuboids, cubes, spheres, cylinders, cones, and frustums of a cone, with a worked example.
What Euler's number, the imaginary unit, Pythagoras' constant, and pi actually represent, including the repeating cycle of powers of i, with a worked example.
What a radian actually measures, the standard-angle conversion table, and converting between degrees and radians, with worked examples.
What makes an expression a polynomial, the monomial/binomial/trinomial distinction, and how to find the degree of a polynomial in one or more variables.
What a prime number is, why 0 and 1 are neither prime nor composite, and what twin primes are, with a worked example checking a number for primality.
The Laplace transform definition, a table of standard transforms, the first shifting theorem, and solving a linear ODE with initial conditions.
The Fourier series formulas for a periodic function, and a fully worked derivation of the Fourier series for f(x) = x on (-π, π).
The path-difference conditions for bright and dark fringes, the fringe width formula, and worked examples finding fringe width and fringe position.
Kirchhoff's Current Law and Voltage Law explained, with a fully worked two-loop circuit solved for all three branch currents.
How a PN junction diode behaves under forward and reverse bias, the diode equation, and a worked example on the 'roughly 60mV per decade' current rule.
The characteristic equation for finding eigenvalues and eigenvectors, with a fully worked 2x2 example verified two ways: by substitution and by the trace/determinant check.
How to take partial derivatives of a multivariable function, with a worked example that also verifies equality of mixed second partial derivatives by direct computation.
The gradient, divergence, and curl operators explained, with worked examples computing each directly from a scalar and a vector field.
Why single-slit diffraction minima and double-slit interference maxima use the exact same-looking formula for opposite reasons, plus a worked diffraction grating example.
Einstein's photoelectric equation, the work function and threshold frequency, and a worked example finding the maximum kinetic energy of emitted electrons.
The de Broglie relation between a particle's momentum and its wavelength, and a worked example deriving the wavelength of an electron accelerated through a known voltage.
How to reduce any linear two-terminal network to a single source and resistor, with a worked example finding the Thevenin and Norton equivalents of the same circuit.
Why a transistor needs a DC bias point before it can amplify, the fixed-bias circuit equations, and a fully worked example finding the operating point.
The ideal op-amp assumptions, the inverting and non-inverting amplifier gain formulas, and worked examples computing the output voltage for each configuration.