HKDSE Maths M2 key terms

A proof method for statements about positive integers: show the statement holds for n = 1, then show that assuming it holds for n = k forces it to hold for n = k + 1.

The expansion of (a + b)^n as a sum of terms involving binomial coefficients; in M2 it is used with positive integral indices and, for general indices, with the condition |x| < 1.

The number C(n, r) = n! / (r!(n − r)!), counting the ways to choose r objects from n; it is the coefficient of a^(n−r) b^r in the binomial expansion.

The expression for the (r + 1)th term of a binomial expansion, namely C(n, r) a^(n−r) b^r, used to pick out one particular coefficient without expanding everything.

A rational function rewritten as a sum of simpler fractions with linear or quadratic denominators; the form of decomposition depends on whether factors are distinct, repeated, or irreducible.

An equation true for all values of the variable; to find unknown constants in a partial fraction decomposition, substitute convenient values or compare coefficients.

A technique for finding unknown constants by equating the coefficients of like powers of x on the two sides of an identity, often used together with substitution.

When a polynomial f(x) is divided by (x − a), the remainder equals f(a); it gives a quick way to evaluate a polynomial or test divisibility.

The statement that (x − a) is a factor of the polynomial f(x) exactly when f(a) = 0; it converts factor questions into root questions.

Rewriting a polynomial as divisor × quotient + remainder; long division or synthetic division is used to lower the degree before solving equations or decomposing fractions.

A function expressible as the quotient of two polynomials; its domain excludes values making the denominator zero, which may produce vertical asymptotes.

A rule expressing each term of a sequence in terms of earlier terms, such as u(n+1) = 2u(n) + 3; solving it means finding an explicit formula for the nth term.

The sum of the first n terms of a geometric sequence with first term a and common ratio r is a(1 − r^n)/(1 − r) for r ≠ 1, valid also when |r| < 1 in the infinite case.

For a geometric series with |r| < 1, the limiting value a/(1 − r) that the partial sums approach as the number of terms grows without bound.

A number of the form a + bi with a, b real and i² = −1; the real part a and imaginary part b locate it in the Argand plane.

For z = a + bi, the conjugate is a − bi, written z̄; multiplying z by z̄ gives the real number a² + b², useful for division and modulus.

The modulus |z| is the distance of z from the origin, and the argument is the angle its position vector makes with the positive real axis, measured in radians.

Writing a complex number as r(cos θ + i sin θ), where r is the modulus and θ the argument; it makes multiplication, division, and powers easy to compute.

The result that [r(cos θ + i sin θ)]^n = r^n(cos nθ + i sin nθ) for integer n, used to raise complex numbers to powers and extract roots.

The n distinct solutions of z^n = w, equally spaced around a circle in the Argand plane and found by adding multiples of 2π/n to the argument.

The coordinate plane used to picture complex numbers, with the real part plotted horizontally and the imaginary part vertically.

The set of points in the plane satisfying a given geometric condition; for complex z, conditions such as |z − z₀| = k describe circles or perpendicular bisectors.

A rectangular array of numbers; in M2, matrices are added, multiplied, and inverted, and square matrices may have determinants and satisfy equations of their own.

A number assigned to a square matrix; a 2 × 2 determinant ad − bc is nonzero exactly when the matrix is invertible, and determinants expand along rows or columns.

The matrix A⁻¹ satisfying AA⁻¹ = A⁻¹A = I; it exists only when det A ≠ 0 and is used to solve matrix equations and linear systems.

A square matrix whose determinant is zero, so it has no inverse; its rows are linearly dependent and it maps distinct vectors to the same image.

Several equations in the same unknowns, solved by elimination or by matrix methods; the number of solutions depends on whether the coefficient matrix is singular.

The value a function or sequence approaches as the variable approaches a point or grows without bound; limits underlie continuity, derivatives, and convergence of series.

A function is continuous at a point when its limit there equals its value there; on an interval this means no breaks, jumps, or holes in the graph.

The instantaneous rate of change of a function, defined as the limit of a difference quotient; geometrically it is the slope of the tangent to the curve.

A function is differentiable at a point when its derivative exists there; differentiability implies continuity, but continuity does not guarantee differentiability, as with |x| at 0.

The rule for differentiating a composite function: dy/dx = dy/du · du/dx, essential for functions such as sin(3x²) or e^(2x+1).

Rules for differentiating products and quotients: (uv)′ = u′v + uv′ and (u/v)′ = (u′v − uv′)/v², applied repeatedly in longer expressions.

Differentiating both sides of an equation relating x and y with respect to x, treating y as a function of x, then solving for dy/dx.

The derivative of the derivative, written d²y/dx²; its sign indicates concavity and it is used to classify stationary points.

A point where dy/dx = 0; the second derivative test or a sign table of the first derivative decides whether it is a maximum, minimum, or point of inflection.

A point where the concavity of a curve changes sign; the second derivative is zero or undefined there, though not every such point is an inflection point.

The family of antiderivatives of a function, written with an arbitrary constant + C; differentiating the result returns the original integrand.

The limit of a sum of areas under a curve between two limits; by the fundamental theorem it equals F(b) − F(a) for any antiderivative F.

Reversing the chain rule: replace a composite expression by a new variable u, convert dx to du, integrate, then substitute back or change the limits.

The rule ∫u dv = uv − ∫v du, used for products such as x·e^x or x·ln x; choosing u and dv well keeps the remaining integral simpler.

The definite integral of a function over an interval; regions below the x-axis contribute negative values, so areas are split at the x-intercepts and signs corrected.

The volume swept out when a curve is rotated about an axis, computed as π∫y² dx about the x-axis or π∫x² dy about the y-axis.