MATSEC AM Pure Mathematics flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Remainder Theorem

When a polynomial f(x) is divided by (x − a), the remainder equals f(a). Setting f(a) = 0 gives the Factor Theorem, so (x − a) is a factor.

Discriminant rule

For ax^2 + bx + c = 0, the value b^2 − 4ac gives two distinct real roots if positive, one repeated root if zero, and no real roots if negative.

Modulus inequality |f(x)| < k

Equivalent to the double inequality −k < f(x) < k when k is positive. For |f(x)| > k, solve f(x) < −k or f(x) > k instead.

Sum of arithmetic series

For n terms with first term a and common difference d, S_n = (n/2)(2a + (n − 1)d), equivalently (n/2)(first + last).

Sum to infinity

A geometric series with first term a and ratio r converges only when |r| < 1, and its sum is a/(1 − r). Otherwise the sum diverges.

General binomial term

In the expansion of (a + b)^n, the term containing b^k is C(n, k) a^(n−k) b^k, where C(n, k) = n!/(k!(n−k)!).

Circle equation

A circle with centre (a, b) and radius r has equation (x − a)^2 + (y − b)^2 = r^2. Expanded, x^2 + y^2 + 2gx + 2fy + c = 0 has centre (−g, −f).

Perpendicular gradients

Two lines are perpendicular exactly when the product of their gradients equals −1, so a line with gradient m has perpendicular gradient −1/m.

Point-to-line distance

The perpendicular distance from (x0, y0) to the line ax + by + c = 0 is |ax0 + by0 + c| divided by √(a^2 + b^2).

Double angle formulae

sin 2A = 2 sin A cos A, and cos 2A = cos^2 A − sin^2 A, which also equals 1 − 2 sin^2 A or 2 cos^2 A − 1.

R sin(θ + α) form

Writing a sin θ + b cos θ as R sin(θ + α) uses R = √(a^2 + b^2) and tan α = b/a. This helps solve equations and find maxima.

Cosine rule

In a triangle with sides a, b, c opposite angles A, B, C: a^2 = b^2 + c^2 − 2bc cos A. It finds a side from two sides and the included angle.

Logarithm laws

For positive x and y: log(xy) = log x + log y, log(x/y) = log x − log y, and log(x^n) = n log x. These turn multiplication into addition.

Change of base

log_a b equals log_c b divided by log_c a for any valid base c. Choosing c = e gives log_a b = ln b / ln a.

Exponential decay half-life

If a quantity follows y = Ae^(−kt) with k positive, its half-life is t = (ln 2)/k, the time for y to fall to half its initial value.

Product rule

If y = uv, where u and v are functions of x, then dy/dx = u(dv/dx) + v(du/dx). Differentiate each factor and hold the other constant.

Quotient rule

If y = u/v, then dy/dx = (v du/dx − u dv/dx)/v^2. The denominator is squared and the order of subtraction matters.

Second derivative test

At a stationary point where dy/dx = 0, a negative second derivative indicates a local maximum and a positive second derivative indicates a local minimum.

Parametric gradient

For a curve given as x = f(t), y = g(t), the gradient is dy/dx = (dy/dt) divided by (dx/dt), provided dx/dt is not zero.

Volume of revolution

Rotating y = f(x) about the x-axis between x = a and x = b sweeps a volume V = π ∫ y^2 dx from a to b.

Integration by parts

The rule ∫ u dv = uv − ∫ v du reverses the product rule. Choose u to simplify on differentiation, often a power of x or a logarithm.

Trapezium rule

With strip width h = (b − a)/n, the integral is approximated by (h/2)[y0 + yn + 2(y1 + … + y_(n−1))], using equally spaced ordinates.

De Moivre's theorem

For z = r(cos θ + i sin θ), z^n = r^n(cos nθ + i sin nθ). This computes powers and roots of complex numbers in polar form.

Modulus and argument

For z = a + bi, the modulus is |z| = √(a^2 + b^2) and the argument is the angle whose tangent is b/a, chosen in the correct quadrant.

Conjugate pair roots

A polynomial equation with real coefficients has non-real roots in conjugate pairs: if a + bi is a root, then a − bi is also a root.

Scalar product formula

a·b = |a||b| cos θ = a1b1 + a2b2 + a3b3. It gives the angle between vectors, and a·b = 0 means the vectors are perpendicular.

Vector equation of a line

A line is written r = a + λd, where a is the position vector of a point on the line, d is a direction vector, and λ is a real parameter.

Equation of a plane

A plane has equation r·n = d, where n is a normal vector perpendicular to the plane and d fixes its distance from the origin.

Newton–Raphson formula

To solve f(x) = 0, iterate x_(n+1) = x_n − f(x_n)/f′(x_n). Convergence is fast near a root but fails where the derivative is small.

Sign-change condition

If f is continuous on [a, b] and f(a) and f(b) have opposite signs, then at least one root of f(x) = 0 lies between a and b.

Binomial distribution mean and variance

If X counts successes in n independent trials each with probability p, then E(X) = np and Var(X) = np(1 − p), with P(X = r) = C(n, r)p^r(1 − p)^(n−r).

Standardising a normal variable

For X ~ N(μ, σ^2), the value z = (x − μ)/σ follows the standard normal distribution with mean 0 and variance 1, allowing table use.