MATSEC IM 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 p(x) is divided by (x − a), the remainder equals p(a); if p(a) = 0 then (x − a) is a factor.

Quadratic discriminant

For ax² + bx + c = 0, the discriminant b² − 4ac determines the roots: positive gives two distinct real roots, zero gives one repeated root, negative gives no real roots.

Sum and product of roots

If α and β are roots of ax² + bx + c = 0, then α + β = −b/a and αβ = c/a; these relations let you form a quadratic from its roots.

Binomial coefficient

C(n, r) = n!/(r!(n−r)!) is the coefficient of a^(n−r)b^r in the expansion of (a+b)^n and counts the number of ways to choose r objects from n.

Inverse function condition

A function f has an inverse only if it is one-to-one on its domain; the graph of f⁻¹ is the reflection of y = f(x) in the line y = x.

Composite function domain

For (f∘g)(x) = f(g(x)), x must lie in the domain of g and g(x) must lie in the domain of f; the range of g becomes input to f.

Horizontal line test

A graph represents a one-to-one function exactly when no horizontal line meets it more than once; such a function therefore has an inverse.

Change of base formula

log_a x = log_b x / log_b a allows logarithms in any base to be rewritten in another base, for example log_a x = ln x / ln a.

Solving a^x = b

Take logarithms of both sides to get x log a = log b, so x = log b / log a; the base used may be 10, e or any other.

Natural exponential inverse

y = e^x and y = ln x are inverse functions: e^(ln x) = x for x > 0 and ln(e^x) = x for all real x.

Geometric series sum to infinity

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

Arithmetic sequence nth term

The nth term of an arithmetic sequence with first term a and common difference d is u_n = a + (n − 1)d.

Sum of first n integers

The sum 1 + 2 + ... + n equals n(n + 1)/2, a standard result useful for summing arithmetic series and in induction proofs.

Cosine rule

In a triangle with sides a, b, c and angle A opposite side a, a² = b² + c² − 2bc cos A; it finds a side or an angle.

Double angle for sine

sin 2A = 2 sin A cos A; it converts products of sine and cosine into a single sine of twice the angle.

Sector area in radians

For a sector of radius r and angle θ measured in radians, the area is ½r²θ and the arc length is rθ.

Tangent function period

The graph of y = tan x has period π, passes through the origin, and has vertical asymptotes where cos x = 0, that is x = π/2 + nπ.

Perpendicular line gradients

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

Circle equation

The circle with centre (a, b) and radius r has equation (x − a)² + (y − b)² = r², obtained from the distance formula.

Distance from point to line

The perpendicular distance from (x₁, y₁) to the line ax + by + c = 0 is |ax₁ + by₁ + c| divided by the square root of a² + b².

Scalar product condition

Two non-zero vectors are perpendicular exactly when their scalar product a·b equals zero; this follows from a·b = |a||b| cos θ.

Unit vector

The unit vector in the direction of a is a/|a|, where |a| is the magnitude; it has length one and is used to specify direction.

Vector line equation

The line through the point with position vector a in direction d has equation r = a + t d, where t is a real parameter.

Modulus of complex number

For z = a + bi the modulus is |z| = √(a² + b²), the distance of the point from the origin on an Argand diagram.

De Moivre's theorem

For z = r(cos θ + i sin θ) and integer n, z^n = r^n(cos nθ + i sin nθ), which makes powers and roots of complex numbers manageable.

Complex conjugate properties

The conjugate of z = a + bi is a − bi; multiplying z by its conjugate gives the real number a² + b², equal to |z|².

Chain rule

If y is a function of u and u is a function of x, then dy/dx = (dy/du)(du/dx); this rule differentiates composite functions.

Second derivative test

At a stationary point where f'(x) = 0, f''(x) < 0 indicates a local maximum while f''(x) > 0 indicates a local minimum.

Quotient rule

For y = u/v with v ≠ 0, dy/dx = (v du/dx − u dv/dx)/v²; the order in the numerator matters and cannot be reversed.

Integration by parts

∫u dv = uv − ∫v du; choose u as the factor that simplifies when differentiated, often a polynomial or logarithm.

Area under a curve

The definite integral of f from a to b gives the signed area between the curve and the x-axis; regions below the axis count as negative.

Volume of revolution

Rotating y = f(x) about the x-axis between x = a and x = b gives volume V = π∫ y² dx, evaluated between those limits.