NSSCAS Mathematics (8227) flashcards

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

Discriminant of a quadratic

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

Change of base formula

log_a x = log_b x / log_b a. It converts a logarithm from one base to another, allowing evaluation with a calculator that offers only base 10 or base e.

Remainder theorem

When polynomial f(x) is divided by (x−a), the remainder equals f(a). If f(a) = 0 then (x−a) is a factor of f(x).

General term of binomial expansion

In (a+b)^n the term containing b^r is C(n,r)·a^(n−r)·b^r, where C(n,r) = n!/(r!(n−r)!). Setting r = 0 gives the first term.

Composite function domain

fg(x) = f(g(x)) is defined only for x in the domain of g whose images g(x) lie in the domain of f. Both conditions must hold.

Inverse function

f⁻¹ reverses f: if f(a) = b then f⁻¹(b) = a. It exists only when f is one-to-one; its domain is the range of f and its graph is f reflected in y = x.

Transformation y=f(x+a)

Shifts the graph of y = f(x) horizontally by a units in the negative direction: a > 0 moves it left, a < 0 moves it right. The shape is unchanged.

Horizontal line test

A function has an inverse function only if every horizontal line meets its graph at most once, meaning the function is one-to-one on its domain.

Perpendicular line gradients

Two straight lines are perpendicular exactly when the product of their gradients is −1, so m₂ = −1/m₁. A horizontal line is perpendicular to a vertical line.

Equation of a circle

(x−a)²+(y−b)² = r² has centre (a,b) and radius r. Expanded as x²+y²+2gx+2fy+c = 0 the centre is (−g,−f) and radius √(g²+f²−c).

Distance from point to line

For the line ax+by+c = 0 and the point (x₀,y₀), the perpendicular distance is |ax₀+by₀+c|/√(a²+b²). The absolute value makes the result non-negative.

Arc length in radians

For a circle of radius r, an arc subtending angle θ radians at the centre has length s = rθ. The angle must be in radians, not degrees.

Sector area in radians

The area of a sector with radius r and angle θ radians is ½r²θ. This equals the fraction θ/(2π) of the whole circle area πr².

Double angle for cosine

cos2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ. Choose the form that matches the other terms in the equation being solved.

Period of tan θ

tanθ repeats every 180° or π radians, so tanθ = k has solutions θ = α + 180°n. Its graph has vertical asymptotes where cosθ = 0, at 90° + 180°n.

Sum of geometric series

For first term a and common ratio r ≠ 1, S_n = a(1−r^n)/(1−r). If |r| < 1 the sum to infinity is a/(1−r).

nth term of an AP

An arithmetic progression with first term a and common difference d has nth term a+(n−1)d. The sum of n terms is n/2(2a+(n−1)d).

Convergence of a geometric series

A geometric series converges to a finite sum only when the common ratio satisfies |r| < 1; otherwise the terms do not shrink to zero and the series diverges.

Chain rule

If y is a function of u and u is a function of x, then dy/dx = (dy/du)(du/dx). It is used for composite functions such as (3x+1)^5.

Quotient rule

For y = u/v, dy/dx = (v·du/dx − u·dv/dx)/v². The denominator is squared, and the order of the subtraction in the numerator matters.

Second derivative test

At a stationary point where dy/dx = 0, d²y/dx² > 0 indicates a minimum and d²y/dx² < 0 indicates a maximum; d²y/dx² = 0 is inconclusive.

Normal to a curve

The normal at a point is perpendicular to the tangent there, so its gradient is −1/m where m is the tangent gradient. Use the point to find its equation.

Volume of revolution

Rotating y = f(x) about the x-axis between x = a and x = b sweeps a volume V = π∫y² dx from a to b. About the y-axis use π∫x² dy.

Definite integral evaluation

If F is an antiderivative of f, then the integral of f(x) from a to b equals F(b) − F(a). The constant of integration cancels, so it is not written.

Area between two curves

For curves where f(x) ≥ g(x) between a and b, the enclosed area is ∫(f(x) − g(x)) dx from a to b. Intersections give the limits.

Reverse chain rule

∫f'(x)[f(x)]^n dx = [f(x)]^(n+1)/(n+1) + C for n ≠ −1. It recognises an integrand that is a derivative multiplied by a power.

Combinations formula

The number of ways to choose r objects from n distinct objects, with order irrelevant, is nCr = n!/(r!(n−r)!). Order-relevant selections use nPr instead.

Variance of discrete random variable

Var(X) = E(X²) − [E(X)]². Compute E(X) and E(X²) using ΣxP(X=x) and Σx²P(X=x), then subtract the square of the mean.

Standardising a normal variable

If X is normally distributed with mean μ and variance σ², then z = (x−μ)/σ follows the standard normal distribution N(0,1). Probabilities come from this z value.

Scalar product

a·b = |a||b|cosθ = x₁x₂ + y₁y₂ + z₁z₂, where θ is the angle between the vectors. The result is a scalar, not a vector.

Unit vector

A unit vector has magnitude 1 and points in the same direction as the original vector. It is obtained by dividing the vector by its magnitude: â = a/|a|.

Condition for perpendicular vectors

Two non-zero vectors are perpendicular exactly when their scalar product is zero, a·b = 0, because cos90° = 0 in the scalar product formula.