LC Ordinary Maths flashcards

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

Quadratic formula

For ax² + bx + c = 0 with a ≠ 0, the roots are x = (−b ± √(b² − 4ac)) / 2a. It solves any quadratic, including ones that do not factorise.

Discriminant test

The value b² − 4ac from ax² + bx + c = 0 decides the nature of the roots: positive gives two real roots, zero gives one repeated root, negative gives no real roots.

Root sum and product

If α and β are the roots of ax² + bx + c = 0, then α + β = −b/a and αβ = c/a. These link coefficients to roots without solving.

Factor theorem

A polynomial f(x) has factor (x − k) exactly when f(k) = 0. More generally, dividing f(x) by (x − k) leaves remainder f(k).

Laws of logarithms

log(ab) = log a + log b, log(a/b) = log a − log b, and log(a^n) = n log a. These turn multiplication into addition and powers into multipliers.

Derivative from first principles

The derivative is the limit f′(x) = lim(h→0) [f(x + h) − f(x)] / h. It measures the slope of the tangent to the curve at x.

Power rule

For any constant a and index n, d/dx (ax^n) = anx^(n−1). Differentiate polynomials term by term; the derivative of a constant is zero.

Chain rule

For a composite function, d/dx f(g(x)) = f′(g(x)) · g′(x). Differentiate the outer function, then multiply by the derivative of the inner function.

Stationary point classification

At a stationary point f′(x) = 0. If f″(x) < 0 it is a local maximum; if f″(x) > 0 it is a local minimum; if f″(x) = 0 the test fails.

Definite integral

The definite integral from a to b of f(x) dx equals F(b) − F(a), where F is any antiderivative of f. It gives the signed area between curve and x-axis.

Area between two curves

Find intersection x-values, then integrate (upper curve − lower curve) between them. The result is positive when the upper curve is correctly identified.

Distance and midpoint formulas

Distance between two points is √((x₂ − x₁)² + (y₂ − y₁)²); midpoint is the average of the coordinates: ((x₁ + x₂)/2, (y₁ + y₂)/2).

Perpendicular line slopes

Two lines are perpendicular exactly when their slopes multiply to −1, so the perpendicular slope is the negative reciprocal. Parallel lines have equal slopes.

Circle equation

A circle with centre (h, k) and radius r has equation (x − h)² + (y − k)² = r². Expanding gives x² + y² + 2gx + 2fy + c = 0 with centre (−g, −f).

Tangent to a circle condition

Substitute the line equation into the circle equation. The resulting quadratic has two real roots for a secant, one repeated root for a tangent, and no real roots otherwise.

Rationalising a denominator

Multiply numerator and denominator by the conjugate of the denominator. For example, 1/(√a + √b) = (√a − √b)/(a − b), which removes surds from the bottom.

Addition rule of probability

P(A ∪ B) = P(A) + P(B) − P(A ∩ B). For mutually exclusive events the intersection is empty, so the probabilities simply add.

Conditional probability

P(A ∩ B) = P(A) · P(B|A), where P(B|A) is the probability of B given A has occurred. For independent events this becomes P(A) · P(B).

Combinations formula

The number of ways to choose r objects from n distinct objects, order irrelevant, is nCr = n! / (r!(n − r)!). Arrangements where order matters use nPr = n!/(n − r)!.

Standard deviation

Standard deviation measures spread about the mean: σ = √(Σ(x − x̄)²/n) for a population. A larger value means data are more widely dispersed.

Correlation coefficient

The correlation coefficient r lies between −1 and 1. Values near 1 show strong positive linear correlation, near −1 strong negative, and near 0 weak linear correlation.

Sine rule

In any triangle, a/sin A = b/sin B = c/sin C. Use it when you know a side and its opposite angle together with one further side or angle.

Cosine rule

a² = b² + c² − 2bc cos A. Use it for two sides with the included angle, or for three sides; rearranged, cos A = (b² + c² − a²)/(2bc).

Triangle area formula

Area = ½ab sin C, where a and b are two sides and C is the angle between them. It works for any triangle, not only right-angled ones.

Arc length and sector area

With angle θ in radians and radius r: arc length s = rθ and sector area A = ½r²θ. Convert degrees to radians by multiplying by π/180.

Pythagorean identity

For any angle θ, sin²θ + cos²θ = 1, and tan θ = sin θ / cos θ wherever cos θ ≠ 0. These identities simplify trigonometric expressions and equations.

Arithmetic sequence

The nth term is Tₙ = a + (n − 1)d, where a is the first term and d the common difference. The sum of n terms is Sₙ = n/2 (first + last).

Geometric series sum

For first term a and ratio r ≠ 1, Sₙ = a(1 − r^n)/(1 − r). The sum to infinity exists only when |r| < 1, and then S∞ = a/(1 − r).

Compound interest formula

A = P(1 + i)^n gives the amount A after n periods when principal P grows at rate i per period, written as a decimal. It models repeated percentage increase.

Present value

The present value of an amount A due in n periods at interest rate i per period is A/(1 + i)^n. It is the money that would grow to A.

Cyclic quadrilateral theorem

In a cyclic quadrilateral, opposite angles sum to 180°. Also, the angle in a semicircle is a right angle, and angles standing on the same arc are equal.

Congruence tests

Triangles are congruent by SSS, SAS, ASA (two angles and a corresponding side), or RHS for right triangles. Congruent triangles have all corresponding sides and angles equal.