CSEC Additional 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). Dividing by (ax − b) gives remainder f(b/a).

Factor Theorem

(x − a) is a factor of polynomial f(x) if and only if f(a) = 0. Used to find unknown coefficients and fully factorise cubics.

Discriminant

For ax² + bx + c = 0, the discriminant b² − 4ac determines root type: positive gives two distinct real roots, zero gives equal roots, 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. Useful for forming equations from given roots.

Composite function fg(x)

fg(x) means f(g(x)): apply g first, then f to the result. The domain of fg is restricted to x values where g(x) lies in f's domain.

Inverse function

f⁻¹ reverses f, so f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. Its graph is the reflection of y = f(x) in the line y = x.

Laws of logarithms

log(xy) = log x + log y; log(x/y) = log x − log y; log xⁿ = n log x. These convert multiplication into addition and powers into coefficients.

Completing the square form

ax² + bx + c = a(x + b/2a)² + (c − b²/4a). The turning point of the curve y = ax² + bx + c is (−b/2a, c − b²/4a).

Distance and midpoint formulas

For points (x₁, y₁) and (x₂, y₂): distance = √((x₂ − x₁)² + (y₂ − y₁)²); midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2).

Perpendicular gradients

Two lines are perpendicular when the product of their gradients equals −1, so m₂ = −1/m₁. Parallel lines have equal gradients.

Equation of a straight line

Forms include y = mx + c and y − y₁ = m(x − x₁) through a known point. The intercept form x/a + y/b = 1 has intercepts a and b.

Circle equation

Centre (h, k), radius r: (x − h)² + (y − k)² = r². In x² + y² + 2gx + 2fy + c = 0 the centre is (−g, −f) and radius √(g² + f² − c).

Sine rule

In any triangle, a/sin A = b/sin B = c/sin C, where each side is opposite its matching angle. Use when given two angles or a side-angle pair.

Cosine rule

a² = b² + c² − 2bc cos A. Use when two sides and the included angle, or all three sides, are known; rearrange to find an angle.

Area of a triangle

Area = ½ab sin C, using two sides and the included angle. Requires the angle between the two chosen sides, not any other angle.

Pythagorean identity

sin²θ + cos²θ = 1 for all angles θ. Dividing by cos²θ gives 1 + tan²θ = sec²θ; used to simplify expressions and solve equations.

Amplitude and period

For y = a sin(bx) + c, amplitude is |a|, period is 360°/b (or 2π/b in radians), and c is the vertical shift of the curve.

Arc length and sector area

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

Derivative from first principles

f′(x) = lim(h→0) [f(x + h) − f(x)]/h. This limit gives the gradient of the tangent and the instantaneous rate of change.

Power rule

d/dx (axⁿ) = anxⁿ⁻¹ for any constant a and exponent n. Differentiate polynomials term by term; constants differentiate to zero.

Product rule

If y = uv, then dy/dx = u′v + uv′, where u′ and v′ are the derivatives of u and v with respect to x.

Quotient rule

If y = u/v with v ≠ 0, then dy/dx = (u′v − uv′)/v². The numerator order matters: derivative of top times bottom minus top times derivative of bottom.

Chain rule

d/dx [f(g(x))] = f′(g(x))·g′(x). For y = (ax + b)ⁿ the derivative is an(ax + b)ⁿ⁻¹.

Stationary points

Points where dy/dx = 0. Classify with d²y/dx²: negative means maximum, positive means minimum; zero requires a sign table test.

Tangent and normal gradients

At x = a the tangent gradient is f′(a); the normal gradient is −1/f′(a). Use y − y₁ = m(x − x₁) for either line.

Indefinite integration

∫axⁿ dx = axⁿ⁺¹/(n + 1) + C for n ≠ −1. Always include the constant of integration C.

Definite integral

∫ₐᵇ f(x) dx = F(b) − F(a), where F is an antiderivative of f. It gives the signed area between the curve and the x-axis.

Compound interest formula

A = P(1 + r/100)ⁿ gives the amount after n periods at r percent per period. Depreciation uses (1 − r/100)ⁿ instead.

Arithmetic progression

nth term = a + (n − 1)d and sum Sₙ = (n/2)[2a + (n − 1)d], where a is the first term and d the common difference.

Geometric progression

nth term = arⁿ⁻¹; sum Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1; sum to infinity S∞ = a/(1 − r) only when |r| < 1.

Addition rule for probability

P(A or B) = P(A) + P(B) − P(A and B). For mutually exclusive events the overlap is zero, so the probabilities simply add.

Linear programming vertices

After graphing the inequalities and shading the feasible region, evaluate the objective function at each vertex; the optimum occurs at a vertex.