Cambridge IAL Mathematics flashcards

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

Discriminant

For ax² + bx + c = 0, Δ = b² − 4ac. It determines the nature of the roots: Δ > 0 two real, Δ = 0 one repeated, Δ < 0 no real roots.

Binomial expansion

(a + b)ⁿ = Σ_{r=0}^{n} C(n, r) a^{n−r} b^r for positive integer n, where C(n, r) = n!/(r!(n−r)!).

Derivative from first principles

f'(x) = lim_{h→0} [f(x+h) − f(x)]/h, provided the limit exists.

Chain rule

If y = f(g(x)), then dy/dx = f'(g(x)) · g'(x).

Product rule

If y = uv, then dy/dx = u dv/dx + v du/dx.

Quotient rule

If y = u/v, then dy/dx = (v du/dx − u dv/dx)/v².

Integration by parts

∫ u dv = uv − ∫ v du. Choose u using LIATE to simplify the integral.

Parametric differentiation

For x = f(t), y = g(t), dy/dx = (dy/dt)/(dx/dt) = g'(t)/f'(t).

Volume of revolution

When y = f(x) is rotated about the x-axis, V = π ∫_a^b [f(x)]² dx.

Modulus–argument form

z = r(cos θ + i sin θ) = r e^{iθ}, where r = |z| and θ = arg(z).

De Moivre's theorem

[r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ) for any integer n.

Newton's second law

For a particle of constant mass, the net force equals mass times acceleration: F_net = ma.

suvat equations

For constant acceleration a: v = u + at, s = ut + ½at², v² = u² + 2as, s = ½(u + v)t.

Moment of a force

The moment of a force F about a point is F × d, where d is the perpendicular distance from the point to the line of action of the force.

Inclined plane components

For a particle on a plane inclined at θ to the horizontal, weight components are mg sin θ parallel to the plane and mg cos θ perpendicular to it.

Friction inequality

The friction force F satisfies F ≤ μR, where μ is the coefficient of friction and R is the normal reaction. At sliding, F = μR.

Projectile motion

For initial speed u at angle θ: x = (u cos θ)t and y = (u sin θ)t − ½gt², taking g ≈ 9.8 m s⁻².

Work–energy principle

The net work done by forces on a particle equals the change in its kinetic energy: W_net = ½mv² − ½mu².

Centre of mass formula

For discrete masses m_i at positions x_i, the centre of mass x̄ = (Σ m_i x_i)/(Σ m_i).

Addition rule

For events A and B, P(A ∪ B) = P(A) + P(B) − P(A ∩ B). If mutually exclusive, P(A ∩ B) = 0.

Conditional probability

P(A | B) = P(A ∩ B)/P(B), provided P(B) > 0.

Independent events

Events A and B are independent if P(A ∩ B) = P(A)P(B).

Binomial distribution

For X ~ B(n, p), P(X = r) = C(n, r) p^r (1−p)^{n−r}, with mean np and variance np(1−p).

Normal distribution

For X ~ N(μ, σ²), standardise to Z = (X − μ)/σ ~ N(0, 1) to use standard normal tables.

Distribution of sample mean

If X ~ N(μ, σ²), then for a sample of size n, X̄ ~ N(μ, σ²/n).

Product moment correlation

The product moment correlation coefficient r measures linear association, with −1 ≤ r ≤ 1.

Regression line

The least squares regression line of y on x is y = a + bx, where b = S_xy/S_xx and a = ȳ − b x̄.

Expectation of discrete variable

For a discrete random variable X, E(X) = Σ x P(X = x).

Variance of discrete variable

Var(X) = E(X²) − [E(X)]².

Permutations and combinations

The number of permutations of r items from n is nPr = n!/(n−r)!; combinations nCr = n!/(r!(n−r)!).

Standard deviation

The standard deviation is the positive square root of the variance: σ = √Var(X).

Poisson distribution

For X ~ Po(λ), P(X = r) = e^{−λ} λ^r / r!, with mean λ and variance λ.