Cambridge IAL Mathematics cheat sheet

For a quadratic ax² + bx + c = 0, the discriminant Δ = b² − 4ac determines the number of real roots: Δ > 0 gives two distinct real roots, Δ = 0 gives one repeated root, Δ < 0 gives no real roots.

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

For a function f(x), the derivative from first principles is f'(x) = lim_{h→0} [f(x+h) − f(x)]/h.

The chain rule: if y = f(g(x)), then dy/dx = f'(g(x)) · g'(x). The product rule: d(uv)/dx = u dv/dx + v du/dx. The quotient rule: d(u/v)/dx = (v du/dx − u dv/dx)/v².

Integration by parts: ∫ u dv = uv − ∫ v du. Choose u according to LIATE (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential) to simplify the integral.

For a curve given parametrically by x = f(t), y = g(t), the gradient dy/dx = (dy/dt)/(dx/dt), and the area under the curve from t = a to t = b is ∫_a^b g(t) f'(t) dt.

The volume of revolution when the curve y = f(x) is rotated about the x-axis through 2π radians is V = π ∫_a^b [f(x)]² dx.

The modulus–argument form of a complex number: 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).

Newton's second law for a particle of constant mass: F_net = ma, where F_net is the vector sum of all forces acting on the particle.

For motion in a straight line with constant acceleration a, use the suvat equations: v = u + at, s = ut + ½at², v² = u² + 2as, s = ½(u + v)t.

The moment of a force about a point is the product of the force magnitude and the perpendicular distance from the point to the line of action of the force: moment = F × d.

For a particle on an inclined plane at angle θ to the horizontal, the component of weight parallel to the plane is mg sin θ, and perpendicular to the plane is mg cos θ.

The coefficient of friction μ satisfies F_friction ≤ μR, where R is the normal reaction; at the point of sliding, F_friction = μR.

For a projectile launched with speed u at angle θ above the horizontal, horizontal motion: x = (u cos θ)t; vertical motion: y = (u sin θ)t − ½gt².

The work done by a constant force F moving a particle through displacement d in the direction of the force is W = Fd. The kinetic energy of a particle is ½mv², and the work–energy principle states that net work done = change in kinetic energy.

For a system of particles, the centre of mass x̄ = (Σ m_i x_i)/(Σ m_i), and for a uniform lamina the centre of mass can be found by symmetry or integration.

For two events A and B, the addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). If A and B are mutually exclusive, P(A ∩ B) = 0.

For conditional probability, P(A | B) = P(A ∩ B)/P(B), provided P(B) > 0. Events A and B are independent if P(A ∩ B) = P(A)P(B).

The binomial distribution B(n, p): P(X = r) = C(n, r) p^r (1−p)^{n−r}, with mean np and variance np(1−p).

The normal distribution N(μ, σ²): the standard normal variable Z = (X − μ)/σ has mean 0 and variance 1. Use standard normal tables to find probabilities.

For a random sample of size n from a normal population with mean μ and variance σ², the sample mean X̄ is normally distributed with mean μ and variance σ²/n.

The product moment correlation coefficient r measures the strength of linear association between two variables; it satisfies −1 ≤ r ≤ 1.

The equation of the least squares regression line of y on x is y = a + bx, where b = S_xy/S_xx and a = ȳ − b x̄, with S_xy = Σ(x_i − x̄)(y_i − ȳ) and S_xx = Σ(x_i − x̄)².

For a discrete random variable X, the expectation E(X) = Σ x P(X = x), and the variance Var(X) = E(X²) − [E(X)]².