H2 Further Mathematics revision reference

This lookup covers the parts of 2027 H2 Further Mathematics (9649) that have lessons here so far. It is not a complete syllabus summary. Examination instructions determine which formulae are supplied.

Complex numbers in polar form

For z ≠ 0 with modulus r > 0 and principal argument -π < θ ≤ π:

• Product: r₁e^iθ₁ · r₂e^iθ₂ = r₁r₂e^(i(θ₁ + θ₂)). Multiplying by z₂ scales by r₂ and rotates anticlockwise through θ₂.
• Quotient: r₁e^iθ₁/(r₂e^iθ₂) = (r₁/r₂)e^(i(θ₁-θ₂)).
• De Moivre, for every integer n: (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ.
• Roots: zⁿ = Re^iφ has the n roots R^(1/n)e^(i(φ + 2kπ)/n), k = 0,1,…,n-1, a turn of 2π/n apart.
• With z = e^iθ: zⁿ + z⁻ⁿ = 2 cos nθ and zⁿ-z⁻ⁿ = 2i sin nθ.

After adding or multiplying arguments, return to the principal interval by adding or subtracting multiples of 2π.

Read more: Polar form and de Moivre’s theorem.

Loci in the Argand diagram

• |z-c| = r: circle, centre c, radius r; ≤ includes the inside, < only the inside.
• |z-a| = |z-b|: the perpendicular bisector of a and b.
• arg(z-a) = α: the half-line from a in direction α, excluding a.

Read more: Loci in the Argand diagram.

Polar curves

x = r cos θ, y = r sin θ, with r ≥ 0 on the stated interval.

Integrate over an interval that traces each part of the curve once.

Read more: Polar curves, area and arc length.

Eigenvalues and diagonalisation

• Mv = λv with v ≠ 0. Eigenvalues solve det(M-λI) = 0.

• For a 2 × 2 matrix, the eigenvalues add to the trace and multiply to the determinant.

• A triangular matrix has its diagonal entries as eigenvalues.

• If the eigenvectors in the columns of Q are independent and D lists their eigenvalues in the same order, then M = QDQ⁻¹ and Mⁿ = QDⁿQ⁻¹.

• For 2 × 2: (a, b; c, d)⁻¹ = 1/(ad-bc) (d, -b; -c, a) when ad-bc ≠ 0.

• Row operations (swap, nonzero multiple, add a multiple of another row) keep the solution set. In echelon form, a row 0 = k with k ≠ 0 means no solution; free unknowns give infinitely many.

• Rank = number of nonzero rows in echelon form; for an n × n matrix, rank + nullity = n. Ax = b is solvable exactly when b is in the column space.

Read more: Row reduction, rank and null space and Eigenvalues, eigenvectors and diagonalisation.

Second-order linear differential equations

For y'' + ay' + by = f(x) with real a, b, the auxiliary equation is m² + am + b = 0.

General solution = complementary function + particular integral. Trial for the particular integral:

• polynomial of degree n → polynomial of degree n (one degree higher if b = 0);
• pe^kx → Ce^kx;
• p cos kx + q sin kx → C cos kx + D sin kx.

If the trial is already in the complementary function, multiply it by x (by x² for a repeated root). Apply initial conditions to the whole general solution.

Read more: Second-order linear differential equations.

Recurrence relations

• uₙ = auₙ₋₁ + b, a ≠ 1: L = b/(1-a) and uₙ = L + (u₀-L)aⁿ. If a = 1, uₙ = u₀ + nb.
• uₙ = puₙ₋₁ + quₙ₋₂: solve λ² = pλ + q. Distinct roots: Aλ₁ⁿ + Bλ₂ⁿ; repeated: (A + Bn)λⁿ; complex ρ e^(± iθ): ρⁿ(C cos nθ + D sin nθ).

Read more: Linear recurrence relations.

Numerical methods for roots

• A sign change of a continuous function on [a,b] locates a root.
• Linear interpolation: x = a + ((-f(a))/(f(b)-f(a)))(b-a).
• Newton–Raphson: xₙ₊₁ = xₙ-f(xₙ)/(f'(xₙ)); it fails where f'(xₙ) = 0 and can cycle or reach a different root.
• xₙ₊₁ = F(xₙ) converges near α when |F'(α)| < 1.

Read more: Locating and approximating roots.

Poisson and geometric distributions

Independent Po(λ) and Po(μ) add to Po(λ + μ). For Geo(p), P(X > x) = (1-p)^x, and the distribution is memoryless.

Read more: Poisson and geometric models.

Applications of integration

Read more: Arc length, volumes and surface area.

Functions of two variables

• ∇ f = (fₓ,fy); Dᵤf = ∇ f · u for a unit vector u.
• Tangent plane at (a,b): z = f(a,b) + fₓ(x-a) + fy(y-b).
• At a stationary point, D = fₓₓf_yy-f_xy²: D > 0, fₓₓ > 0 minimum; D > 0, fₓₓ < 0 maximum; D < 0 saddle; D = 0 inconclusive.

Read more: Partial derivatives and stationary points.

Continuous random variables

• P(a ≤ X ≤ b) = ∫ₐ^bf(x) dx; E(g(X)) = ∫ g(x)f(x) dx; F(x) = P(X ≤ x) and f = F'; the median solves F(m) = 1/2.
• Uniform on [a,b]: mean (a + b)/2, variance ((b-a)²)/12.
• Exp(λ): P(T > t) = e^(-λ t), mean 1/λ, variance 1/λ², median (ln 2)/λ; the waiting time to the first event of a Poisson process of rate λ.

Read more: Density functions and exponential models.

Tests and confidence intervals

• Small sample, normal population, unknown variance: T = (X bar -μ)/(S/square root of n)∼ tₙ₋₁, with s² = 1/(n-1)∑(xᵢ-x bar)².
• Interval for μ: x bar ± tₙ₋₁ s/square root of n. A two-tailed test at level α rejects μ₀ exactly when it is outside the 100(1-α)% interval.
• Sign test: under H₀, the number above the median is B(n,1/2). Wilcoxon: rank |d|, compare the smaller signed-rank sum T with the critical value; reject if T is at most it.

Read more: t-tests and confidence intervals and Sign and Wilcoxon tests.

First-order equations and growth models

• dy/(dx) + p(x)y = q(x): multiply by I = e^(∫ p dx), then Iy = ∫ Iq dx (add the constant before dividing).
• Equilibria of dP/(dt) = f(P) solve f(P) = 0; stable if nearby solutions move towards them.
• Logistic: dP/(dt) = rP(1-P/K), stable equilibrium K; with constant harvest h, an equilibrium exists only while h ≤ rK/4.

Read more: First-order equations and growth models.

Linear transformations and numerical integration

• The columns of the matrix of T are the images of the standard basis vectors; “A then B” is BA.
• A subspace contains 0 and is closed under addition and scalar multiplication; a basis is an independent spanning set.
• Trapezium: h/2[y₀ + 2(y₁ + … + yₙ₋₁) + yₙ]. Simpson (n even): h/3[y₀ + 4(odd) + 2(even) + yₙ].
• Euler: yₙ₊₁ = yₙ + hf(xₙ,yₙ); improved Euler averages the gradients at both ends of the step.

Read more: Linear transformations and linear spaces and Numerical integration and Euler methods.

Paired, two-sample and χ² tests

• Paired: a one-sample t-test on the differences, with n-1 degrees of freedom.
• Two independent samples: z = (x bar ₁-x bar ₂)/(square root of (σ₁²/n₁ + σ₂²/n₂)) (sample variances for large samples).
• χ²: X² = ∑((O-E)²)/E, all E ≥ 5. Goodness of fit: ν = classes -1- estimated parameters; independence: ν = (r-1)(c-1), E = (row × column)/(total).

Read more: Paired and two-sample tests and χ² tests.