GCE A-Level Further Mathematics cheat sheet

i² = −1; division uses multiplication by the conjugate.

|z| = √(a² + b²), and the argument must be placed in the correct quadrant.

Modulus–argument form is re^(iθ); multiplication adds arguments.

Real-coefficient polynomials have non-real roots in conjugate pairs.

|z − z₀| = r is a circle centred at z₀.

Consistency requires rank A = rank [A|b]; uniqueness also needs full column rank.

det A = 0 means no unique solution, not necessarily no solution.

Trace = sum of eigenvalues; determinant = product of eigenvalues.

a×b is perpendicular to both vectors; its magnitude is the parallelogram area.

Distance from point to plane: |n·P − d| / |n|.

Linear first order: μ = exp(∫P dx) turns the left side into (μy)'.

Repeated characteristic root: multiply the second term by x.

Complementary + particular, then apply initial conditions.

For resonance, multiply the trial by x to the multiplicity of the corresponding characteristic root.

Sign of f(y) between equilibria determines stability on the phase line.

AP sum: n/2 × (first + last) or n/2(2a + (n − 1)d).

GP sum to infinity exists only for |r| < 1 and equals a/(1 − r).

The general binomial term C(n, r)a^(n−r)b^r gives any coefficient directly.

Count the number of terms carefully when the index starts at zero.

A negative common ratio alternates signs, so check the parity of the power.

Chain rule: never drop the inner derivative g'(x).

Integration by parts: choose u so du is simpler than u.

Change limits immediately when substituting in a definite integral.

Improper integrals require an explicit limit, not direct substitution.

Implicit differentiation collects dy/dx terms after differentiating both sides.

Resultant force vector = mass times acceleration vector; draw the force diagram first.

Smooth pulley: a = (m₁ − m₂)g/(m₁ + m₂) for masses on either side.

Tension is found by applying F = ma to one body after the acceleration is known.

Static friction obeys |F_s| ≤ μ_sR and reaches equality at impending slip; sliding uses a separate kinetic-friction model.

On an incline resolve as mg sin θ along the slope and mg cos θ perpendicular to it.

Median resists outliers; mean does not.

P(A ∪ B) subtracts the intersection to avoid double counting.

Conditional probability divides the joint probability by the condition's probability.

Combinations count selections; permutations count arrangements.

z = (x − μ)/σ places a value on the standard normal scale.

Random assignment, not random sampling alone, supports causal conclusions.