Concours Advance - independent subject practice cheat sheet

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.

State domain restrictions before simplifying; cancelled factors still exclude values.

Composition f(g(x)) is evaluated inside-out and is generally not commutative.

Inverses require one-to-one functions; domain and range swap.

Horizontal transformations act opposite to the sign or factor shown.

Log laws convert sums to products, enabling solution of log equations.

Always check candidate solutions against original domain restrictions.

Similarity scales lengths by k, areas by k², volumes by k³.

Inscribed angle equals half the intercepted central arc.

Sine law requires a known side–opposite-angle pair; watch the ambiguous case.

Cosine law handles SAS and SSS configurations.

Arc length and sector area formulas require radians.

Pythagorean identity sin²θ + cos²θ = 1 underpins most simplifications.

Bayes: posterior ∝ likelihood × prior; always compute P(+) first.

Standard error of the mean is σ/√n, not σ.

p-value = P(data at least as extreme | H₀), never P(H₀ true).

Binomial variance np(1 − p); Poisson mean and variance both λ.

Sample variance uses n − 1 to be unbiased.

Read modals, conditions, negation and logical connectors carefully.

Keep units, denominators and percentage-point distinctions.

Paraphrases must preserve qualification and scope.

A written technical exercise does not certify practical or oral competence.