HKDSE Maths M2 cheat sheet

Mathematical induction: prove P(1) true, then assume P(k) true and derive P(k+1); the assumption step must actually be used, otherwise the argument is incomplete.

Binomial theorem: (a+b)^n = Σ(r=0 to n) C(n,r) a^(n−r) b^r, with general term C(n,r) a^(n−r) b^r; the term independent of x is found by setting the exponent of x to zero.

Identity C(n,r) = C(n,n−r) and Pascal relation C(n+1,r) = C(n,r−1) + C(n,r) are the standard tools for coefficient sums and identities.

Sum of a finite geometric series: a(1−r^n)/(1−r) for r ≠ 1; for |r| < 1 the infinite sum converges to a/(1−r).

System of linear equations in three unknowns: consistency is decided by comparing ranks of the coefficient matrix and augmented matrix; infinitely many solutions arise when the ranks are equal but less than the number of unknowns.

Division algorithm for polynomials: dividend = divisor × quotient + remainder, with deg(remainder) < deg(divisor); remainder theorem gives remainder f(a) on division by (x−a), and factor theorem gives f(a)=0 ⇔ (x−a) is a factor.

A quadratic equation with real coefficients has non-real roots only in conjugate pairs; the discriminant b²−4ac decides real, repeated, or complex roots.

Euclidean algorithm on polynomials: repeated division of divisor by remainder terminates at the gcd (up to a non-zero constant factor).

Inequalities: multiplying or dividing by a negative number reverses the sign; |x| < a ⇔ −a < x < a, while |x| > a ⇔ x > a or x < −a.

Derivative from first principles: f′(x) = lim(h→0) [f(x+h) − f(x)]/h; this limit definition is required when the question says 'from first principles'.

Product rule (uv)′ = u′v + uv′; quotient rule (u/v)′ = (u′v − uv′)/v²; chain rule dy/dx = dy/du · du/dx.

Standard derivatives: d/dx(x^n) = nx^(n−1), d/dx(e^x) = e^x, d/dx(ln x) = 1/x, d/dx(sin x) = cos x, d/dx(cos x) = −sin x, d/dx(tan x) = sec²x.

Implicit differentiation: differentiate both sides with respect to x, applying the chain rule to every y-term, then solve for dy/dx.

Second derivative test: at a stationary point, f″(x) > 0 gives a local minimum, f″(x) < 0 gives a local maximum, and f″(x) = 0 is inconclusive.

Definite integral: ∫(a to b) f(x) dx = F(b) − F(a) where F′ = f; reversing the limits changes the sign, and splitting at an interior point c is valid.

Area between curves: integrate (upper curve − lower curve) over the interval between intersection points; for curves given as x = g(y), integrate with respect to y.

Substitution in integrals: replace the inner function by u, convert dx to du/(du/dx), and change the limits when the integral is definite.

Volume of revolution about the x-axis: V = π∫(a to b) y² dx; about the y-axis with x expressed in terms of y: V = π∫(c to d) x² dy.