MATSEC AM Pure Mathematics cheat sheet

Remainder Theorem: the remainder when f(x) is divided by (x − a) is f(a).

Factor Theorem: (x − a) is a factor of f(x) if and only if f(a) = 0.

For a rational function, vertical asymptotes occur where the denominator is zero but the numerator is not.

Partial fractions: a repeated linear factor (x − a)^2 needs terms A/(x − a) + B/(x − a)^2.

An improper fraction must be divided out before decomposing: degree of numerator must be less than degree of denominator.

The discriminant b^2 − 4ac of ax^2 + bx + c determines the number of real roots: positive gives two, zero gives one, negative gives none.

For |f(x)| < k (k > 0) solve −k < f(x) < k; for |f(x)| > k solve f(x) < −k or f(x) > k.

A function is even if f(−x) = f(x) and odd if f(−x) = −f(x); the graph of an even function is symmetric about the y-axis.

Arithmetic sequence: nth term a + (n − 1)d; sum of n terms S_n = (n/2)(2a + (n − 1)d).

Geometric sequence: nth term ar^(n−1); sum S_n = a(1 − r^n)/(1 − r) for r ≠ 1.

Sum to infinity of a geometric series exists only when |r| < 1, and equals a/(1 − r).

Binomial theorem: (a + b)^n = Σ from k = 0 to n of C(n,k) a^(n−k) b^k, where C(n,k) = n!/(k!(n−k)!).

General term in the expansion of (a + b)^n is C(n,k) a^(n−k) b^k.

Partial sums: telescoping series collapse when consecutive terms cancel.

For an arithmetic series the sum of terms equidistant from the ends is constant.

Method of differences: express a term as f(r) − f(r + 1) to evaluate a summation.

Distance between (x1, y1) and (x2, y2) is √((x2 − x1)^2 + (y2 − y1)^2).

Gradient of a line through two points is (y2 − y1)/(x2 − x1); perpendicular gradients satisfy m1·m2 = −1.

Equation of a circle centre (a, b), radius r: (x − a)^2 + (y − b)^2 = r^2.

General circle x^2 + y^2 + 2gx + 2fy + c = 0 has centre (−g, −f) and radius √(g^2 + f^2 − c).

Distance from point (x0, y0) to line ax + by + c = 0 is |ax0 + by0 + c|/√(a^2 + b^2).

A line is tangent to a circle when substituting the line into the circle equation gives a repeated root (discriminant zero).

Midpoint of (x1, y1) and (x2, y2) is ((x1 + x2)/2, (y1 + y2)/2).

Two circles intersect where their equations, subtracted, give the radical-axis line, solved with either circle.

Pythagorean identity: sin^2 θ + cos^2 θ = 1; also 1 + tan^2 θ = sec^2 θ and 1 + cot^2 θ = cosec^2 θ.

Compound angles: sin(A ± B) = sin A cos B ± cos A sin B; cos(A ± B) = cos A cos B ∓ sin A sin B.

Double angle: sin 2A = 2 sin A cos A; cos 2A = cos^2 A − sin^2 A = 1 − 2 sin^2 A = 2 cos^2 A − 1.

The equation a sin θ + b cos θ = c can be written R sin(θ + α) with R = √(a^2 + b^2).

Sine rule: a/sin A = b/sin B = c/sin C; cosine rule: a^2 = b^2 + c^2 − 2bc cos A.

Area of a triangle = (1/2)ab sin C.

For sin θ = k, solutions in 0° to 360° are θ and 180° − θ; for cos θ = k they are θ and 360° − θ.

Radian measure: arc length s = rθ and sector area A = (1/2)r^2 θ, with θ in radians.

log_a(xy) = log_a x + log_a y; log_a(x/y) = log_a x − log_a y; log_a(x^n) = n log_a x.

Change of base: log_a b = log_c b / log_c a.

a^x = e^(x ln a), so any exponential can be written with base e.

To solve a^x = b take logarithms: x = log b / log a.

The graph of y = e^x has gradient equal to its value at every point; d/dx(e^x) = e^x.

ln x is defined only for x > 0, and ln 1 = 0, ln e = 1.

An exponential model y = Ae^(kt) becomes linear in ln y: ln y = ln A + kt.

A decay model y = Ae^(−kt) has half-life t = (ln 2)/k.

Product rule: d/dx(uv) = u dv/dx + v du/dx.

Quotient rule: d/dx(u/v) = (v du/dx − u dv/dx)/v^2.

Chain rule: dy/dx = dy/du · du/dx.

d/dx(sin x) = cos x; d/dx(cos x) = −sin x; d/dx(tan x) = sec^2 x.

d/dx(ln x) = 1/x; d/dx(e^(kx)) = k e^(kx).

At a stationary point dy/dx = 0; the second derivative decides maximum (negative) or minimum (positive).

Parametric curves: dy/dx = (dy/dt)/(dx/dt).

Implicit differentiation: differentiate each term with respect to x, applying the chain rule to y terms.

∫x^n dx = x^(n+1)/(n+1) + C for n ≠ −1; ∫(1/x) dx = ln|x| + C.

∫e^(kx) dx = (1/k)e^(kx) + C; ∫sin x dx = −cos x + C; ∫cos x dx = sin x + C.

Definite integral equals F(b) − F(a) where F is any antiderivative.

Area between curve and x-axis: integrate |f(x)|, splitting where the curve crosses the axis.

Volume of revolution about the x-axis: V = π∫y^2 dx between limits.

Integration by substitution: replace dx by (dx/du) du and change the limits for definite integrals.

Integration by parts: ∫u dv = uv − ∫v du.

Trapezium rule: ∫f(x)dx ≈ (h/2)[y0 + 2(y1 + … + y_(n−1)) + yn] with h = (b − a)/n.

i^2 = −1; for z = a + bi, modulus |z| = √(a^2 + b^2) and argument arg z = arctan(b/a) adjusted for quadrant.

Conjugate of a + bi is a − bi; z·z̄ = |z|^2.

Multiplication in polar form multiplies moduli and adds arguments; division subtracts arguments.

De Moivre: (r(cos θ + i sin θ))^n = r^n(cos nθ + i sin nθ).

Roots of z^n = w are equally spaced on a circle of radius |w|^(1/n), separated by 2π/n in argument.

Euler form: z = r e^(iθ), so e^(iπ) + 1 = 0.

For a polynomial with real coefficients, non-real roots occur in conjugate pairs.

Division: multiply numerator and denominator by the conjugate of the denominator.

Scalar product: a·b = |a||b| cos θ = a1b1 + a2b2 + a3b3; perpendicular vectors have a·b = 0.

Vector product magnitude |a × b| = |a||b| sin θ; the result is perpendicular to both a and b.

Line: r = a + λd, where a is a position vector on the line and d a direction vector.

Plane: r·n = d, where n is a normal vector to the plane.

Angle between two lines uses the angle between their direction vectors via the scalar product.

Shortest distance from point P to a line through A with direction d is |AP × d|/|d|.

Distance from point P to plane r·n = d is |(p − a)·n̂|, where a is a point in the plane.

Two lines intersect when their parametric equations give a common point for some parameters.

Sign-change test: if f is continuous on [a, b] and f(a), f(b) have opposite signs, a root lies in (a, b).

Newton–Raphson iteration: x_(n+1) = x_n − f(x_n)/f′(x_n).

Fixed-point iteration x_(n+1) = g(x_n) converges when |g′(x)| < 1 near the root.

Linear interpolation (false position): x ≈ a − f(a)(b − a)/(f(b) − f(a)).

Trapezium rule error decreases roughly with h^2 as the number of strips increases.

Simpson's rule needs an even number of strips: (h/3)[y0 + 4(odd y) + 2(even y) + yn].

An iteration diverges if successive values move away from the root rather than settling.

Bisection halves the interval each step, so accuracy doubles in binary terms each iteration.

P(A ∪ B) = P(A) + P(B) − P(A ∩ B); for mutually exclusive events P(A ∩ B) = 0.

Conditional probability: P(A|B) = P(A ∩ B)/P(B); independent events satisfy P(A ∩ B) = P(A)P(B).

Bayes: P(A|B) = P(B|A)P(A)/P(B).

Binomial: P(X = r) = C(n, r) p^r (1 − p)^(n−r), mean np, variance np(1 − p).

Normal distribution standardised by z = (x − μ)/σ.

Mean of a discrete random variable: E(X) = Σ x P(X = x); Var(X) = E(X^2) − (E(X))^2.

For a sample, variance uses divisor (n − 1); for a population, divisor n.

Poisson: P(X = r) = e^(−λ) λ^r / r!, with mean and variance both λ.