FY13CE Pure Mathematics cheat sheet

Direct proof: assume the hypothesis P, derive the conclusion Q by valid algebraic or logical steps; a proof is complete only when every step follows from axioms, definitions or previously proved results.

Contrapositive: P ⇒ Q is logically equivalent to ¬Q ⇒ ¬P; proving the contrapositive is often easier when the conclusion is a negative statement (for example, irrationality).

Contradiction: assume P ∧ ¬Q and derive a falsehood; conclude ¬(P ∧ ¬Q), which is equivalent to P ⇒ Q.

Counterexample: a single case where P is true and Q is false disproves the universal statement ∀x, P(x) ⇒ Q(x); it never proves the statement.

Necessary vs sufficient: P is sufficient for Q when P ⇒ Q; P is necessary for Q when Q ⇒ P; 'if and only if' requires both implications proved separately.

Quantifiers: ¬(∀x, P(x)) ≡ ∃x, ¬P(x) and ¬(∃x, P(x)) ≡ ∀x, ¬P(x); order matters - ∀x∃y differs from ∃y∀x.

Exhaustion and cases: if the domain splits into finitely many exhaustive cases, a proof covering every case is complete; ensure cases are mutually exclusive or overlap harmlessly.

Standard facts to quote: √2 is irrational; infinitely many primes (Euclid's argument); between any two distinct reals there is a rational; the product of two odd integers is odd.

Quadratic: roots x = (−b ± √(b² − 4ac))/(2a); discriminant Δ = b² − 4ac gives two real roots (Δ>0), one repeated (Δ=0), complex conjugate pair (Δ<0); vertex at x = −b/(2a).

Remainder Theorem: when f(x) is divided by (x − a) the remainder is f(a); Factor Theorem: (x − a) is a factor exactly when f(a) = 0; for (ax − b) use f(b/a).

Rational functions: vertical asymptotes where the denominator vanishes (unless cancelled), horizontal asymptote y = (leading coefficient ratio) when degrees match, y = 0 when denominator degree is larger.

Partial fractions: distinct linear factors give A/(x−a); repeated factors need A/(x−a) + B/(x−a)²; an irreducible quadratic factor needs (Cx + D)/(quadratic).

Modulus: |x| = x if x ≥ 0 and −x if x < 0; |x − a| < b ⇔ a − b < x < a + b; |x − a| > b ⇔ x < a − b or x > a + b, for b > 0.

Transformations: y = f(x − a) shifts right by a; y = af(x) stretches vertically by factor a; y = f(kx) stretches horizontally by factor 1/k; y = −f(x) reflects in the x-axis.

Inequalities: multiplying or dividing by a negative quantity reverses the inequality sign; solve rational inequalities by combining to a single fraction and using a sign table.

Exponentials and logs as inverses: aˣ = e^(x ln a); the domain of ln is x > 0; solving a^(f(x)) = b typically requires taking logs of both sides.

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

Geometric: nth term ar^(n−1); Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1; S∞ = a/(1 − r) only when |r| < 1.

Convergence of geometric series requires |r| < 1; the sum to infinity diverges otherwise, and a divergent series has no finite sum.

Binomial theorem for positive integer n: (a + b)ⁿ = Σ from k=0 to n of C(n,k) a^(n−k) b^k, with C(n,k) = n!/(k!(n−k)!); the general term is C(n,k) a^(n−k) b^k.

Method of differences: if the general term can be written as f(r) − f(r+1), then the sum telescopes to f(1) − f(n+1).

Sigma notation: Σ from r=1 to n of 1 = n; Σ r = n(n+1)/2; Σ r² = n(n+1)(2n+1)/6; Σ r³ = [n(n+1)/2]².

Recurrence relations: a sequence defined by u_{n+1} = f(uₙ) is increasing if u_{n+1} > uₙ for all n; a limit L must satisfy L = f(L).

Proof by induction for series: verify the base case, assume true for n = k, then show true for n = k + 1 using the assumed sum plus the (k+1)th term.

Straight line: gradient m = (y₂ − y₁)/(x₂ − x₁); forms y = mx + c, y − y₁ = m(x − x₁), ax + by + c = 0; parallel lines have equal m, perpendicular lines satisfy m₁m₂ = −1.

Distance between points is √((x₂ − x₁)² + (y₂ − y₁)²); the midpoint is ((x₁ + x₂)/2, (y₁ + y₂)/2).

Circle centre (a, b), radius r: (x − a)² + (y − b)² = r²; expanded form x² + y² + 2gx + 2fy + c = 0 has centre (−g, −f) and r² = g² + f² − c.

Perpendicular from the centre to a chord bisects the chord; the tangent at a point is perpendicular to the radius at that point.

Distance from point (x₁, y₁) to line ax + by + c = 0 is |ax₁ + by₁ + c|/√(a² + b²).

Parabola y² = 4ax has focus (a, 0) and directrix x = −a; every point on it is equidistant from focus and directrix.

Parametric curves: eliminate the parameter to obtain a Cartesian equation; dy/dx = (dy/dt)/(dx/dt) when dx/dt ≠ 0.

Intersections: substitute one equation into the other and use the discriminant to decide whether the line meets, misses or touches the curve (tangency when Δ = 0).

Radians: π radians = 180°; arc length s = rθ; sector area A = ½r²θ, with θ in radians.

Identities: sin²θ + cos²θ = 1; tanθ = sinθ/cosθ; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ.

Compound angles: sin(A ± B) = sinA cosB ± cosA sinB; cos(A ± B) = cosA cosB ∓ sinA sinB; tan(A ± B) = (tanA ± tanB)/(1 ∓ tanA tanB).

Double angle: sin2θ = 2 sinθ cosθ; cos2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ; tan2θ = 2tanθ/(1 − tan²θ).

Forms R sin(θ + α) or R cos(θ − α): R = √(a² + b²), with α determined by tanα = b/a; the maximum value of a sinθ + b cosθ is R.

General solutions: sinθ = k gives θ = α + 360°n or θ = 180° − α + 360°n; cosθ = k gives θ = ±α + 360°n; tanθ = k gives θ = α + 180°n.

Exact values: sin30° = ½, cos30° = √3/2, tan30° = 1/√3; sin45° = cos45° = 1/√2; sin60° = √3/2, cos60° = ½, tan60° = √3.

Reciprocal functions: secθ = 1/cosθ, cosecθ = 1/sinθ, cotθ = 1/tanθ; undefined where the corresponding denominator vanishes.

Laws of logs: log(ab) = log a + log b; log(a/b) = log a − log b; log(aⁿ) = n log a; log_a a = 1; log_a 1 = 0.

Change of base: log_a b = (log_c b)/(log_c a); consequently log_a b · log_b a = 1.

e^x is its own derivative; d/dx ln x = 1/x for x > 0; ∫ e^x dx = e^x + c; ∫ (1/x) dx = ln|x| + c.

Solving a^x = b: take logs of both sides to get x = ln b / ln a; equations reducible to a quadratic in e^x are solved by substitution.

Exponential growth and decay: y = Ae^(kt) with k > 0 growth, k < 0 decay; the doubling time is ln2/k and the half-life is ln2/|k|.

Graph shapes: y = e^x passes through (0, 1) with horizontal asymptote y = 0; y = ln x passes through (1, 0) with vertical asymptote x = 0 and is the reflection of e^x in y = x.

Linearising data: if y = axⁿ then log y = log a + n log x (a straight line of gradient n); if y = ab^x then log y = log a + x log b.

The domain of ln(f(x)) requires f(x) > 0; always check that solutions satisfy this condition.

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

Product rule: (uv)′ = u′v + uv′; quotient rule: (u/v)′ = (u′v − uv′)/v².

Chain rule: dy/dx = dy/du · du/dx; for a function of a function, differentiate the outer function and multiply by the derivative of the inner function.

Implicit differentiation: differentiate both sides with respect to x, treating y as a function of x, so d/dx(y²) = 2y dy/dx.

Parametric: dy/dx = (dy/dt)/(dx/dt); the second derivative is d/dt(dy/dx) divided by dx/dt.

Stationary points occur where dy/dx = 0; classify with the second derivative (positive = minimum, negative = maximum) or by a sign change in dy/dx.

Tangents and normals: tangent gradient is m = f′(a); normal gradient is −1/m at that point.

Connected rates of change: dV/dt = (dV/dr)(dr/dt); related rates are handled by the chain rule linking the variables.

Standard integrals: ∫ xⁿ dx = x^(n+1)/(n+1) + c for n ≠ −1; ∫ (1/x) dx = ln|x| + c; ∫ sin x dx = −cos x + c; ∫ cos x dx = sin x + c.

Definite integral: ∫ from a to b f(x) dx = F(b) − F(a); reversing the limits changes the sign; the area between a curve and the x-axis is computed by splitting at the roots and taking absolute values.

Substitution: change the variable and the limits; ∫ f(g(x))g′(x) dx = ∫ f(u) du with u = g(x).

Integration by parts: ∫ u dv = uv − ∫ v du; choose u to be the factor that simplifies when differentiated (often a power of x or a logarithm).

Partial fractions allow integration of rational functions; an irreducible quadratic factor produces a logarithm plus an arctangent term.

Area between curves: ∫ (upper − lower) dx over the interval between the intersection points.

Volume of revolution about the x-axis: V = π∫ y² dx between x = a and x = b; about the y-axis use V = π∫ x² dy.

Trapezium rule: ∫ f dx ≈ (h/2)[y₀ + 2(y₁ + … + y_{n−1}) + yₙ] with h = (b − a)/n; error decreases as n increases.

Vector between points: AB = b − a; magnitude |a| = √(a₁² + a₂² + a₃²); unit vector is a/|a|.

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

Vector equation of a line: r = a + λd, where a is a point on the line and d is a direction vector; λ is a scalar parameter.

Skew lines: two lines in 3D that are neither parallel nor intersecting; solve a + λd = b + μe for λ and μ, and if no consistent solution exists while d and e are not parallel, the lines are skew.

Angle between two lines uses cosθ = (d₁·d₂)/(|d₁||d₂|); the acute angle is found with the absolute value of the scalar product.

Plane equation: r·n = d, or n₁x + n₂y + n₃z = d, where n is a normal vector; the plane through point a with normal n is (r − a)·n = 0.

Distance from point p to plane r·n = d is |p·n − d|/|n|; the foot of the perpendicular is p minus the signed distance times the unit normal.

Line and plane: substitute the line into the plane equation to find λ; if the direction satisfies d·n = 0 the line is parallel to the plane (or lies in it if the point also satisfies the plane).

Forms: z = a + bi with i² = −1; modulus |z| = √(a² + b²); argument arg z = arctan(b/a) adjusted to the correct quadrant.

Modulus-argument form: z = r(cosθ + i sinθ) = re^(iθ); multiplication multiplies moduli and adds arguments; division divides moduli and subtracts arguments.

De Moivre: [r(cosθ + i sinθ)]ⁿ = rⁿ(cos nθ + i sin nθ); useful for powers and for multiple-angle identities.

Complex conjugate: z̄ = a − bi; z z̄ = |z|²; for a quadratic with real coefficients, non-real roots occur in conjugate pairs.

nth roots of z = re^(iθ) are r^(1/n)(cos((θ + 2kπ)/n) + i sin((θ + 2kπ)/n)) for k = 0, 1, …, n − 1; they lie equally spaced on a circle.

Loci: |z − a| = r is a circle centre a radius r; |z − a| = |z − b| is the perpendicular bisector of the segment joining a and b; arg(z − a) = θ is a half-line from a.

Euler's relation: e^(iθ) = cosθ + i sinθ; consequently cosθ = (e^(iθ) + e^(−iθ))/2 and sinθ = (e^(iθ) − e^(−iθ))/(2i).

Roots of unity: the nth roots of 1 are e^(2πik/n); their sum is 0 and their product is (−1)^(n−1).

Determinant of a 2×2 matrix [[a, b], [c, d]] is ad − bc; a 2×2 matrix has an inverse exactly when the determinant is non-zero.

Inverse of [[a, b], [c, d]] is (1/(ad − bc))[[d, −b], [−c, a]]; verify by multiplying to get the identity.

For 3×3 matrices the determinant can be found by expanding along a row or column using cofactors; a zero determinant means the matrix is singular.

Transformations: [[a, b], [c, d]] maps the unit square to a parallelogram of area |det| times the original area; a negative determinant indicates a reflection.

Common matrices: rotation by θ is [[cosθ, −sinθ], [sinθ, cosθ]]; reflection in the x-axis is [[1, 0], [0, −1]]; enlargement by k is kI.

Composition of transformations corresponds to matrix multiplication, and order matters: applying A then B is represented by BA.

Eigenvalues satisfy det(A − λI) = 0; eigenvectors satisfy (A − λI)v = 0 and are non-zero.

Solving linear equations with matrices: AX = B has the unique solution X = A⁻¹B when A is invertible; otherwise the system has no solution or infinitely many.

Sign-change rule: if f is continuous on [a, b] and f(a) and f(b) have opposite signs, then f has at least one root in (a, b); this locates a root but does not guarantee uniqueness.

Fixed-point iteration x_{n+1} = g(xₙ) converges to a root of x = g(x) when |g′(x)| < 1 near the root; divergence occurs when |g′(x)| > 1.

Newton–Raphson: x_{n+1} = xₙ − f(xₙ)/f′(xₙ); it converges rapidly but fails when f′(xₙ) is zero or near zero.

Linear interpolation (false position): x = (a f(b) − b f(a))/(f(b) − f(a)) estimates the root between a and b where signs differ.

Trapezium rule error is reduced by increasing the number of strips n; the approximation overestimates for a convex-up curve and underestimates for a concave-up curve.

Simpson's rule (with an even number of strips): ∫ f dx ≈ (h/3)[y₀ + 4(y₁ + y₃ + …) + 2(y₂ + y₄ + …) + yₙ].

Iteration diagrams: a staircase diagram arises when 0 < g′(x) < 1 and a cobweb diagram when −1 < g′(x) < 0.

Numerical accuracy: keep more significant figures than required during intermediate steps and round only at the end to avoid accumulated error.