Balearic Islands PAU Mathematics II cheat sheet

Types of matrix: square (rows = columns), identity I, zero matrix, diagonal, triangular, symmetric (A = Aᵗ), skew-symmetric (A = −Aᵗ); A is regular or invertible only if it is square with |A| ≠ 0.

Product of matrices: A·B exists only when columns of A equal rows of B; the product is generally non-commutative (AB ≠ BA), and AB = 0 does not imply A = 0 or B = 0, so cancellation is invalid.

Transpose rules: (Aᵗ)ᵗ = A, (A + B)ᵗ = Aᵗ + Bᵗ, (AB)ᵗ = BᵗAᵗ, (A⁻¹)ᵗ = (Aᵗ)⁻¹; a matrix is symmetric when A = Aᵗ and orthogonal when A·Aᵗ = I.

Determinant of order 2: |A| = ad − bc for A = [[a, b], [c, d]]. Order 3: expand by any row or column using cofactors, or apply the rule of Sarrus; the determinant of a triangular matrix is the product of its diagonal entries.

Determinant properties: |AB| = |A||B|, |Aᵗ| = |A|, |kA| = kⁿ|A| for order n, |A⁻¹| = 1/|A|; swapping two rows changes the sign, two equal rows give 0, and adding a multiple of one row to another leaves the determinant unchanged.

Rank: the rank of A is the largest order of a non-zero minor, equivalently the number of non-zero rows after Gaussian elimination. For an n×n matrix, rank n ⇔ |A| ≠ 0 ⇔ the columns are linearly independent.

Inverse by the adjugate: A⁻¹ = adj(A)ᵗ / |A|, where the adjugate collects the cofactors of the elements; valid only when |A| ≠ 0. For order 2, [[a, b], [c, d]]⁻¹ = (1/(ad − bc))·[[d, −b], [−c, a]].

Rouché–Capelli: the system AX = B is consistent ⇔ rank(A) = rank(A|B). If rank(A) = rank(A|B) = n (unknowns) the solution is unique; if rank(A) = rank(A|B) = r < n the system is indeterminate with n − r free parameters.

Cramer's rule for a square system with |A| ≠ 0: xᵢ = |Aᵢ|/|A|, where Aᵢ is A with column i replaced by the constant vector B. Homogeneous system AX = 0 always has the trivial solution and has non-trivial solutions only if |A| = 0.

Matrix equations: solve AX = B as X = A⁻¹B (left multiplication), and XA = B as X = BA⁻¹ (right multiplication); the order matters and requires A to be invertible.

Vector operations: given u = (u₁, u₂, u₃) and v = (v₁, v₂, v₃), the scalar product is u·v = u₁v₁ + u₂v₂ + u₃v₃ and the cross product u × v is the formal determinant of rows (i, j, k), (u₁, u₂, u₃), (v₁, v₂, v₃).

Orthogonality and parallelism: u ⊥ v ⇔ u·v = 0; u ∥ v ⇔ u × v = 0 ⇔ components are proportional. The angle between vectors satisfies cos θ = (u·v)/(|u||v|), so the angle is acute if u·v > 0 and obtuse if u·v < 0.

Geometric meanings: |u × v| is the area of the parallelogram spanned by u and v, and |u·(v × w)| is the volume of the parallelepiped; three vectors are coplanar ⇔ u·(v × w) = 0.

Line forms: vector form (x, y, z) = P + t·d, parametric x = p₁ + t·d₁ etc., and continuous form (x − p₁)/d₁ = (y − p₂)/d₂ = (z − p₃)/d₃; the direction vector d is read directly from the coefficients.

Plane forms: general Ax + By + Cz + D = 0 with normal n = (A, B, C); point-normal A(x − x₀) + B(y − y₀) + C(z − z₀) = 0; a plane is fixed by a point and two independent direction vectors via the determinant equation.

Relative positions: two lines are coincident, parallel, intersecting or skew, decided by comparing direction vectors (parallel or not) and then testing whether the vector joining two points lies in the plane of the directions; two planes are parallel (same normal up to a scalar, different D) or secant along a line.

Angles: angle between lines uses their direction vectors; angle between planes uses their normals; angle between a line and a plane uses sin θ = |d·n|/(|d||n|). The acute angle is always reported, so take the absolute value.

Distance from a point P to a plane Ax + By + Cz + D = 0 is |Ax₀ + By₀ + Cz₀ + D| / √(A² + B² + C²); distance from a point to a line is |PQ × d|/|d|, where Q is a point on the line.

Distance between skew lines: |(Q − P)·(d₁ × d₂)| / |d₁ × d₂|, where P and Q lie on the two lines; it is zero exactly when the lines intersect or are parallel. The distance between parallel planes equals the distance from any point of one to the other.

Symmetric point and projection: the projection of P onto a plane is found by intersecting the line through P with direction n; the symmetric point is P′ = 2H − P, where H is the projection. The same construction with a line's direction gives the projection onto a line.

Indeterminate forms and tools: 0/0 and ∞/∞ are resolved with L'Hôpital's rule (differentiate numerator and denominator separately) or by factoring; 1^∞ is resolved with the number e via lim (1 + f)^(1/f) = e when f → 0, and ∞ − ∞ by common denominators or conjugates.

Continuity at x = a requires three things: f(a) exists, lim f(x) as x → a exists, and the two are equal. A function is continuous on an interval when it is continuous at every interior point plus the appropriate one-sided continuity at the endpoints.

Derivative from the definition: f′(a) = lim (f(a + h) − f(a))/h as h → 0. Differentiability at a point implies continuity there, but continuity does not imply differentiability (for example |x| at 0).

Standard derivatives: (xⁿ)′ = nxⁿ⁻¹, (eˣ)′ = eˣ, (ln x)′ = 1/x, (sin x)′ = cos x, (cos x)′ = −sin x, (tan x)′ = 1/cos²x, (arcsin x)′ = 1/√(1 − x²), (arctan x)′ = 1/(1 + x²).

Rules: product (uv)′ = u′v + uv′; quotient (u/v)′ = (u′v − uv′)/v²; chain rule [f(g(x))]′ = f′(g(x))·g′(x). For f(x) = g(x)^h(x), take logarithms: f′/f = h′·ln g + h·g′/g.

Tangent line at x = a: y = f(a) + f′(a)(x − a). The normal line has slope −1/f′(a). Rolle's theorem needs continuity on [a, b], differentiability on (a, b) and f(a) = f(b); the mean value theorem drops the equality condition.

Monotonicity: f′ > 0 on an interval means increasing, f′ < 0 means decreasing. Critical points are where f′ = 0 or f′ does not exist; a sign change of f′ at a critical point produces a local extremum (first-derivative test).

Second-derivative test: at a critical point with f″(a) > 0 there is a local minimum, f″(a) < 0 gives a local maximum, and f″(a) = 0 is inconclusive. Concavity is upward where f″ > 0 and downward where f″ < 0; inflection points require a sign change of f″.

Asymptotes: vertical where the denominator vanishes without cancellation; horizontal when the limit at ±∞ is finite; oblique y = mx + n with m = lim f(x)/x and n = lim [f(x) − mx], both finite and m ≠ 0.

Global optimisation on a closed interval [a, b]: evaluate f at all critical points inside the interval and at both endpoints a and b; the largest value is the global maximum and the smallest is the global minimum. With parameters, split the analysis by the position of the critical points relative to the interval.

Immediate antiderivatives: ∫xⁿ dx = xⁿ⁺¹/(n + 1) + C for n ≠ −1, ∫(1/x) dx = ln|x| + C, ∫eˣ dx = eˣ + C, ∫aˣ dx = aˣ/ln a + C, ∫sin x dx = −cos x + C, ∫cos x dx = sin x + C, ∫1/(1 + x²) dx = arctan x + C.

Substitution: ∫f(g(x))g′(x) dx = ∫f(u) du with u = g(x). For definite integrals, change the limits to the new variable instead of returning to x; this avoids errors at the boundary.

Integration by parts: ∫u dv = uv − ∫v du. Choose u by the order inverse, logarithmic, algebraic, trigonometric, exponential (ILATE); typical cases are xⁿ·eˣ, xⁿ·ln x, xⁿ·sin x and arctan x.

Rational functions: if the degree of the numerator is at least that of the denominator, divide first. Then decompose into partial fractions according to the factors of the denominator (distinct linear, repeated linear, irreducible quadratic), and integrate each term with logarithms and arctangents.

Fundamental theorem of calculus: if F is an antiderivative of f, then ∫ from a to b of f(x) dx = F(b) − F(a). The function G(x) = ∫ from a to x of f(t) dt satisfies G′(x) = f(x) for continuous f.

Properties of the definite integral: linearity, additivity over adjacent intervals, ∫ from a to a = 0, ∫ from a to b = −∫ from b to a. If f is odd on [−a, a] the integral is 0; if f is even the integral is 2∫ from 0 to a.

Area between two curves: ∫ from a to b of |f(x) − g(x)| dx, where a and b are the intersection abscissas. If the curves cross inside the interval, split the integral at every crossing point and add the absolute values of the pieces.

Area with respect to the y-axis: ∫ from c to d of |x_right(y) − x_left(y)| dy. Choose the axis that gives a single integral whenever possible.

Volume of revolution about the x-axis: V = π∫ from a to b of [f(x)]² dx (disc method). About the y-axis with the shell method: V = 2π∫ from a to b of x·f(x) dx. For a region between two curves, subtract the inner solid from the outer one.

Volumes by cross-sections: if the section perpendicular to the x-axis at position x has area A(x), then V = ∫ from a to b of A(x) dx; for a solid whose base is a region and whose sections are squares of side f(x), the volume is ∫[f(x)]² dx.

Improper integrals: ∫ from a to ∞ of f(x) dx = lim as t → ∞ of ∫ from a to t of f(x) dx. The integral converges when the limit is finite; ∫ from 1 to ∞ of 1/xᵖ dx converges for p > 1 and diverges for p ≤ 1.

Counting: permutations without repetition P(n, r) = n!/(n − r)!; combinations C(n, r) = n!/(r!(n − r)!), used when order does not matter; permutations with repetition nʳ when choosing r items from n types with replacement. The binomial theorem gives the coefficients of (a + b)ⁿ as C(n, k).

Probability axioms: 0 ≤ P(A) ≤ 1, P(E) = 1, and for disjoint events P(A ∪ B) = P(A) + P(B). In general P(A ∪ B) = P(A) + P(B) − P(A ∩ B); the complement satisfies P(Ā) = 1 − P(A).

Conditional probability: P(A|B) = P(A ∩ B)/P(B) for P(B) > 0. Events are independent when P(A ∩ B) = P(A)P(B), equivalently P(A|B) = P(A); mutually exclusive events with positive probability are never independent.

Total probability: if B₁, …, Bₙ form a partition of the sample space, P(A) = Σ P(Bᵢ)P(A|Bᵢ). Bayes' theorem reverses the conditioning: P(Bᵢ|A) = P(Bᵢ)P(A|Bᵢ) / Σ P(Bⱼ)P(A|Bⱼ).

Binomial distribution B(n, p): P(X = k) = C(n, k)pᵏ(1 − p)ⁿ⁻ᵏ. Mean μ = np, variance σ² = np(1 − p). It models the number of successes in n independent trials with constant probability p.

Normal distribution N(μ, σ): standardise with z = (x − μ)/σ and read probabilities from the standard normal table. Key values: P(−1 < z < 1) ≈ 0.6826, P(−2 < z < 2) ≈ 0.9544, P(−3 < z < 3) ≈ 0.9974; P(z ≤ 0) = 0.5 and P(z ≤ −a) = 1 − P(z ≤ a).

Normal approximation to the binomial: valid when np > 5 and n(1 − p) > 5; use μ = np and σ = √(np(1 − p)) and apply the continuity correction by adding or subtracting 0.5 at the boundaries.

Sampling distributions: for a sample of size n from a population with mean μ and standard deviation σ, the sample mean has mean μ and standard error σ/√n; for large n the central limit theorem makes the distribution approximately normal regardless of the population shape.

Confidence interval for the mean with known σ: x̄ ± z_(α/2)·σ/√n, where z_(α/2) = 1.96 for 95% and 2.575 for 99%. The margin of error is z·σ/√n, so quadrupling the sample size halves it.

Hypothesis testing for the mean: state H₀: μ = μ₀ and H₁ (one- or two-tailed), compute the statistic z = (x̄ − μ₀)/(σ/√n), and compare it with the critical value of the chosen significance level; reject H₀ when the statistic falls in the critical region, and interpret the p-value as the probability of a result at least as extreme given H₀.