UNSA ETF Math Entrance Exam cheat sheet

Division algorithm: for integers a and b>0 there are unique q, r with a = bq + r and 0 ≤ r < b.

Euclid's algorithm: gcd(a,b) = gcd(b, a mod b); iterate until the remainder is 0, then the last nonzero remainder is the gcd.

gcd(a,b)·lcm(a,b) = |ab| for positive integers a and b; use it to find the lcm from the gcd.

Integer a is divisible by 3 (or 9) iff the sum of its decimal digits is; divisible by 4 iff its last two digits form a multiple of 4.

A number is prime iff it has exactly two positive divisors; test primality by trial division up to √n only.

Fundamental theorem of arithmetic: every integer n > 1 factors uniquely into primes, up to order.

a ≡ b (mod m) means m divides a − b; congruences add, subtract, and multiply, and cancel only by a factor coprime to m.

Number of divisors: if n = p^a·q^b·…, then d(n) = (a+1)(b+1)…; count ordered factor pairs as d(n)/2 when n is not a perfect square.

In base 10, a number with digits d_k…d_0 equals Σ d_i·10^i; converting to base b uses repeated division by b reading remainders upward.

Quadratic formula: roots of ax² + bx + c = 0 are x = (−b ± √(b² − 4ac)) / (2a); discriminant Δ = b² − 4ac gives 2, 1, or 0 real roots as Δ > 0, = 0, < 0.

Vieta's formulas: for ax² + bx + c = 0, sum of roots = −b/a and product = c/a.

Difference of squares a² − b² = (a − b)(a + b); sum/difference of cubes a³ ± b³ = (a ± b)(a² ∓ ab + b²).

Binomial theorem: (a + b)^n = Σ_{k=0}^{n} C(n,k) a^{n−k} b^k, with general term C(n,k) a^{n−k} b^k.

Exponent rules: a^m·a^n = a^{m+n}, (a^m)^n = a^{mn}, a^{−n} = 1/a^n, a^{1/n} = ⁿ√a.

Logarithm rules: log(xy) = log x + log y, log(x/y) = log x − log y, log(x^k) = k log x, log_b b = 1, log_b 1 = 0.

For |x| < 1, the geometric series sums to 1/(1 − x); solving inequalities with absolute values splits into two cases.

Rational expressions simplify only after factoring numerator and denominator; domain excludes values making any denominator zero.

A function assigns each input exactly one output; the vertical line test detects non-functions on a graph.

Domain of f(x) = √(g(x)) requires g(x) ≥ 0; domain of 1/g(x) requires g(x) ≠ 0; log requires a positive argument.

f is even if f(−x) = f(x) (symmetric about the y-axis); odd if f(−x) = −f(x) (symmetric about the origin).

Transformations: y = f(x − h) + k shifts right by h and up by k; y = −f(x) reflects across the x-axis; y = f(−x) across the y-axis.

A function has an inverse only if it is one-to-one; the graph of f⁻¹ is the reflection of f across the line y = x, and domains and ranges swap.

Composition (f∘g)(x) = f(g(x)); the domain of f∘g excludes x not in dom g or with g(x) not in dom f.

Exponential f(x) = a^x (a > 0, a ≠ 1) is increasing for a > 1, decreasing for 0 < a < 1, and has horizontal asymptote y = 0.

Logarithm f(x) = log_a x is the inverse of a^x, has domain x > 0 and vertical asymptote x = 0.

Pythagorean theorem: in a right triangle with legs a, b and hypotenuse c, a² + b² = c²; triples include 3-4-5 and 5-12-13.

Area formulas: triangle ½bh, trapezoid ½(b₁ + b₂)h, circle πr², sector ½r²θ with θ in radians.

Circle: circumference 2πr, arc length s = rθ, and inscribed angle equals half the central angle subtending the same arc.

Radian-degree conversion: 180° = π radians; so degrees × π/180 gives radians.

Right-triangle ratios: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.

Pythagorean identity sin²θ + cos²θ = 1; also 1 + tan²θ = sec²θ and 1 + cot²θ = csc²θ.

Law of sines a/sin A = b/sin B = c/sin C = 2R; law of cosines c² = a² + b² − 2ab cos C.

Sum formulas: sin(A ± B) = sin A cos B ± cos A sin B; cos(A ± B) = cos A cos B ∓ sin A sin B.

Volume and surface: sphere V = (4/3)πr³, S = 4πr²; cylinder V = πr²h; cone V = (1/3)πr²h.

Arithmetic sequence: a_n = a_1 + (n − 1)d; sum S_n = n(a_1 + a_n)/2 = n[2a_1 + (n − 1)d]/2.

Geometric sequence: a_n = a_1 r^{n−1}; sum S_n = a_1(1 − r^n)/(1 − r) for r ≠ 1; infinite sum a_1/(1 − r) only when |r| < 1.

Sum of first n integers: Σ_{k=1}^{n} k = n(n + 1)/2; Σ k² = n(n + 1)(2n + 1)/6; Σ k³ = [n(n + 1)/2]².

Limit laws: lim(f ± g) = lim f ± lim g, lim(fg) = (lim f)(lim g), and lim(f/g) = lim f / lim g when lim g ≠ 0.

For a rational function as x → ∞, compare degrees: equal degrees give the ratio of leading coefficients; numerator degree larger gives ±∞; denominator larger gives 0.

lim_{x→0} (sin x)/x = 1 and lim_{x→0} (1 − cos x)/x = 0; lim_{x→∞} (1 + 1/x)^x = e.

Indeterminate forms 0/0 and ∞/∞ require factoring, rationalizing, or dividing by the dominant term before evaluating.

A series converges if its partial sums approach a finite limit; the geometric series converges exactly when |r| < 1, and the harmonic series diverges.

Multiplication rule: if a task has m ways then n ways, the combined task has mn ways; addition rule applies to mutually exclusive alternatives.

Permutations: P(n,r) = n!/(n − r)! orders r items from n; combinations C(n,r) = n!/[r!(n − r)!] ignores order.

Arrangements of a multiset with repeated letters: n!/(n₁!·n₂!·…).

Circular arrangements of n distinct objects: (n − 1)! up to rotation.

Probability of an event A is P(A) = favorable outcomes / total equally likely outcomes, always between 0 and 1.

Complement rule P(Aᶜ) = 1 − P(A); addition rule P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

Independent events: P(A ∩ B) = P(A)P(B); conditional probability P(A|B) = P(A ∩ B)/P(B).

Binomial probability: P(k successes in n trials) = C(n,k) p^k (1 − p)^{n−k} for independent trials with success probability p.

Expected value of a discrete variable: E(X) = Σ x·P(X = x); variance is E(X²) − [E(X)]².

Imaginary unit: i² = −1; powers of i cycle with period 4: i¹ = i, i² = −1, i³ = −i, i⁴ = 1.

For z = a + bi, modulus |z| = √(a² + b²) and conjugate z̄ = a − bi; z·z̄ = a² + b² is real and nonnegative.

Division: multiply numerator and denominator by the conjugate of the denominator to rationalize a + bi over c + di.

Polar form z = r(cos θ + i sin θ) with r = |z| and θ = arg z; multiplication multiplies moduli and adds arguments.

De Moivre's theorem: [r(cos θ + i sin θ)]^n = r^n(cos nθ + i sin nθ).

Euler's formula e^{iθ} = cos θ + i sin θ, so z = re^{iθ} and e^{iπ} + 1 = 0.

nth roots of z = r cis θ are r^{1/n} cis[(θ + 2πk)/n] for k = 0, 1, …, n − 1, spaced evenly on a circle.

Quadratic with negative discriminant Δ < 0 has complex roots (−b ± i√(−Δ))/(2a), which are conjugates.

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

Slope m = (y₂ − y₁)/(x₂ − x₁); parallel lines have equal slopes, perpendicular lines have slopes with product −1.

Line forms: slope-intercept y = mx + b, point-slope y − y₁ = m(x − x₁), general Ax + By + C = 0.

Circle centered at (h, k) with radius r: (x − h)² + (y − k)² = r².

Parabola y = ax² + bx + c has vertex x = −b/(2a); completing the square converts it to vertex form.

Vectors u = (u₁, u₂), v = (v₁, v₂): addition is componentwise, and |u| = √(u₁² + u₂²).

Dot product u·v = u₁v₁ + u₂v₂ = |u||v|cos θ; u ⊥ v exactly when u·v = 0.

Projection of u onto v is (u·v / v·v)·v; the 2D cross product magnitude |u₁v₂ − u₂v₁| equals the area of the parallelogram spanned by u and v.