UNSA ETF Math Entrance Exam flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Division algorithm

For integers a and positive b, there exist unique integers q and r with a = bq + r and 0 ≤ r < b. Here q is the quotient and r the remainder.

Euclidean algorithm

Repeatedly replace the larger number by its remainder upon division by the smaller; the last nonzero remainder equals the greatest common divisor of the original pair.

Fundamental theorem of arithmetic

Every integer greater than 1 factors into primes, and that prime factorization is unique apart from the order of the factors.

Divisibility by 9

An integer is divisible by 9 exactly when the sum of its decimal digits is divisible by 9. The same digit-sum test works for divisibility by 3.

Quadratic formula

The solutions of ax² + bx + c = 0 with a ≠ 0 are x = (−b ± √(b² − 4ac)) divided by 2a.

Discriminant rule

For ax² + bx + c = 0, the discriminant Δ = b² − 4ac determines the roots: two distinct real roots if Δ > 0, one repeated real root if Δ = 0, and no real roots if Δ < 0.

Vieta's formulas

For a quadratic ax² + bx + c = 0, the sum of the two roots equals −b/a and their product equals c/a, without solving the equation.

Logarithm product rule

For positive x and y, log(xy) = log x + log y. Related rules: log(x/y) = log x − log y and log(x^k) = k log x.

Even and odd functions

A function is even when f(−x) = f(x), giving symmetry about the y-axis, and odd when f(−x) = −f(x), giving symmetry about the origin.

Inverse function condition

A function has an inverse only when it is one-to-one. The inverse's graph reflects the original across the line y = x, and domain and range swap.

Graph shift rule

Replacing x by x − h shifts a graph horizontally by h, and adding k outside the function shifts it vertically by k, giving y = f(x − h) + k.

Exponential function range

For f(x) = a^x with a > 0 and a ≠ 1, the domain is all reals, the range is positive reals, and y = 0 is a horizontal asymptote.

Pythagorean identity

For every angle θ, sin²θ + cos²θ = 1. Dividing by cos²θ gives 1 + tan²θ = sec²θ, and dividing by sin²θ gives 1 + cot²θ = csc²θ.

Law of cosines

In any triangle with sides a, b, c opposite angles A, B, C: c² = a² + b² − 2ab cos C. It reduces to the Pythagorean theorem when C = 90°.

Arc length formula

For a circle of radius r, an arc subtending a central angle θ measured in radians has length s = rθ. The sector area is ½r²θ.

Radian conversion

A straight angle equals π radians, so multiply degrees by π/180 to obtain radians and multiply radians by 180/π to obtain degrees.

Arithmetic series sum

For an arithmetic sequence with first term a₁, common difference d, and n terms, the sum is n(2a₁ + (n − 1)d)/2, equivalently n(a₁ + a_n)/2.

Infinite geometric series

The series a + ar + ar² + … converges to a/(1 − r) exactly when |r| < 1. If |r| ≥ 1 the partial sums do not approach a finite limit.

Sum of squares

The sum 1² + 2² + … + n² equals n(n + 1)(2n + 1)/6, useful for evaluating limits of sums written in sigma notation.

Rational function limit at infinity

As x → ∞, a rational function behaves like the ratio of leading terms: equal degrees give the coefficient ratio, larger numerator degree diverges, larger denominator degree tends to zero.

Combinations formula

The number of ways to choose r objects from n distinct objects, ignoring order, is C(n,r) = n!/[r!(n − r)!]. Orderings count P(n,r) = n!/(n − r)!.

Addition rule for probability

For any two events, P(A ∪ B) = P(A) + P(B) − P(A ∩ B). When the events are mutually exclusive, the intersection term is zero.

Conditional probability

If P(B) > 0, the probability of A given B is P(A|B) = P(A ∩ B)/P(B). Events are independent exactly when P(A|B) = P(A).

Binomial probability

In n independent trials each with success probability p, the probability of exactly k successes is C(n,k) p^k (1 − p)^(n − k).

Powers of i

Since i² = −1, the powers repeat with period four: i, −1, −i, 1. To compute i^n, reduce n modulo 4 and read the corresponding value.

Complex modulus and conjugate

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

De Moivre's theorem

For z = r(cos θ + i sin θ), the nth power is z^n = r^n(cos nθ + i sin nθ), so powers multiply moduli and multiply angles.

nth roots of a complex number

The n distinct nth roots of r cis θ are r^(1/n) cis[(θ + 2πk)/n] for k = 0, 1, …, n − 1, equally spaced around a circle.

Perpendicular line slopes

Two nonvertical lines are perpendicular exactly when their slopes multiply to −1, so the perpendicular slope is the negative reciprocal of the given slope.

Circle equation

The circle with center (h, k) and radius r has equation (x − h)² + (y − k)² = r². Expanding it gives a general quadratic in x and y.

Dot product

For u = (u₁, u₂) and v = (v₁, v₂), u·v = u₁v₁ + u₂v₂ = |u||v|cos θ. The vectors are perpendicular exactly when the dot product is zero.

Vector projection

The projection of u onto nonzero v is the vector (u·v / v·v)·v, pointing along v with length |u·v|/|v|.