Bulgaria DZI Mathematics flashcards

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

De Morgan's laws

For sets A and B, the complement of their union equals the intersection of their complements, and the complement of their intersection equals the union of their complements.

Contrapositive equivalence

The implication P ⇒ Q is logically equivalent to its contrapositive ¬Q ⇒ ¬P; both have the same truth value in every case.

Triangle inequality for reals

For any real numbers a and b, the absolute value of their sum is at most the sum of their absolute values: |a + b| ≤ |a| + |b|.

Quadratic formula

The real roots of ax² + bx + c = 0 with a ≠ 0 are x = [−b ± √(b² − 4ac)] / (2a), provided the discriminant b² − 4ac is non-negative.

Vieta's formulas

For the quadratic equation ax² + bx + c = 0 with roots x₁ and x₂, the sum of the roots is −b/a and the product is c/a.

Absolute value equation

The equation |ax + b| = c with c ≥ 0 is equivalent to the two linear equations ax + b = c and ax + b = −c; if c < 0 there is no solution.

Even and odd functions

A function f is even if f(−x) = f(x) for all x in its domain, and odd if f(−x) = −f(x); even graphs are symmetric about the y-axis, odd graphs about the origin.

Vertex of a parabola

For f(x) = ax² + bx + c with a ≠ 0, the vertex lies on the axis of symmetry x = −b/(2a); its y-coordinate is f(−b/(2a)).

Inverse function condition

A function f has an inverse function f⁻¹ exactly when f is one-to-one; the graph of f⁻¹ is the reflection of the graph of f across the line y = x.

Logarithm definition

For a > 0, a ≠ 1, and b > 0, log_a b is the exponent c such that a^c = b; equivalently, log_a b = c if and only if a^c = b.

Product rule for logarithms

For positive x and y and base a > 0, a ≠ 1, the logarithm of a product equals the sum of the logarithms: log_a(xy) = log_a x + log_a y.

Change of base formula

For positive a, b, c with a ≠ 1 and c ≠ 1, log_a b = log_c b / log_c a; in particular, log_a b = 1 / log_b a.

Pythagorean identity

For every real angle α, the sum of the squares of sine and cosine equals one: sin² α + cos² α = 1.

Cosine rule

In a triangle with sides a, b, c and angle A opposite side a, a² = b² + c² − 2bc cos A; equivalently cos A = (b² + c² − a²)/(2bc).

Sine rule

In a triangle with sides a, b, c opposite angles A, B, C, the ratios a/sin A = b/sin B = c/sin C are equal and equal 2R, where R is the circumradius.

Double-angle formula for sine

For any angle α, sin 2α = 2 sin α cos α; this expresses the sine of twice an angle in terms of the sine and cosine of the angle.

Arithmetic progression nth term

In an arithmetic progression with first term a₁ and common difference d, the nth term is a_n = a₁ + (n − 1)d.

Geometric progression sum

For a geometric progression with first term a₁ and ratio q ≠ 1, the sum of the first n terms is S_n = a₁(1 − q^n)/(1 − q).

Infinite geometric series sum

If |q| < 1, the sum of the infinite geometric series with first term a₁ and ratio q is S = a₁/(1 − q); if |q| ≥ 1 the series diverges.

Combinations formula

The number of ways to choose k objects from n distinct objects without regard to order is C(n, k) = n! / [k!(n − k)!].

Conditional probability

For events A and B with P(B) > 0, the probability of A given B is P(A | B) = P(A ∩ B) / P(B).

Independent events

Two events A and B are independent exactly when P(A ∩ B) = P(A) · P(B); equivalently, P(A | B) = P(A) when P(B) > 0.

Variance of data

For data values x₁, …, x_n with mean x̄, the variance is σ² = [(x₁ − x̄)² + … + (x_n − x̄)²]/n; the standard deviation is σ = √σ².

Area of a triangle

The area of a triangle equals half the product of two sides and the sine of the included angle: S = (1/2)ab sin C.

Inscribed angle theorem

An angle inscribed in a circle has measure equal to half the measure of its intercepted arc; an angle inscribed in a semicircle is a right angle.

Tangent-secant theorem

If a tangent from an external point P touches a circle at T and a secant from P meets the circle at A and B, then PT² = PA · PB.

Volume of a pyramid

The volume of a pyramid equals one third of the product of its base area B and its height h: V = (1/3)B · h.

Volume of a sphere

The volume of a sphere of radius r is V = (4/3)πr³, and its surface area is S = 4πr².

Perpendicular line and plane

A line is perpendicular to a plane if it is perpendicular to every line in the plane passing through its foot; it suffices that it be perpendicular to two intersecting lines in the plane.

Distance between two points

For points A(x₁, y₁) and B(x₂, y₂) in the plane, the distance AB = √[(x₂ − x₁)² + (y₂ − y₁)²].

Perpendicular lines slopes

Two non-vertical lines are perpendicular exactly when the product of their slopes is −1; parallel non-vertical lines have equal slopes.

Circle equation

The circle with centre (a, b) and radius r has equation (x − a)² + (y − b)² = r²; expanded, it is x² + y² + Dx + Ey + F = 0.