Državna Matura Math A cheat sheet

For real numbers a, b: |a| is the distance from a to 0; |ab| = |a||b|, |a/b| = |a|/|b| (b ≠ 0), and |a + b| ≤ |a| + |b| (triangle inequality).

Laws of exponents: a^m · a^n = a^{m+n}, (a^m)^n = a^{mn}, (ab)^n = a^n b^n, a^{m/n} = \sqrt[n]{a^m} (a > 0 for non-integer exponents).

Quadratic formula: solutions of ax^2 + bx + c = 0 (a ≠ 0) are x = (-b ± √(b^2 - 4ac)) / (2a); discriminant D = b^2 - 4ac determines number of real roots: D > 0 two distinct, D = 0 one double, D < 0 none.

Vieta's formulas: if x1 and x2 are roots of ax^2 + bx + c = 0, then x1 + x2 = -b/a and x1 x2 = c/a.

Solving inequalities: multiplying or dividing by a negative number reverses the inequality sign; for rational inequalities, use a sign chart on the intervals determined by zeros and poles.

Absolute value equations: |x| = a (a ≥ 0) gives x = a or x = -a; |x| < a gives -a < x < a; |x| > a gives x < -a or x > a.

Logarithm laws: log_a(xy) = log_a x + log_a y, log_a(x/y) = log_a x - log_a y, log_a(x^r) = r log_a x, log_a a = 1, log_a 1 = 0, and change of base log_a x = log_b x / log_b a.

Binomial expansion: (a + b)^n = Σ_{k=0}^n C(n,k) a^{n-k} b^k, where C(n,k) = n! / (k!(n-k)!).

Factorisation identities: a^2 - b^2 = (a-b)(a+b), a^3 ± b^3 = (a ± b)(a^2 ∓ ab + b^2), and a^3 + b^3 + c^3 - 3abc = (a+b+c)(a^2+b^2+c^2 - ab - bc - ca).

A function f: A → B assigns to each x in A exactly one f(x) in B; the domain is A, the range is {f(x) : x ∈ A}.

Linear function f(x) = kx + n has graph a line with slope k and y-intercept n; it is increasing if k > 0, decreasing if k < 0, constant if k = 0.

Quadratic function f(x) = ax^2 + bx + c (a ≠ 0) has vertex at x = -b/(2a); opens upward if a > 0, downward if a < 0; axis of symmetry x = -b/(2a).

Exponential function f(x) = a^x (a > 0, a ≠ 1): domain ℝ, range (0, ∞), increasing if a > 1, decreasing if 0 < a < 1; passes through (0,1).

Logarithmic function f(x) = log_a x (a > 0, a ≠ 1): domain (0, ∞), range ℝ; it is the inverse of a^x; passes through (1,0).

Transformations: y = f(x) + c shifts graph vertically by c; y = f(x - c) shifts horizontally by c; y = -f(x) reflects across x-axis; y = f(-x) reflects across y-axis; y = a f(x) stretches vertically by factor |a|.

Composition: (f ∘ g)(x) = f(g(x)); the domain of f ∘ g is all x in domain of g such that g(x) is in domain of f.

Inverse function: f^{-1} exists iff f is one-to-one; its graph is the reflection of the graph of f across the line y = x; f^{-1}(f(x)) = x and f(f^{-1}(y)) = y.

Polynomial of degree n has at most n real roots; rational function f(x) = p(x)/q(x) has vertical asymptotes where q(x) = 0 (and p(x) ≠ 0), and horizontal asymptote determined by degrees of p and q.

Triangle area: A = (1/2) b h; Heron's formula: A = √(s(s-a)(s-b)(s-c)) where s = (a+b+c)/2.

Pythagorean theorem: in a right triangle with legs a, b and hypotenuse c, a^2 + b^2 = c^2.

Circle: circumference C = 2πr, area A = πr^2; arc length s = rθ (θ in radians), sector area = (1/2) r^2 θ.

Volume and surface area: cylinder V = πr^2 h, S = 2πr^2 + 2πrh; cone V = (1/3)πr^2 h, S = πr^2 + πrℓ (ℓ slant height); sphere V = (4/3)πr^3, S = 4πr^2.

Trigonometric ratios in right triangle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.

Fundamental identity: sin^2 θ + cos^2 θ = 1; tan θ = sin θ / cos θ; 1 + tan^2 θ = sec^2 θ.

Sine rule: a/sin A = b/sin B = c/sin C = 2R (R circumradius); cosine rule: c^2 = a^2 + b^2 - 2ab cos C.

Exact values: sin 30° = 1/2, cos 30° = √3/2, tan 30° = √3/3; sin 45° = cos 45° = √2/2, tan 45° = 1; sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3.

Radian-degree conversion: π radians = 180°; 1 radian = 180/π degrees; 1° = π/180 radians.

Arithmetic sequence: a_n = a_1 + (n-1)d; sum of first n terms S_n = (n/2)(2a_1 + (n-1)d) = (n/2)(a_1 + a_n).

Geometric sequence: a_n = a_1 q^{n-1}; sum of first n terms S_n = a_1 (1 - q^n)/(1 - q) for q ≠ 1, S_n = n a_1 for q = 1.

Infinite geometric series converges iff |q| < 1; its sum is S = a_1 / (1 - q).

Arithmetic mean of a and b is (a+b)/2; geometric mean of positive a and b is √(ab).

If a, b, c are consecutive terms of an arithmetic sequence, then 2b = a + c; if geometric, then b^2 = ac (for positive terms).

Sum of first n natural numbers: Σ_{k=1}^n k = n(n+1)/2; sum of squares: Σ k^2 = n(n+1)(2n+1)/6; sum of cubes: Σ k^3 = [n(n+1)/2]^2.

Recursive sequences: a_{n+1} = f(a_n) with given a_1; terms are found by iteration.

Simple and compound interest: simple interest A = P(1 + rt); compound interest A = P(1 + r)^t (annual compounding).

Distance between points A(x1,y1) and B(x2,y2): d = √((x2-x1)^2 + (y2-y1)^2); midpoint coordinates ((x1+x2)/2, (y1+y2)/2).

Line equation forms: slope-intercept y = kx + n; point-slope y - y1 = k(x - x1); general Ax + By + C = 0; slope k = (y2-y1)/(x2-x1).

Parallel lines have equal slopes (k1 = k2); perpendicular lines have slopes satisfying k1 k2 = -1 (if neither is vertical).

Distance from point (x0,y0) to line Ax + By + C = 0 is |Ax0 + By0 + C| / √(A^2 + B^2).

Circle with centre (p,q) and radius r: (x-p)^2 + (y-q)^2 = r^2; general form x^2 + y^2 + Dx + Ey + F = 0.

Parabola y^2 = 2px has focus (p/2, 0) and directrix x = -p/2; parabola y = ax^2 + bx + c has vertex (-b/(2a), c - b^2/(4a)).

Ellipse x^2/a^2 + y^2/b^2 = 1 (a > b > 0): foci at (±c, 0) with c^2 = a^2 - b^2; hyperbola x^2/a^2 - y^2/b^2 = 1 has foci (±c, 0) with c^2 = a^2 + b^2.

Vectors in plane: u = (u1,u2), v = (v1,v2); addition u+v = (u1+v1, u2+v2); scalar multiplication k u = (k u1, k u2); magnitude |u| = √(u1^2 + u2^2); dot product u·v = u1 v1 + u2 v2 = |u||v| cos θ.

Combinatorics: number of permutations of n distinct objects is n!; number of combinations of n objects taken k at a time is C(n,k) = n!/(k!(n-k)!).

Number of permutations of n objects taken k at a time: P(n,k) = n!/(n-k)!. Permutations with repetition: n^r.

Probability of event A: P(A) = number of favourable outcomes / total number of equally likely outcomes; 0 ≤ P(A) ≤ 1.

Complement rule: P(A') = 1 - P(A); addition rule: P(A ∪ B) = P(A) + P(B) - P(A ∩ B); for mutually exclusive events, P(A ∪ B) = P(A) + P(B).

Conditional probability: P(A|B) = P(A ∩ B)/P(B) (P(B) > 0); independent events satisfy P(A ∩ B) = P(A)P(B).

Binomial distribution: probability of exactly k successes in n independent trials with success probability p is C(n,k) p^k (1-p)^{n-k}.

Descriptive statistics: mean = (Σ x_i)/n; median is middle value of ordered data; mode is most frequent value; range = max - min.

Variance: σ^2 = (Σ (x_i - μ)^2)/n (population) or s^2 = (Σ (x_i - x̄)^2)/(n-1) (sample); standard deviation is the square root of variance.

Quartiles: Q1 is median of lower half, Q3 median of upper half; interquartile range IQR = Q3 - Q1; outliers are values below Q1 - 1.5 IQR or above Q3 + 1.5 IQR.

Limit definition of derivative: f'(x) = lim_{h→0} (f(x+h) - f(x))/h; derivative represents instantaneous rate of change and slope of tangent.

Derivative rules: (x^n)' = n x^{n-1}; (e^x)' = e^x; (ln x)' = 1/x; (sin x)' = cos x; (cos x)' = -sin x; (tan x)' = 1/cos^2 x.

Product rule: (uv)' = u'v + uv'; quotient rule: (u/v)' = (u'v - uv')/v^2; chain rule: (f(g(x)))' = f'(g(x)) g'(x).

Critical points occur where f'(x) = 0 or f'(x) is undefined; first derivative test: sign change from + to - gives local maximum, - to + gives local minimum.

Second derivative test: if f'(c) = 0 and f''(c) > 0, then f has local minimum at c; if f''(c) < 0, local maximum.

Indefinite integral: ∫ x^n dx = x^{n+1}/(n+1) + C (n ≠ -1); ∫ (1/x) dx = ln|x| + C; ∫ e^x dx = e^x + C; ∫ cos x dx = sin x + C; ∫ sin x dx = -cos x + C.

Definite integral: ∫_a^b f(x) dx = F(b) - F(a) where F is any antiderivative of f (Fundamental Theorem of Calculus).

Area between curves: if f(x) ≥ g(x) on [a,b], area = ∫_a^b (f(x) - g(x)) dx.

L'Hôpital's rule: for limits of form 0/0 or ∞/∞, lim f(x)/g(x) = lim f'(x)/g'(x), provided the latter limit exists.