NMT / ZNO

Formulas and reference

A compact mathematics reference in two levels: the NMT exam sheet and the full ZNO formula set.

NMT — exam reference sheet · Algebra · 63 formulas

NMT mode Shows only the compact set matching the NMT-2026 reference materials.

Numbers and absolute value

Absolute value

Absolute value of a number

\[|a|=\begin{cases}a,&a\ge0\\-a,&a\lt 0\end{cases}\]

Powers and roots

Powers

Zero exponent

\[a^0=1\]

Condition \(a\ne0\)

Negative integer exponent

\[a^{-n}=\frac1{a^n}\]

Condition \(a\ne0,\ n\in\mathbb N\)

Product of powers

\[a^m a^n=a^{m+n}\]

Quotient of powers

\[\frac{a^m}{a^n}=a^{m-n}\]

Condition \(a\ne0\)

Power of a power

\[(a^m)^n=a^{mn}\]

Power of a product

\[(ab)^n=a^n b^n\]

Power of a quotient

\[\left(\frac ab\right)^n=\frac{a^n}{b^n}\]

Condition \(b\ne0\)

Rational exponent

\[a^{\frac mn}=\sqrt[n]{a^m}\]

Condition \(a\gt 0,\ m\in\mathbb Z,\ n\in\mathbb N,\ n\ge2\)

Powers and roots

Roots

Square root of a square

\[\sqrt{a^2}=|a|\]

Algebraic identities

Difference of squares

\[a^2-b^2=(a-b)(a+b)\]

Square of a sum

\[(a+b)^2=a^2+2ab+b^2\]

Square of a difference

\[(a-b)^2=a^2-2ab+b^2\]

Quadratic equation

\[ax^2+bx+c=0\]

Condition \(a\ne0\)

Discriminant

\[D=b^2-4ac\]

Roots for D>0

\[x_1=\frac{-b-\sqrt D}{2a}\]

\[x_2=\frac{-b+\sqrt D}{2a}\]

Condition \(D\gt 0\)

Root for D=0

\[x_1=x_2=-\frac b{2a}\]

Condition \(D=0\)

Factorization of a quadratic trinomial

\[ax^2+bx+c=a(x-x_1)(x-x_2)\]

Sequences

Arithmetic progression

n-th term of an arithmetic progression

\[a_n=a_1+d(n-1)\]

Sum of the first n terms of an arithmetic progression

\[S_n=\frac{a_1+a_n}{2}\,n\]

Sequences

Geometric progression

n-th term of a geometric progression

\[b_n=b_1q^{\,n-1}\]

Sum of the first n terms of a geometric progression

\[S_n=\frac{b_1(q^n-1)}{q-1}\]

Condition \(q\ne1\)

Exponential expressions and logarithms

Logarithms

Logarithm of one

\[\log_a1=0\]

Condition \(a\gt 0,\ a\ne1\)

Logarithm of the base

\[\log_a a=1\]

Condition \(a\gt 0,\ a\ne1\)

Fundamental logarithmic identity

\[a^{\log_a b}=b\]

Condition \(a\gt 0,\ a\ne1,\ b\gt 0\)

Logarithm of a product

\[\log_a(bc)=\log_ab+\log_ac\]

Condition \(a\gt 0,\ a\ne1,\ b\gt 0,\ c\gt 0\)

Logarithm of a quotient

\[\log_a\frac bc=\log_ab-\log_ac\]

Condition \(a\gt 0,\ a\ne1,\ b\gt 0,\ c\gt 0\)

Logarithm of a power

\[\log_a b^n=n\log_a b\]

Condition \(a\gt 0,\ a\ne1,\ b\gt 0\)

Logarithm of a root

\[\log_a\sqrt[k]{b}=\frac1k\log_a b\]

Condition \(a\gt 0,\ a\ne1,\ b\gt 0,\ k\ne0\)

Trigonometry

Fundamental trigonometric identity

\[\sin^2\alpha+\cos^2\alpha=1\]

Tangent

\[\tg\alpha=\frac{\sin\alpha}{\cos\alpha}\]

Condition \(\cos\alpha\ne0\)

Tangent identity

\[1+\tg^2\alpha=\frac1{\cos^2\alpha}\]

Condition \(\cos\alpha\ne0\)

Sine of a double angle

\[\sin2\alpha=2\sin\alpha\cos\alpha\]

Cosine of a double angle

\[\cos2\alpha=\cos^2\alpha-\sin^2\alpha\]

\[\cos2\alpha=2\cos^2\alpha-1\]

\[\cos2\alpha=1-2\sin^2\alpha\]

Formulas for 90°+α

\[\sin(90^\circ+\alpha)=\cos\alpha\]

\[\cos(90^\circ+\alpha)=-\sin\alpha\]

\[\tg(90^\circ+\alpha)=-\frac1{\tg\alpha}\]

Formulas for 180°−α

\[\sin(180^\circ-\alpha)=\sin\alpha\]

\[\cos(180^\circ-\alpha)=-\cos\alpha\]

\[\tg(180^\circ-\alpha)=-\tg\alpha\]

Derivative

Derivatives

Derivative of a constant

\[(C)'=0\]

Derivative of x

\[x'=1\]

Derivative of a power function

\[(x^\alpha)'=\alpha x^{\alpha-1}\]

Derivative of the square root

\[(\sqrt{x})'=\frac1{2\sqrt{x}}\]

Derivative of e^x

\[(e^x)'=e^x\]

Derivative of ln x

\[(\ln x)'=\frac1x\]

Derivative of sin x

\[(\sin x)'=\cos x\]

Derivative of cos x

\[(\cos x)'=-\sin x\]

Derivative of tg x

\[(\tg x)'=\frac1{\cos^2x}\]

Derivative

Differentiation rules

Derivative of a sum

\[(u+v)'=u'+v'\]

Derivative of a difference

\[(u-v)'=u'-v'\]

Derivative of a product

\[(uv)'=u'v+uv'\]

Derivative of a quotient

\[\left(\frac uv\right)'=\frac{u'v-uv'}{v^2}\]

Condition \(v\ne0\)

Constant factor rule

\[(Cu)'=Cu'\]

Antiderivatives and integrals

Antiderivatives

Antiderivative of 1

\[\int 1\,dx=x+C\]

Antiderivative of a power function

\[\int x^\alpha dx=\frac{x^{\alpha+1}}{\alpha+1}+C,\quad \alpha\ne-1\]

Antiderivative of 1/x

\[\int\frac{dx}{x}=\ln|x|+C\]

Antiderivative of e^x

\[\int e^x dx=e^x+C\]

Antiderivative of sin x

\[\int\sin x\,dx=-\cos x+C\]

Antiderivative of cos x

\[\int\cos x\,dx=\sin x+C\]

Antiderivative of 1/cos²x

\[\int\frac{dx}{\cos^2x}=\tg x+C\]

Antiderivatives and integrals

Definite integral

Newton–Leibniz formula

\[\int_a^b f(x)\,dx=F(b)-F(a)\]

Combinatorics

Factorial

\[n!=1\cdot2\cdot\ldots\cdot n\]

Permutations

\[P_n=n!\]

k-permutations

\[A_n^k=\frac{n!}{(n-k)!}\]

Combinations

\[C_n^k=\frac{n!}{k!(n-k)!}\]

Probability and statistics

Probability

Classical probability

\[P(A)=\frac mn\]

m — number of favorable equally likely outcomes; n — total number of outcomes.

Trigonometric values table

α	0°	30°	45°	60°	90°	180°	270°	360°
sin α	\(0\)	\(1/2\)	\(\sqrt2/2\)	\(\sqrt3/2\)	\(1\)	\(0\)	\(-1\)	\(0\)
cos α	\(1\)	\(\sqrt3/2\)	\(\sqrt2/2\)	\(1/2\)	\(0\)	\(-1\)	\(0\)	\(1\)
tg α	\(0\)	\(1/\sqrt3\)	\(1\)	\(\sqrt3\)	\(\text{—}\)	\(0\)	\(\text{—}\)	\(0\)

Squares table 10–49

10² = 100 11² = 121 12² = 144 13² = 169 14² = 196 15² = 225 16² = 256 17² = 289 18² = 324 19² = 361 20² = 400 21² = 441 22² = 484 23² = 529 24² = 576 25² = 625 26² = 676 27² = 729 28² = 784 29² = 841 30² = 900 31² = 961 32² = 1024 33² = 1089 34² = 1156 35² = 1225 36² = 1296 37² = 1369 38² = 1444 39² = 1521 40² = 1600 41² = 1681 42² = 1764 43² = 1849 44² = 1936 45² = 2025 46² = 2116 47² = 2209 48² = 2304 49² = 2401