Formula sheet for the Mathematics Matura

Logical operations

\(\top\) – true, \(\bot\) – false

Logical laws

\(\displaystyle \neg(\neg p)\Leftrightarrow p\) double negation

\(\displaystyle \neg(p\wedge q)\Leftrightarrow \neg p\vee\neg q\) De Morgan’s laws

\(\displaystyle \neg(p\vee q)\Leftrightarrow \neg p\wedge\neg q\)

\(\displaystyle (p\Rightarrow q)\Leftrightarrow(\neg p\vee q)\) replacing the implication

\(\displaystyle (p\Rightarrow q)\Leftrightarrow(\neg q\Rightarrow\neg p)\) contrapositive

\(\displaystyle p\vee\neg p\) excluded middle

\(\displaystyle \neg(p\wedge\neg p)\) non-contradiction

\(\displaystyle p\wedge q\Leftrightarrow q\wedge p,\quad p\vee q\Leftrightarrow q\vee p\) commutative law

\(\displaystyle (p\wedge q)\wedge r\Leftrightarrow p\wedge(q\wedge r)\) associative law (also for \(\vee\))

\(\displaystyle p\wedge(q\vee r)\Leftrightarrow(p\wedge q)\vee(p\wedge r)\) distributive laws

\(\displaystyle p\vee(q\wedge r)\Leftrightarrow(p\vee q)\wedge(p\vee r)\)

\(\displaystyle \big((p\Rightarrow q)\wedge p\big)\Rightarrow q\) modus ponens

\(\displaystyle \big((p\Rightarrow q)\wedge\neg q\big)\Rightarrow\neg p\) modus tollens

\(\displaystyle \big((p\Rightarrow q)\wedge(q\Rightarrow r)\big)\Rightarrow(p\Rightarrow r)\) hypothetical syllogism

Sets

\(\displaystyle A\cup B=\{x\mid x\in A\vee x\in B\}\) union

\(\displaystyle A\cap B=\{x\mid x\in A\wedge x\in B\}\) intersection

\(\displaystyle A\setminus B=\{x\mid x\in A\wedge x\notin B\}\) difference

\(\displaystyle A'=U\setminus A\) complement

\(\displaystyle (A\cup B)'=A'\cap B',\quad (A\cap B)'=A'\cup B'\) De Morgan’s laws

\(\displaystyle A\cap(B\cup C)=(A\cap B)\cup(A\cap C)\)

\(\displaystyle A\cup(B\cap C)=(A\cup B)\cap(A\cup C)\)

\(\displaystyle n(A\cup B)=n(A)+n(B)-n(A\cap B)\) number of elements

\(\displaystyle 2^n\) subsets of a set with \(n\) elements

Numbers and divisibility

\(\displaystyle \mathbb N\subset\mathbb Z\subset\mathbb Q\subset\mathbb R\subset\mathbb C\)

\(\displaystyle \mathbb Q=\left\{\tfrac mn\;\middle|\; m\in\mathbb Z,\ n\in\mathbb N\right\}\)

\(\displaystyle \gcd(a,b)\cdot \operatorname{lcm}(a,b)=a\cdot b\)

Divisibility: by 2 – the last digit is even; by 3 (9) – the sum of the digits is divisible by 3 (9); by 4 – the number formed by the last two digits is divisible by 4; by 5 – it ends in 0 or 5; by 10 – it ends in 0.

Absolute value and intervals

\(\displaystyle |x|=\begin{cases}x, & x\ge 0\\ -x, & x<0\end{cases}\)

\(\displaystyle |a\cdot b|=|a|\cdot|b|,\quad \left|\frac ab\right|=\frac{|a|}{|b|}\)

\(\displaystyle |x|<a\ \Leftrightarrow\ -a<x<a\) for \(a>0\)

\(\displaystyle |x|>a\ \Leftrightarrow\ x<-a\ \vee\ x>a\) for \(a>0\)

\(\displaystyle d(A,B)=|b-a|\) distance between \(A(a)\) and \(B(b)\) on the number line

\(\displaystyle (a,b)=\{x\mid a<x<b\},\quad [a,b]=\{x\mid a\le x\le b\}\)

\(\displaystyle [a,b)=\{x\mid a\le x<b\},\quad (a,+\infty)=\{x\mid x>a\}\)

Roots

\(\displaystyle \sqrt{a\cdot b}=\sqrt a\cdot\sqrt b,\quad \sqrt{\frac ab}=\frac{\sqrt a}{\sqrt b}\)

\(\displaystyle \left(\sqrt a\right)^2=a,\quad \sqrt{a^2}=|a|\)

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

\(\displaystyle \frac{1}{\sqrt a}=\frac{\sqrt a}{a},\quad \frac{1}{\sqrt a-\sqrt b}=\frac{\sqrt a+\sqrt b}{a-b}\) rationalising the denominator

Complex numbers

\(\displaystyle i^2=-1,\quad z=a+bi,\quad a=\operatorname{Re}z,\ b=\operatorname{Im}z\) real and imaginary part

\(\displaystyle \bar z=a-bi,\quad -z=-a-bi\) conjugate and opposite

\(\displaystyle a+bi=c+di\ \Leftrightarrow\ a=c\ \wedge\ b=d\)

\(\displaystyle (a+bi)\pm(c+di)=(a\pm c)+(b\pm d)i\)

\(\displaystyle (a+bi)(c+di)=(ac-bd)+(ad+bc)i\)

\(\displaystyle \frac{a+bi}{c+di}=\frac{(a+bi)(c-di)}{c^2+d^2}\)

\(\displaystyle |z|=\sqrt{a^2+b^2},\quad z\cdot\bar z=|z|^2\) modulus

\(\displaystyle i^{4k}=1,\ \ i^{4k+1}=i,\ \ i^{4k+2}=-1,\ \ i^{4k+3}=-i\)

The number \(z=a+bi\) is the point \((a,b)\) in the complex plane.

Proportions

\(\displaystyle a:b=c:d\ \Leftrightarrow\ a\cdot d=b\cdot c\)

\(\displaystyle x:y:z=a:b:c\ \Rightarrow\ x=ak,\ y=bk,\ z=ck\) extended proportion

\(\displaystyle y=k\cdot x\quad\left(\tfrac yx=k\right)\) direct proportion

\(\displaystyle y=\frac kx\quad(x\cdot y=k)\) inverse proportion

\(\displaystyle k=\frac{S}{a+b+c}\) sharing in a ratio: \(S\) is shared in the ratio \(a:b:c\) into \(ak, bk, ck\)

Percentages and interest

\(\displaystyle D=\frac{G\cdot p}{100}\) \(G\) – base value, \(p\) – percentage rate, \(D\) – percentage amount

\(\displaystyle G\left(1+\frac{p}{100}\right),\quad G\left(1-\frac{p}{100}\right)\) increase and decrease by \(p\%\)

\(\displaystyle G=\frac{100\,(G+D)}{100+p}\) “above a hundred”: the increased value \(G+D\) is known

\(\displaystyle G=\frac{100\,(G-D)}{100-p}\) “below a hundred”: the decreased value \(G-D\) is known

\(\displaystyle I=\frac{K\cdot p\cdot t}{100},\quad I=\frac{K\cdot p\cdot m}{1200},\quad I=\frac{K\cdot p\cdot d}{36500}\) simple interest: \(K\) – principal, \(p\) – yearly rate, \(t\) years, \(m\) months, \(d\) days

\(\displaystyle K_n=K\left(1+\frac{p}{100}\right)^n\) compound interest after \(n\) years

Powers

\(\displaystyle a^m\cdot a^n=a^{m+n},\quad a^m:a^n=a^{m-n}\)

\(\displaystyle \left(a^m\right)^n=a^{m\cdot n}\)

\(\displaystyle (a\cdot b)^n=a^n\cdot b^n,\quad \left(\frac ab\right)^n=\frac{a^n}{b^n}\)

\(\displaystyle a^0=1,\quad a^{-n}=\frac{1}{a^n}\) for \(a\neq 0\)

\(\displaystyle (-a)^{2n}=a^{2n},\quad (-a)^{2n+1}=-a^{2n+1}\)

Special products

\(\displaystyle (a\pm b)^2=a^2\pm 2ab+b^2\)

\(\displaystyle (a\pm b)^3=a^3\pm 3a^2b+3ab^2\pm b^3\)

\(\displaystyle a^2-b^2=(a-b)(a+b)\)

\(\displaystyle a^3\pm b^3=(a\pm b)\left(a^2\mp ab+b^2\right)\)

\(\displaystyle (a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ca\)

Algebraic fractions

\(\displaystyle \frac ab\pm\frac cd=\frac{ad\pm bc}{bd}\)

\(\displaystyle \frac ab\cdot\frac cd=\frac{ac}{bd},\quad \frac ab:\frac cd=\frac{ad}{bc}\)

It is defined when the denominator is not zero.

Linear equations and inequalities

\(\displaystyle ax=b,\ a\neq0\ \Rightarrow\ x=\frac ba\) one solution

\(\displaystyle a=0,\ b=0\) every \(x\) is a solution

\(\displaystyle a=0,\ b\neq0\) no solution

\(\displaystyle a<b,\ c<0\ \Rightarrow\ a\cdot c>b\cdot c\) multiplying or dividing by a negative number reverses the inequality

Linear function

\(\displaystyle y=kx+n\) \(k\) – slope, \(n\) – \(y\)-intercept

\(\displaystyle x_0=-\frac nk\) zero of the function; axis intercepts \((x_0,0)\) and \((0,n)\)

\(\displaystyle k>0\) increasing; \(k<0\) – decreasing; \(k=0\) – constant

\(\displaystyle k=\frac{y_2-y_1}{x_2-x_1},\quad y-y_1=k(x-x_1)\) line through two points

\(\displaystyle k_1=k_2\) parallel lines

\(\displaystyle k_1\cdot k_2=-1\) perpendicular lines

\(\displaystyle \frac xm+\frac yn=1\) intercept form

System of linear equations

\(\displaystyle \begin{cases}a_1x+b_1y=c_1\\ a_2x+b_2y=c_2\end{cases}\)

\(\displaystyle D=\begin{vmatrix}a_1&b_1\\a_2&b_2\end{vmatrix}=a_1b_2-a_2b_1\)

\(\displaystyle D_x=c_1b_2-c_2b_1,\quad D_y=a_1c_2-a_2c_1\)

\(\displaystyle D\neq0:\ x=\frac{D_x}{D},\ y=\frac{D_y}{D}\) one solution (the lines intersect)

\(\displaystyle D=0,\ D_x\neq0\ \vee\ D_y\neq0\) no solution (parallel lines)

\(\displaystyle D=D_x=D_y=0\) infinitely many solutions (the lines coincide)

Quadratic equation

\(\displaystyle ax^2+bx+c=0,\ \ a\neq0\)

\(\displaystyle D=b^2-4ac,\quad x_{1,2}=\frac{-b\pm\sqrt D}{2a}\) discriminant

\(\displaystyle D>0\) two different real roots; \(D=0\) – one double real root \(x_1=x_2=-\frac{b}{2a}\); \(D<0\) – two complex conjugate roots

\(\displaystyle ax^2+bx=0\ \Rightarrow\ x_1=0,\ x_2=-\frac ba\) incomplete equations

\(\displaystyle ax^2+c=0\ \Rightarrow\ x^2=-\frac ca\)

\(\displaystyle ax^2+bx+c=a(x-x_1)(x-x_2)\) factorising the trinomial

Vieta’s formulas

\(\displaystyle x_1+x_2=-\frac ba,\quad x_1\cdot x_2=\frac ca\)

\(\displaystyle x^2+px+q=0:\ \ x_1+x_2=-p,\ \ x_1x_2=q\)

\(\displaystyle x^2-(x_1+x_2)\,x+x_1x_2=0\) equation with roots \(x_1\) and \(x_2\)

\(\displaystyle x_1^2+x_2^2=(x_1+x_2)^2-2x_1x_2\)

\(\displaystyle \frac{1}{x_1}+\frac{1}{x_2}=\frac{x_1+x_2}{x_1x_2}\)

Quadratic function

\(\displaystyle y=ax^2+bx+c,\ \ a\neq0\)

\(\displaystyle T\left(-\frac{b}{2a},\ \frac{4ac-b^2}{4a}\right)\) vertex of the parabola

\(\displaystyle y=a(x-\alpha)^2+\beta,\quad \alpha=-\frac{b}{2a},\ \beta=-\frac{D}{4a}\) vertex form

\(\displaystyle a>0\) opens upwards, minimum \(\beta\); \(a<0\) – downwards, maximum \(\beta\)

\(\displaystyle x=\alpha\) axis of symmetry; \(y\)-intercept \((0,c)\)

\(\displaystyle a>0,\ D>0:\ \ ax^2+bx+c<0\ \Leftrightarrow\ x_1<x<x_2\) quadratic inequality (\(x_1<x_2\)); \(>0\) for \(x<x_1\) or \(x>x_2\)

\(\displaystyle a>0,\ D<0:\ \ ax^2+bx+c>0\) for every real \(x\)

Triangle

\(\displaystyle \alpha+\beta+\gamma=180^\circ,\quad \alpha_1=\beta+\gamma\) an exterior angle equals the sum of the two opposite interior angles

\(\displaystyle |a-b|<c<a+b\) triangle inequality

\(\displaystyle L=a+b+c,\quad P=\frac{a\cdot h_a}{2}\)

\(\displaystyle P=\sqrt{s(s-a)(s-b)(s-c)},\quad s=\frac{a+b+c}{2}\) Heron’s formula

\(\displaystyle P=\frac{a\cdot b\cdot\sin\gamma}{2}\)

\(\displaystyle r=\frac Ps,\quad R=\frac{abc}{4P}\) radius of the inscribed and circumscribed circle

Right and equilateral triangle

\(\displaystyle c^2=a^2+b^2\) Pythagoras’ theorem (\(c\) – hypotenuse)

\(\displaystyle P=\frac{a\cdot b}{2},\quad h_c=\frac{a\cdot b}{c}\)

\(\displaystyle a^2=c\cdot p,\quad b^2=c\cdot q,\quad h_c^2=p\cdot q\) \(p, q\) – projections of the legs on the hypotenuse

\(\displaystyle R=\frac c2,\quad r=\frac{a+b-c}{2}\)

\(\displaystyle h=\frac{a\sqrt3}{2},\quad P=\frac{a^2\sqrt3}{4}\) equilateral triangle

\(\displaystyle R=\frac{a\sqrt3}{3},\quad r=\frac{a\sqrt3}{6}\)

Quadrilaterals

\(\displaystyle L=4a,\quad P=a^2,\quad d=a\sqrt2\) square

\(\displaystyle L=2(a+b),\quad P=a\cdot b,\quad d=\sqrt{a^2+b^2}\) rectangle

\(\displaystyle L=2(a+b),\quad P=a\cdot h_a=b\cdot h_b\) parallelogram

\(\displaystyle L=4a,\quad P=a\cdot h=\frac{d_1\cdot d_2}{2}\) rhombus

\(\displaystyle P=\frac{(a+b)\cdot h}{2}=m\cdot h,\quad m=\frac{a+b}{2}\) trapezoid (\(m\) – midline)

\(\displaystyle P=\frac{d_1\cdot d_2}{2}\) kite and any quadrilateral with perpendicular diagonals

\(\displaystyle \alpha+\beta+\gamma+\delta=360^\circ\) general quadrilateral: split it into two triangles with a diagonal to find the area

Polygons

\(\displaystyle (n-2)\cdot180^\circ\) sum of the interior angles; the exterior angles add up to \(360^\circ\)

\(\displaystyle D_n=\frac{n(n-3)}{2}\) number of diagonals; from one vertex \(n-3\)

\(\displaystyle \alpha=\frac{(n-2)\cdot180^\circ}{n},\quad \varphi=\frac{360^\circ}{n}\) regular polygon: interior and central angle

\(\displaystyle P=\frac{n\cdot a\cdot r}{2}\) \(r\) – radius of the inscribed circle

\(\displaystyle P=\frac{3a^2\sqrt3}{2},\quad R=a\) regular hexagon

Circle

\(\displaystyle L=2r\pi,\quad P=r^2\pi\)

\(\displaystyle l=\frac{r\pi\alpha}{180^\circ}\) arc length

\(\displaystyle P_i=\frac{r^2\pi\alpha}{360^\circ}=\frac{l\cdot r}{2}\) sector

\(\displaystyle P_o=P_i-P_\triangle\) segment

\(\displaystyle P=(R^2-r^2)\pi\) annulus

\(\displaystyle \alpha=\frac{\varphi}{2}\) an inscribed angle is half the central angle on the same arc; on a diameter it is \(90^\circ\)

Vectors

\(\displaystyle \overrightarrow{AB}=(x_2-x_1,\ y_2-y_1)\) for \(A(x_1,y_1)\), \(B(x_2,y_2)\)

\(\displaystyle |\vec a|=\sqrt{a_1^2+a_2^2}\) magnitude

\(\displaystyle \vec a\pm\vec b=(a_1\pm b_1,\ a_2\pm b_2),\quad k\vec a=(ka_1,\ ka_2)\)

\(\displaystyle \overrightarrow{AB}+\overrightarrow{BC}=\overrightarrow{AC},\quad \overrightarrow{BA}=-\overrightarrow{AB}\) triangle rule; opposite vector

\(\displaystyle \vec a=k\cdot\vec b\) collinear vectors

\(\displaystyle d(A,B)=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\) distance between two points

\(\displaystyle M\left(\frac{x_1+x_2}{2},\ \frac{y_1+y_2}{2}\right)\) midpoint of a segment

Prism

\(\displaystyle P=2B+M,\quad V=B\cdot H\)

\(\displaystyle P=6a^2,\quad V=a^3,\quad D=a\sqrt3\) cube

\(\displaystyle P=2(ab+bc+ca),\quad V=abc,\quad D=\sqrt{a^2+b^2+c^2}\) cuboid

Pyramid

\(\displaystyle P=B+M,\quad V=\frac{B\cdot H}{3}\)

\(\displaystyle M=\frac{n\cdot a\cdot h}{2}\) regular pyramid (\(h\) – slant height of a face)

\(\displaystyle P=B_1+B_2+M,\quad V=\frac H3\left(B_1+\sqrt{B_1B_2}+B_2\right)\) frustum of a pyramid

Cylinder and cone

\(\displaystyle M=2r\pi H,\quad P=2r\pi(r+H),\quad V=r^2\pi H\) cylinder; equilateral: \(H=2r\)

\(\displaystyle s^2=r^2+H^2,\quad M=r\pi s\) cone

\(\displaystyle P=r\pi(r+s),\quad V=\frac{r^2\pi H}{3}\)

\(\displaystyle s^2=H^2+(R-r)^2,\quad M=\pi s(R+r)\) frustum of a cone

\(\displaystyle P=\pi\left(R^2+r^2+s(R+r)\right),\quad V=\frac{\pi H}{3}\left(R^2+Rr+r^2\right)\)

Sphere

\(\displaystyle P=4R^2\pi,\quad V=\frac{4R^3\pi}{3}\)

\(\displaystyle P=2R\pi h\) spherical cap and zone (\(h\) – height)

\(\displaystyle V=\frac{\pi h^2(3R-h)}{3}\) spherical segment

\(\displaystyle V=\frac{2R^2\pi h}{3}\) spherical sector

\(\displaystyle \rho=\sqrt{R^2-d^2}\) radius of the cross-section by a plane at distance \(d\) from the centre

Right triangle

\(\displaystyle \sin\alpha=\frac ac,\quad \cos\alpha=\frac bc\)

\(\displaystyle \tan \alpha=\frac ab,\quad \cot \alpha=\frac ba\) \(a\) – opposite side, \(b\) – adjacent side, \(c\) – hypotenuse

\(\displaystyle \alpha_{\text{rad}}=\frac{\alpha^\circ\cdot\pi}{180^\circ},\quad 180^\circ=\pi\) degrees and radians

Identities

\(\displaystyle \sin^2\alpha+\cos^2\alpha=1\)

\(\displaystyle \tan \alpha=\frac{\sin\alpha}{\cos\alpha},\quad \cot \alpha=\frac{\cos\alpha}{\sin\alpha}\)

\(\displaystyle \tan \alpha\cdot \cot \alpha=1\)

\(\displaystyle 1+\tan^2\alpha=\frac{1}{\cos^2\alpha},\quad 1+\cot^2\alpha=\frac{1}{\sin^2\alpha}\)

\(\displaystyle \sin(90^\circ-\alpha)=\cos\alpha,\quad \cos(90^\circ-\alpha)=\sin\alpha\) complementary angles

\(\displaystyle \tan(90^\circ-\alpha)=\cot \alpha,\quad \cot(90^\circ-\alpha)=\tan \alpha\)

Values