### Teorija

Formulas optimālā līmeņa matemātikas valsts pārbaudes darbam (Skola2030)
1. daļa

 Saīsinātās reizināšanas formulas  $\begin{array}{l}{\left(a±b\right)}^{3}={a}^{3}±3{a}^{2}b+3a{b}^{2}±{b}^{3}\\ {a}^{3}+{b}^{3}=\left(a+b\right)\left({a}^{2}-\mathit{ab}+{b}^{2}\right)\\ {a}^{3}-{b}^{3}=\left(a-b\right)\left({a}^{2}+\mathit{ab}+{b}^{2}\right)\end{array}$ Kvadrāttrinoms, kvadrātvienādojums  $\begin{array}{l}a{x}^{2}+\mathit{bx}+c=a\left(x-{x}_{1}\right)\left(x-{x}_{2}\right)\\ {x}^{2}+\mathit{px}+q\\ \left\{\begin{array}{l}{x}_{1}+{x}_{2}=-p\\ {x}_{1}\cdot {x}_{2}=q\end{array}\right\\end{array}$ Aritmētiskā progresija  $\begin{array}{l}{a}_{n}={a}_{1}+\left(n-1\right)\cdot d\phantom{\rule{0.147em}{0ex}}\\ {S}_{n}=\frac{\left({a}_{1}+{a}_{n}\right)\cdot n}{2}\phantom{\rule{0.147em}{0ex}}\\ {a}_{k}=\frac{{a}_{k-1}+{a}_{k+1}}{2}\phantom{\rule{0.147em}{0ex}}\end{array}$ Ģeometriskā progresija$\begin{array}{l}{b}_{n}={b}_{1}\cdot {q}^{n-1}\phantom{\rule{0.147em}{0ex}}\\ {S}_{n}=\frac{{b}_{1}\cdot \left({q}^{n}-1\right)}{q-1}\phantom{\rule{0.147em}{0ex}}\\ {b}_{k}^{2}={b}_{k-1}\cdot {b}_{k+1\phantom{\rule{0.147em}{0ex}}}\\ \mathit{Ja}\phantom{\rule{0.147em}{0ex}}\left|q\right|<1,\mathit{tad}\phantom{\rule{0.147em}{0ex}}S=\frac{{b}_{1}}{1-q}\end{array}$ Pakāpju īpašības    $\begin{array}{l}{a}^{0}=1\phantom{\rule{0.147em}{0ex}}\left(a\ne 0\right)\\ {a}^{-n}=\frac{1}{{a}^{n}}\\ {a}^{\frac{m}{n}}=\sqrt[n]{{a}^{m}}\\ {a}^{m}\cdot {a}^{n}={a}^{m+n}\\ {a}^{m}:{a}^{n}={a}^{m-n}\\ {\left({a}^{m}\right)}^{n}={a}^{\mathit{mn}}\\ {a}^{m}\cdot {b}^{m}={\left(\mathit{ab}\right)}^{m}\\ \frac{{a}^{n}}{{b}^{n}}={\left(\frac{a}{b}\right)}^{n}\end{array}$ Sakņu īpašības    $\begin{array}{l}\sqrt[n]{a}\cdot \sqrt[n]{b}=\sqrt[n]{a\cdot b}\\ \frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}\\ \sqrt[n\cdot m]{{a}^{k\cdot m}}=\sqrt[n]{{a}^{k}}\\ \sqrt[n]{\sqrt[m]{a}}=\sqrt[n\cdot m]{a}\\ \\ \sqrt{{a}^{2}}=\left|a\right|\end{array}$ Logaritmu īpašības    $\begin{array}{l}{a}^{{\mathrm{log}}_{a}b}=b\\ \\ {\mathrm{log}}_{a}\left(x\cdot y\right)={\mathrm{log}}_{a}x+{\mathrm{log}}_{a}y\\ {\mathrm{log}}_{a}\frac{x}{y}={\mathrm{log}}_{a}x-{\mathrm{log}}_{a}y\phantom{\rule{0.147em}{0ex}}\\ {\mathrm{log}}_{a}{x}^{k}=k\cdot {\mathrm{log}}_{a}x\phantom{\rule{0.147em}{0ex}}\\ {\mathrm{log}}_{a}b=\frac{{\mathrm{log}}_{c}b}{{\mathrm{log}}_{c}a}\end{array}$ Trigonometrija Kombinatorika$\begin{array}{l}{P}_{n}=n!\phantom{\rule{0.441em}{0ex}}{\phantom{\rule{0.147em}{0ex}}A}_{n}^{k}=\frac{n!}{\left(n-k\right)!}\phantom{\rule{0.588em}{0ex}}{\overline{A}}_{n}^{k}={n}^{k}\\ {\phantom{\rule{0.147em}{0ex}}A}_{n}^{k}=n\left(n-1\right)\left(n-2\right)\cdot ...\cdot \left(n-k+1\right)\\ {C}_{n}^{k}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\frac{n!}{k!\left(n-k\right)!}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}{C}_{n}^{k}=\frac{{A}_{n}^{k}}{k!}\\ \\ {C}_{n}^{m}={C}_{n}^{n-m}\\ \\ {C}_{n}^{0}+{C}_{n}^{1}+{C}_{n}^{2}+...+{C}_{n}^{n-1}+\phantom{\rule{0.147em}{0ex}}{C}_{n}^{n}={2}^{n}\end{array}$ Trigonometrija   $\begin{array}{l}{\mathrm{sin}}^{2}\mathrm{\alpha }+{\mathrm{cos}}^{2}\mathrm{\alpha }=1\\ \\ \mathrm{sin}\phantom{\rule{0.147em}{0ex}}2\mathrm{\alpha }\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}2\phantom{\rule{0.147em}{0ex}}\mathrm{sin}\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }\phantom{\rule{0.147em}{0ex}}\cdot \phantom{\rule{0.147em}{0ex}}\mathrm{cos}\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }\phantom{\rule{0.147em}{0ex}}\\ \mathrm{cos}\phantom{\rule{0.147em}{0ex}}2\mathrm{\alpha }\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}{\mathrm{cos}}^{2}\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }\phantom{\rule{0.147em}{0ex}}-\phantom{\rule{0.147em}{0ex}}{\mathrm{sin}}^{2}\mathrm{\alpha }\\ \\ \mathrm{sin}\phantom{\rule{0.147em}{0ex}}\left(\mathrm{\alpha }±\mathrm{\beta }\right)\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\mathrm{sin}\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }\phantom{\rule{0.147em}{0ex}}\mathrm{cos}\phantom{\rule{0.147em}{0ex}}\mathrm{\beta }\phantom{\rule{0.147em}{0ex}}±\mathrm{cos}\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }\phantom{\rule{0.147em}{0ex}}\mathrm{sin}\phantom{\rule{0.147em}{0ex}}\mathrm{\beta }\\ \mathrm{cos}\phantom{\rule{0.147em}{0ex}}\left(\mathrm{\alpha }+\mathrm{\beta }\right)\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\mathrm{cos}\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }\phantom{\rule{0.147em}{0ex}}\mathrm{cos}\phantom{\rule{0.147em}{0ex}}\mathrm{\beta }\phantom{\rule{0.147em}{0ex}}-\phantom{\rule{0.147em}{0ex}}\mathrm{sin}\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }\phantom{\rule{0.147em}{0ex}}\mathrm{sin}\phantom{\rule{0.147em}{0ex}}\mathrm{\beta }\\ \mathrm{cos}\phantom{\rule{0.147em}{0ex}}\left(\mathrm{\alpha }-\mathrm{\beta }\right)\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\mathrm{cos}\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }\phantom{\rule{0.147em}{0ex}}\mathrm{cos}\phantom{\rule{0.147em}{0ex}}\mathrm{\beta }\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}\mathrm{sin}\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }\phantom{\rule{0.147em}{0ex}}\mathrm{sin}\phantom{\rule{0.147em}{0ex}}\mathrm{\beta }\end{array}$ Varbūtību teorija  Ja $A$ un $B$ - nesavienojami notikumi, $P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)$ Ja $A$ un $B$ - neatkarīgi notikumi, tad$P\left(A\cap B\right)=P\left(A\right)\cdot P\left(B\right)$ Ja $A$ un $B$ - atkarīgi notikumi, tad$P\left(A\cap B\right)=P\left(A\right)\cdot P\left(B|A\right)=P\left(B\right)\cdot P\left(A|B\right)$ Statistika${s}^{2}=\frac{1}{n}\sum _{i=1}^{n}{\left({x}_{i}-\overline{x}\right)}^{2},$kur ${s}^{2}$ - dispersija, $$s$$ - standartnovirze nesagrupētai izlasei ${\mathrm{\sigma }}^{2}=\frac{1}{n-1}\sum _{i=1}^{n}{\left({x}_{i}-\overline{x}\right)}^{2},$kur ${\mathrm{\sigma }}^{2}$ - dispersija, $\mathrm{\sigma }$ - standartnovirze populācijai, aprēķinot tās no izlases. Vektori plaknē  $\begin{array}{l}\mathit{Ja}\phantom{\rule{0.147em}{0ex}}A\left({x}_{1};{y}_{1}\right)\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\mathit{un}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}B\left({x}_{2};{y}_{2}\right),\phantom{\rule{0.147em}{0ex}}\mathit{tad}\phantom{\rule{0.147em}{0ex}}\\ \stackrel{\to }{\mathit{AB}}=\left({x}_{2}-{x}_{1};\phantom{\rule{0.147em}{0ex}}{y}_{2}-{y}_{1}\right)\\ \\ \mathit{Ja}\phantom{\rule{0.147em}{0ex}}\stackrel{\to }{a}=\left({a}_{x};{a}_{y}\right)\phantom{\rule{0.147em}{0ex}}\mathit{un}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\stackrel{\to }{b}=\left({b}_{x};{b}_{y}\right),\mathit{tad}\\ \stackrel{\to }{a}±\stackrel{\to }{b}=\left({a}_{x}±{b}_{x};\phantom{\rule{0.147em}{0ex}}{a}_{y}±{b}_{y}\right)\\ k\cdot \stackrel{\to }{a}=\left(k\cdot {a}_{x};\phantom{\rule{0.147em}{0ex}}k\cdot {a}_{y}\right)\\ \\ \left|\stackrel{\to }{a}\right|=\sqrt{{a}_{x}^{2}+{a}_{y}^{2}}\text{}\end{array}$ Attālums starp punktiem, nogriežņa viduspunkts  $\begin{array}{l}\mathit{Ja}\phantom{\rule{0.147em}{0ex}}A\left({x}_{1};{y}_{1}\right)\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\mathit{un}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}B\left({x}_{2};{y}_{2}\right),\phantom{\rule{0.147em}{0ex}}\mathit{tad}\phantom{\rule{0.147em}{0ex}}\\ \left|\stackrel{\to }{\mathit{AB}}\right|=\sqrt{{\left({x}_{2}-{x}_{1}\right)}^{2}+\phantom{\rule{0.147em}{0ex}}{\left({y}_{2}-{y}_{1}\right)}^{2}}\end{array}$ $$[AB]$$ viduspunkts ir $C\left(\frac{{x}_{1}+{x}_{2}}{2};\frac{{y}_{1}+{y}_{2}}{2}\right)$____________________________________ Riņķa līnijas vienādojums  Ja centrs $O\left({x}_{o};{y}_{o}\right)$ un rādiuss $$R$$, tad ${\left(x-{x}_{o}\right)}^{2}+{\left(y-{y}_{o}\right)}^{2}={R}^{2}$ Vektori telpā$\begin{array}{l}\mathit{Ja}\phantom{\rule{0.147em}{0ex}}A\left({x}_{1};{y}_{1};{z}_{1}\right)\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\mathit{un}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}B\left({x}_{2};{y}_{2};{z}_{2}\right),\phantom{\rule{0.147em}{0ex}}\mathit{tad}\phantom{\rule{0.147em}{0ex}}\\ \stackrel{\to }{\mathit{AB}}=\left({x}_{2}-{x}_{1};\phantom{\rule{0.147em}{0ex}}{y}_{2}-{y}_{1};{z}_{2}-{z}_{1}\phantom{\rule{0.147em}{0ex}}\right)\\ \\ \mathit{Ja}\phantom{\rule{0.147em}{0ex}}\stackrel{\to }{a}=\left({a}_{x};{a}_{y};{a}_{z}\right)\phantom{\rule{0.147em}{0ex}}\mathit{un}\phantom{\rule{0.147em}{0ex}}\stackrel{\to }{b}=\left({b}_{x};{b}_{y};{b}_{z}\right),\mathit{tad}\\ \stackrel{\to }{a}±\stackrel{\to }{b}=\left({a}_{x}±{b}_{x};\phantom{\rule{0.147em}{0ex}}{a}_{y}±{b}_{y};\phantom{\rule{0.147em}{0ex}}{a}_{z}±{b}_{z}\right)\\ k\cdot \stackrel{\to }{a}=\left(k\cdot {a}_{x};\phantom{\rule{0.147em}{0ex}}k\cdot {a}_{y};\phantom{\rule{0.147em}{0ex}}k\cdot {a}_{z}\right)\\ \\ \left|\stackrel{\to }{a}\right|=\sqrt{{a}_{x}^{2}+{a}_{y}^{2}+{a}_{z}^{2}}\text{}\end{array}$ Taisnes vienādojumsVienādojums taisnei, kas iet caur punktiem ${P}_{1}\left({x}_{1};{y}_{1}\right)\phantom{\rule{0.147em}{0ex}}\mathit{un}\phantom{\rule{0.147em}{0ex}}{P}_{2}\left({x}_{2};{y}_{2}\right)$: $\frac{x-{x}_{1}}{{x}_{2}-{x}_{1}}=\frac{y-{y}_{1}}{{y}_{2}-{y}_{1}}$ Taisnes $$y=kx+b$$ virziena koeficients $k=\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}$ Taisnes $y={k}_{1}x+{b}_{1}$ un $y={k}_{2}x+{b}_{2}$ ir:paralēlas, ja ${k}_{1}={k}_{2}$perpendikulāras, ja ${k}_{1}\cdot {k}_{2}=-1$
Atsauce:
© Valsts izglītības satura centrs | ESF projekts Nr. 8.3.1.1/16/I/002 Kompetenču pieeja mācību saturā, Matemātika optimālajā mācību satura apguves līmenī. Valsts pārbaudes darba programma