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MATeMAtyka 1. Zakres podstawowy to podręcznik wydany przez Nowa Era, przeznaczony dla uczniów liceum ogólnokształcącego oraz technikum, na pierwszym roku nauki. Jako podręcznik do matematyki, obejmuje on kluczowe zagadnienia, które przygotowują uczniów do dalszego etapu edukacyjnego.

Spis treści książki zawiera m.in. wprowadzenie do języka matematyki, zadania dotyczące funkcji, układy równań, a także szczegółowe omówienie zbiorów rozłącznych i przedziałów liczbowych. Autorzy podają różnorodne przykłady i ćwiczenia praktyczne, które pomagają w zrozumieniu oraz utrwaleniu materiału.

Oferujemy odrobione odpowiedzi do podręcznika, włącznie z gotowym PDF z odpowiedziami, które ułatwią naukę i przygotowania do testów i egzaminów. Dostępne formaty umożliwiają łatwe i szybkie znalezienie potrzebnych rozwiązań. Odrabiamy zadania do podręcznika MATeMAtyka 1. Zakres podstawowy, pomagając uczniom w skutecznym opanowaniu wymaganej wiedzy.

","longDescription","

Podręcznik MATeMAtyka 1. Zakres podstawowy od wydawnictwa Nowa Era, dedykowany dla uczniów liceum oraz technikum, stanowi idealne źródło wiedzy dla wszystkich, którzy pragną zrozumieć i opanować podstawy matematyki na poziomie ponadpodstawowym. Dostosowany do aktualnych wymogów edukacyjnych, podręcznik ten oferuje klarowne wyjaśnienia, możliwość sprawdzenia nabytych umiejętności oraz pomoc w przygotowaniach do egzaminów. Dzięki stronie gotowiec.pl uczniowie mogą znaleźć odrobione zadania do podręcznika MATeMAtyka 1. Zakres podstawowy, co jest nieocenioną pomocą w procesie edukacji.

Podręcznik oswaja uczniów z kluczowymi zagadnieniami matematycznymi, począwszy od zrozumienia definicji i własności funkcji, poprzez szkicowanie wykresów, aż do rozwiązywania złożonych układów równań. Każdy rozdział rozpoczyna się od krótkiego wprowadzenia teoretycznego, co stanowi solidne fundamenty do dalszej nauki. Rozdział dotyczący języka matematyki omawia, jak można przedstawiać i interpretować informacje w formie matematycznej, co jest kluczowe w rozwijaniu umiejętności analitycznych i problemowych.

W szczególności, sekcja na temat przystawania i symetrii trójkątów pokazuje, jak ważne jest zrozumienie tych koncepcji w geometrii. Uzupełnieniem są ćwiczenia praktyczne i przykłady, które pozwalają na natychmiastowe zastosowanie teoretycznych wywodów. To idealna okazja, aby przekonać się, jak rozumienie teorii przekłada się na praktykę, szczególnie kiedy można skorzystać z gotowych odpowiedzi do książki MATeMAtyka 1. Zakres podstawowy dostępnych na gotowiec.pl.

Podręcznik skupia się również na omawianiu stężeń procentowych i roztwarów w rzeczywistych sytuacjach życiowych, co ma na celu pokazanie praktycznego stosowania matematyki w codziennym życiu. Rozdział zadaniowy łączy teorię z praktyką, zadania tekstowe umożliwiają wykorzystanie matematyki do rozwiązywania realnych problemów, co jest częściej spotykane na egzaminach i w prawdziwym życiu.

Warto również zwrócić uwagę na bogate źródło zadań, które uczniowie mogą przerabiać, mając dostęp do PDF z odpowiedziami do książki MATeMAtyka 1. Zakres podstawowy. To doskonałe narzędzie dla tych, którzy chcą solidnie przygotować się do przyszłych wyzwań edukacyjnych, zarówno na poziomie liceum, jak i technikum.

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Rysunek z zadania:

\r\n\r\n

\r\nZauważmy, że:

\r\n\r\n

→ punkt jest środkiem odcinka , zatem dzieli ten odcinek na dwie równe części
\r\n → kąty wierzchołkowe, zatem muszą mieć równą miarę
\r\n → kąty

","stylesheet","mjx-c.mjx-c7C::before {\n padding: 0.75em 0.278em 0.249em 0;\n content: \"|\";\n}\n\nmjx-c.mjx-c44::before {\n padding: 0.683em 0.764em 0 0;\n content: \"D\";\n}\n\nmjx-c.mjx-c1D437.TEX-I::before {\n padding: 0.683em 0.828em 0 0;\n content: \"D\";\n}\n\nmjx-c.mjx-c2222.TEX-A::before {\n padding: 0.551em 0.722em 0.051em 0;\n content: \"\\2222\";\n}\n\nmjx-c.mjx-c45::before {\n padding: 0.68em 0.681em 0 0;\n content: \"E\";\n}\n\nmjx-c.mjx-cB0::before {\n padding: 0.715em 0.5em 0 0;\n content: \"\\B0\";\n}\n\nmjx-c.mjx-c2206::before {\n padding: 0.716em 0.833em 0 0;\n content: \"\\394\";\n}\n\nmjx-c.mjx-c2261::before {\n padding: 0.464em 0.778em 0 0;\n content: \"\\2261\";\n}\n\n\n\nmjx-stretchy-h.mjx-c2192 mjx-ext mjx-c::before {\n content: \"\\2212\";\n padding: 0.583em 0 0.082em 0;\n}\n\nmjx-stretchy-h.mjx-c2192 mjx-end mjx-c::before {\n content: \"\\2192\";\n padding: 0.583em 0 0.082em 0;\n}\n\nmjx-c.mjx-c42::before {\n padding: 0.683em 0.708em 0 0;\n content: \"B\";\n}\n\nmjx-c.mjx-c1D442.TEX-I::before {\n padding: 0.704em 0.763em 0.022em 0;\n content: \"O\";\n}\n\nmjx-c.mjx-c5B.TEX-S2::before {\n padding: 1.15em 0.472em 0.649em 0;\n content: \"[\";\n}\n\nmjx-c.mjx-c5D.TEX-S2::before {\n padding: 1.15em 0.472em 0.649em 0;\n content: \"]\";\n}\n\nmjx-c.mjx-c4B::before {\n padding: 0.683em 0.778em 0 0;\n content: \"K\";\n}\n\nmjx-c.mjx-c1D43E.TEX-I::before {\n padding: 0.683em 0.889em 0 0;\n content: \"K\";\n}\n\nmjx-c.mjx-c1D466.TEX-I::before {\n padding: 0.442em 0.49em 0.205em 0;\n content: \"y\";\n}\n\nmjx-c.mjx-c78::before {\n padding: 0.431em 0.528em 0 0;\n content: \"x\";\n}\n\nmjx-c.mjx-c2260::before {\n padding: 0.716em 0.778em 0.215em 0;\n content: \"\\2260\";\n}\n\nmjx-c.mjx-c1D435.TEX-I::before {\n padding: 0.683em 0.759em 0 0;\n content: \"B\";\n}\n\nmjx-c.mjx-c7B.TEX-S3::before {\n padding: 1.45em 0.75em 0.949em 0;\n content: \"{\";\n}\n\n\n\nmjx-c.mjx-cD7::before {\n padding: 0.491em 0.778em 0 0;\n content: \"\\D7\";\n}\n\nmjx-c.mjx-c1D451.TEX-I::before {\n padding: 0.694em 0.52em 0.01em 0;\n content: \"d\";\n}\n\nmjx-c.mjx-c21D2::before {\n padding: 0.525em 1em 0.024em 0;\n content: \"\\21D2\";\n}\n\nmjx-c.mjx-c48::before {\n padding: 0.683em 0.75em 0 0;\n content: \"H\";\n}\n\nmjx-c.mjx-c58::before {\n padding: 0.683em 0.75em 0 0;\n content: \"X\";\n}\n\nmjx-c.mjx-c1D43B.TEX-I::before {\n padding: 0.683em 0.888em 0 0;\n content: \"H\";\n}\n\nmjx-c.mjx-c1D44B.TEX-I::before {\n padding: 0.683em 0.852em 0 0;\n content: \"X\";\n}\n\nmjx-c.mjx-c7A::before {\n padding: 0.431em 0.444em 0 0;\n content: \"z\";\n}\n\nmjx-c.mjx-c25::before {\n padding: 0.75em 0.833em 0.056em 0;\n content: \"%\";\n}\n\n\n\nmjx-c.mjx-c1D45F.TEX-I::before {\n padding: 0.442em 0.451em 0.011em 0;\n content: \"r\";\n}\n\nmjx-c.mjx-c2223::before {\n padding: 0.75em 0.278em 0.249em 0;\n content: \"\\2223\";\n}\n\nmjx-c.mjx-c1D450.TEX-I::before {\n padding: 0.442em 0.433em 0.011em 0;\n content: \"c\";\n}\n\nmjx-mfenced {\n display: inline-block;\n text-align: left;\n}\n\nmjx-c.mjx-c1D44F.TEX-I::before {\n padding: 0.694em 0.429em 0.011em 0;\n content: \"b\";\n}\n\nmjx-c.mjx-c62::before {\n padding: 0.694em 0.556em 0.011em 0;\n content: \"b\";\n}\n\nmjx-c.mjx-c2265::before {\n padding: 0.636em 0.778em 0.138em 0;\n content: \"\\2265\";\n}\n\nmjx-c.mjx-c3E::before {\n padding: 0.54em 0.778em 0.04em 0;\n content: \">\";\n}\n\nmjx-c.mjx-c2208::before {\n padding: 0.54em 0.667em 0.04em 0;\n content: \"\\2208\";\n}\n\nmjx-c.mjx-c1D445.TEX-I::before {\n padding: 0.683em 0.759em 0.021em 0;\n content: \"R\";\n}\n\nmjx-c.mjx-c28.TEX-S1::before {\n padding: 0.85em 0.458em 0.349em 0;\n content: \"(\";\n}\n\nmjx-c.mjx-c29.TEX-S1::before {\n padding: 0.85em 0.458em 0.349em 0;\n content: \")\";\n}\n\nmjx-c.mjx-c5B.TEX-S1::before {\n padding: 0.85em 0.417em 0.349em 0;\n content: \"[\";\n}\n\nmjx-c.mjx-c5D.TEX-S1::before {\n padding: 0.85em 0.417em 0.349em 0;\n content: \"]\";\n}\n\nmjx-c.mjx-c2264::before {\n padding: 0.636em 0.778em 0.138em 0;\n content: \"\\2264\";\n}\n\n\n\nmjx-c.mjx-c67::before {\n padding: 0.453em 0.5em 0.206em 0;\n content: \"g\";\n}\n\nmjx-c.mjx-c1D439.TEX-I::before {\n padding: 0.68em 0.749em 0 0;\n content: \"F\";\n}\n\nmjx-c.mjx-c1D454.TEX-I::before {\n padding: 0.442em 0.477em 0.205em 0;\n content: \"g\";\n}\n\nmjx-c.mjx-c2F::before {\n padding: 0.75em 0.5em 0.25em 0;\n content: \"/\";\n}\n\nmjx-c.mjx-c3A::before {\n padding: 0.43em 0.278em 0 0;\n content: \":\";\n}\n\nmjx-c.mjx-c68::before {\n padding: 0.694em 0.556em 0 0;\n content: \"h\";\n}\n\nmjx-menclose {\n display: inline-block;\n text-align: left;\n position: relative;\n}\n\nmjx-menclose > mjx-dstrike {\n display: inline-block;\n left: 0;\n top: 0;\n position: absolute;\n border-top: 0.067em solid;\n transform-origin: top left;\n}\n\nmjx-menclose > mjx-ustrike {\n display: inline-block;\n left: 0;\n bottom: 0;\n position: absolute;\n border-top: 0.067em solid;\n transform-origin: bottom left;\n}\n\nmjx-menclose > mjx-hstrike {\n border-top: 0.067em solid;\n position: absolute;\n left: 0;\n right: 0;\n bottom: 50%;\n transform: translateY(0.034em);\n}\n\nmjx-menclose > mjx-vstrike {\n border-left: 0.067em solid;\n position: absolute;\n top: 0;\n bottom: 0;\n right: 50%;\n transform: translateX(0.034em);\n}\n\nmjx-menclose > mjx-rbox {\n position: absolute;\n top: 0;\n bottom: 0;\n right: 0;\n left: 0;\n border: 0.067em solid;\n border-radius: 0.267em;\n}\n\nmjx-menclose > mjx-cbox {\n position: absolute;\n top: 0;\n bottom: 0;\n right: 0;\n left: 0;\n border: 0.067em solid;\n border-radius: 50%;\n}\n\nmjx-menclose > mjx-arrow {\n position: absolute;\n left: 0;\n bottom: 50%;\n height: 0;\n width: 0;\n}\n\nmjx-menclose > mjx-arrow > * {\n display: block;\n position: absolute;\n transform-origin: bottom;\n border-left: 0.268em solid;\n border-right: 0;\n box-sizing: border-box;\n}\n\nmjx-menclose > mjx-arrow > mjx-aline {\n left: 0;\n top: -0.034em;\n right: 0.201em;\n height: 0;\n border-top: 0.067em solid;\n border-left: 0;\n}\n\nmjx-menclose > mjx-arrow[double] > mjx-aline {\n left: 0.201em;\n height: 0;\n}\n\nmjx-menclose > mjx-arrow > mjx-rthead {\n transform: skewX(0.464rad);\n right: 0;\n bottom: -1px;\n border-bottom: 1px solid transparent;\n border-top: 0.134em solid transparent;\n}\n\nmjx-menclose > mjx-arrow > mjx-rbhead {\n transform: skewX(-0.464rad);\n transform-origin: top;\n right: 0;\n top: -1px;\n border-top: 1px solid transparent;\n border-bottom: 0.134em solid transparent;\n}\n\nmjx-menclose > mjx-arrow > mjx-lthead {\n transform: skewX(-0.464rad);\n left: 0;\n bottom: -1px;\n border-left: 0;\n border-right: 0.268em solid;\n border-bottom: 1px solid transparent;\n border-top: 0.134em solid transparent;\n}\n\nmjx-menclose > mjx-arrow > mjx-lbhead {\n transform: skewX(0.464rad);\n transform-origin: top;\n left: 0;\n top: -1px;\n border-left: 0;\n border-right: 0.268em solid;\n border-top: 1px solid transparent;\n border-bottom: 0.134em solid transparent;\n}\n\nmjx-menclose > dbox {\n position: absolute;\n top: 0;\n bottom: 0;\n left: -0.3em;\n width: 0.6em;\n border: 0.067em solid;\n border-radius: 50%;\n clip-path: inset(0 0 0 0.3em);\n box-sizing: border-box;\n}\n\nmjx-mmultiscripts {\n display: inline-block;\n text-align: left;\n}\n\nmjx-prescripts {\n display: inline-table;\n padding-left: .05em;\n}\n\nmjx-scripts {\n display: inline-table;\n padding-right: .05em;\n}\n\nmjx-prescripts > mjx-row > mjx-cell {\n text-align: right;\n}\n\n[script-align=\"left\"] > mjx-row > mjx-cell {\n text-align: left;\n}\n\n[script-align=\"center\"] > mjx-row > mjx-cell {\n text-align: center;\n}\n\n[script-align=\"right\"] > mjx-row > mjx-cell {\n text-align: right;\n}\n\nmjx-mspace {\n display: inline-block;\n text-align: left;\n}\n\nmjx-c.mjx-c1D463.TEX-I::before {\n padding: 0.443em 0.485em 0.011em 0;\n content: \"v\";\n}\n\nmjx-c.mjx-c6B::before {\n padding: 0.694em 0.528em 0 0;\n content: \"k\";\n}\n\nmjx-c.mjx-c210E.TEX-I::before {\n padding: 0.694em 0.576em 0.011em 0;\n content: \"h\";\n}\n\nmjx-c.mjx-c1D45B.TEX-I::before {\n padding: 0.442em 0.6em 0.011em 0;\n content: \"n\";\n}\n\nmjx-c.mjx-c1D452.TEX-I::before {\n padding: 0.442em 0.466em 0.011em 0;\n content: \"e\";\n}\n\nmjx-c.mjx-cB7::before {\n padding: 0.31em 0.278em 0 0;\n content: \"\\22C5\";\n}\n\nmjx-c.mjx-c3F::before {\n padding: 0.705em 0.472em 0 0;\n content: \"?\";\n}\n\nmjx-c.mjx-c1D45E.TEX-I::before {\n padding: 0.442em 0.46em 0.194em 0;\n content: \"q\";\n}\n\nmjx-c.mjx-c41::before {\n padding: 0.716em 0.75em 0 0;\n content: \"A\";\n}\n\nmjx-msqrt {\n display: inline-block;\n text-align: left;\n}\n\nmjx-root {\n display: inline-block;\n white-space: nowrap;\n}\n\nmjx-surd {\n display: inline-block;\n vertical-align: top;\n}\n\nmjx-sqrt {\n display: inline-block;\n padding-top: .07em;\n}\n\nmjx-sqrt > mjx-box {\n border-top: .07em solid;\n}\n\nmjx-sqrt.mjx-tall > mjx-box {\n padding-left: .3em;\n margin-left: -.3em;\n}\n\nmjx-TeXAtom {\n display: inline-block;\n text-align: left;\n}\n\nmjx-c.mjx-c221A::before {\n padding: 0.8em 0.853em 0.2em 0;\n content: \"\\221A\";\n}\n\nmjx-c.mjx-c63::before {\n padding: 0.448em 0.444em 0.011em 0;\n content: \"c\";\n}\n\nmjx-c.mjx-c6F::before {\n padding: 0.448em 0.5em 0.01em 0;\n content: \"o\";\n}\n\nmjx-c.mjx-c2061::before {\n padding: 0 0 0 0;\n content: \"\";\n}\n\nmjx-c.mjx-c28.TEX-S2::before {\n padding: 1.15em 0.597em 0.649em 0;\n content: \"(\";\n}\n\nmjx-c.mjx-c29.TEX-S2::before {\n padding: 1.15em 0.597em 0.649em 0;\n content: \")\";\n}\n\nmjx-c.mjx-c5B::before {\n padding: 0.75em 0.278em 0.25em 0;\n content: \"[\";\n}\n\nmjx-c.mjx-c5D::before {\n padding: 0.75em 0.278em 0.25em 0;\n content: \"]\";\n}\n\nmjx-c.mjx-c1D440.TEX-I::before {\n padding: 0.683em 1.051em 0 0;\n content: \"M\";\n}\n\nmjx-c.mjx-c64::before {\n padding: 0.694em 0.556em 0.011em 0;\n content: \"d\";\n}\n\nmjx-c.mjx-c52::before {\n padding: 0.683em 0.736em 0.022em 0;\n content: \"R\";\n}\n\nmjx-c.mjx-c1D465.TEX-I::before {\n padding: 0.442em 0.572em 0.011em 0;\n content: \"x\";\n}\n\nmjx-c.mjx-c1D460.TEX-I::before {\n padding: 0.442em 0.469em 0.01em 0;\n content: \"s\";\n}\n\nmjx-c.mjx-c72::before {\n padding: 0.442em 0.392em 0 0;\n content: \"r\";\n}\n\nmjx-c.mjx-c1D44E.TEX-I::before {\n padding: 0.441em 0.529em 0.01em 0;\n content: \"a\";\n}\n\nmjx-c.mjx-c76::before {\n padding: 0.431em 0.528em 0.011em 0;\n content: \"v\";\n}\n\nmjx-c.mjx-c74::before {\n padding: 0.615em 0.389em 0.01em 0;\n content: \"t\";\n}\n\nmjx-c.mjx-c1D45A.TEX-I::before {\n padding: 0.442em 0.878em 0.011em 0;\n content: \"m\";\n}\n\nmjx-c.mjx-c46::before {\n padding: 0.68em 0.653em 0 0;\n content: \"F\";\n}\n\nmjx-c.mjx-c47::before {\n padding: 0.705em 0.785em 0.022em 0;\n content: \"G\";\n}\n\nmjx-c.mjx-c4D::before {\n padding: 0.683em 0.917em 0 0;\n content: \"M\";\n}\n\nmjx-mtable {\n display: inline-block;\n text-align: center;\n vertical-align: .25em;\n position: relative;\n box-sizing: border-box;\n border-spacing: 0;\n border-collapse: collapse;\n}\n\nmjx-mstyle[size=\"s\"] mjx-mtable {\n vertical-align: .354em;\n}\n\nmjx-labels {\n position: absolute;\n left: 0;\n top: 0;\n}\n\nmjx-table {\n display: inline-block;\n vertical-align: -.5ex;\n box-sizing: border-box;\n}\n\nmjx-table > mjx-itable {\n vertical-align: middle;\n text-align: left;\n box-sizing: border-box;\n}\n\nmjx-labels > mjx-itable {\n position: absolute;\n top: 0;\n}\n\nmjx-mtable[justify=\"left\"] {\n text-align: left;\n}\n\nmjx-mtable[justify=\"right\"] {\n text-align: right;\n}\n\nmjx-mtable[justify=\"left\"][side=\"left\"] {\n padding-right: 0 ! important;\n}\n\nmjx-mtable[justify=\"left\"][side=\"right\"] {\n padding-left: 0 ! important;\n}\n\nmjx-mtable[justify=\"right\"][side=\"left\"] {\n padding-right: 0 ! important;\n}\n\nmjx-mtable[justify=\"right\"][side=\"right\"] {\n padding-left: 0 ! important;\n}\n\nmjx-mtable[align] {\n vertical-align: baseline;\n}\n\nmjx-mtable[align=\"top\"] > mjx-table {\n vertical-align: top;\n}\n\nmjx-mtable[align=\"bottom\"] > mjx-table {\n vertical-align: bottom;\n}\n\nmjx-mtable[side=\"right\"] mjx-labels {\n min-width: 100%;\n}\n\nmjx-mtr {\n display: table-row;\n text-align: left;\n}\n\nmjx-mtr[rowalign=\"top\"] > mjx-mtd {\n vertical-align: top;\n}\n\nmjx-mtr[rowalign=\"center\"] > mjx-mtd {\n vertical-align: middle;\n}\n\nmjx-mtr[rowalign=\"bottom\"] > mjx-mtd {\n vertical-align: bottom;\n}\n\nmjx-mtr[rowalign=\"baseline\"] > mjx-mtd {\n vertical-align: baseline;\n}\n\nmjx-mtr[rowalign=\"axis\"] > mjx-mtd {\n vertical-align: .25em;\n}\n\nmjx-mtd {\n display: table-cell;\n text-align: center;\n padding: .215em .4em;\n}\n\nmjx-mtd:first-child {\n padding-left: 0;\n}\n\nmjx-mtd:last-child {\n padding-right: 0;\n}\n\nmjx-mtable > * > mjx-itable > *:first-child > mjx-mtd {\n padding-top: 0;\n}\n\nmjx-mtable > * > mjx-itable > *:last-child > mjx-mtd {\n padding-bottom: 0;\n}\n\nmjx-tstrut {\n display: inline-block;\n height: 1em;\n vertical-align: -.25em;\n}\n\nmjx-labels[align=\"left\"] > mjx-mtr > mjx-mtd {\n text-align: left;\n}\n\nmjx-labels[align=\"right\"] > mjx-mtr > mjx-mtd {\n text-align: right;\n}\n\nmjx-mtd[extra] {\n padding: 0;\n}\n\nmjx-mtd[rowalign=\"top\"] {\n vertical-align: top;\n}\n\nmjx-mtd[rowalign=\"center\"] {\n vertical-align: middle;\n}\n\nmjx-mtd[rowalign=\"bottom\"] {\n vertical-align: bottom;\n}\n\nmjx-mtd[rowalign=\"baseline\"] {\n vertical-align: baseline;\n}\n\nmjx-mtd[rowalign=\"axis\"] {\n vertical-align: .25em;\n}\n\nmjx-mtext {\n display: inline-block;\n text-align: left;\n}\n\nmjx-msup {\n display: inline-block;\n text-align: left;\n}\n\nmjx-mfrac {\n display: inline-block;\n text-align: left;\n}\n\nmjx-frac {\n display: inline-block;\n vertical-align: 0.17em;\n padding: 0 .22em;\n}\n\nmjx-frac[type=\"d\"] {\n vertical-align: .04em;\n}\n\nmjx-frac[delims] {\n padding: 0 .1em;\n}\n\nmjx-frac[atop] {\n padding: 0 .12em;\n}\n\nmjx-frac[atop][delims] {\n padding: 0;\n}\n\nmjx-dtable {\n display: inline-table;\n width: 100%;\n}\n\nmjx-dtable > * {\n font-size: 2000%;\n}\n\nmjx-dbox {\n display: block;\n font-size: 5%;\n}\n\nmjx-num {\n display: block;\n text-align: center;\n}\n\nmjx-den {\n display: block;\n text-align: center;\n}\n\nmjx-mfrac[bevelled] > mjx-num {\n display: inline-block;\n}\n\nmjx-mfrac[bevelled] > mjx-den {\n display: inline-block;\n}\n\nmjx-den[align=\"right\"], mjx-num[align=\"right\"] {\n text-align: right;\n}\n\nmjx-den[align=\"left\"], mjx-num[align=\"left\"] {\n text-align: left;\n}\n\nmjx-nstrut {\n display: inline-block;\n height: .054em;\n width: 0;\n vertical-align: -.054em;\n}\n\nmjx-nstrut[type=\"d\"] {\n height: .217em;\n vertical-align: -.217em;\n}\n\nmjx-dstrut {\n display: inline-block;\n height: .505em;\n width: 0;\n}\n\nmjx-dstrut[type=\"d\"] {\n height: .726em;\n}\n\nmjx-line {\n display: block;\n box-sizing: border-box;\n min-height: 1px;\n height: .06em;\n border-top: .06em solid;\n margin: .06em -.1em;\n overflow: hidden;\n}\n\nmjx-line[type=\"d\"] {\n margin: .18em -.1em;\n}\n\nmjx-c.mjx-c1D448.TEX-I::before {\n padding: 0.683em 0.767em 0.022em 0;\n content: \"U\";\n}\n\nmjx-c.mjx-c3D::before {\n padding: 0.583em 0.778em 0.082em 0;\n content: \"=\";\n}\n\nmjx-c.mjx-c30::before {\n padding: 0.666em 0.5em 0.022em 0;\n content: \"0\";\n}\n\nmjx-c.mjx-c56::before {\n padding: 0.683em 0.75em 0.022em 0;\n content: \"V\";\n}\n\nmjx-c.mjx-c1D443.TEX-I::before {\n padding: 0.683em 0.751em 0 0;\n content: \"P\";\n}\n\nmjx-c.mjx-c38::before {\n padding: 0.666em 0.5em 0.022em 0;\n content: \"8\";\n}\n\nmjx-c.mjx-c35::before {\n padding: 0.666em 0.5em 0.022em 0;\n content: \"5\";\n}\n\nmjx-c.mjx-c57::before {\n padding: 0.683em 1.028em 0.022em 0;\n content: \"W\";\n}\n\nmjx-c.mjx-c1D449.TEX-I::before {\n padding: 0.683em 0.769em 0.022em 0;\n content: \"V\";\n}\n\nmjx-c.mjx-c31::before {\n padding: 0.666em 0.5em 0 0;\n content: \"1\";\n}\n\nmjx-c.mjx-c6C::before {\n padding: 0.694em 0.278em 0 0;\n content: \"l\";\n}\n\nmjx-c.mjx-c394::before {\n padding: 0.716em 0.833em 0 0;\n content: \"\\394\";\n}\n\nmjx-c.mjx-c2218::before {\n padding: 0.444em 0.5em 0 0;\n content: \"\\2218\";\n}\n\nmjx-c.mjx-c1D444.TEX-I::before {\n padding: 0.704em 0.791em 0.194em 0;\n content: \"Q\";\n}\n\nmjx-c.mjx-c34::before {\n padding: 0.677em 0.5em 0 0;\n content: \"4\";\n}\n\nmjx-c.mjx-c4A::before {\n padding: 0.683em 0.514em 0.022em 0;\n content: \"J\";\n}\n\nmjx-c.mjx-c2C::before {\n padding: 0.121em 0.278em 0.194em 0;\n content: \",\";\n}\n\nmjx-c.mjx-c22C5::before {\n padding: 0.31em 0.278em 0 0;\n content: \"\\22C5\";\n}\n\nmjx-c.mjx-c1D43C.TEX-I::before {\n padding: 0.683em 0.504em 0 0;\n content: \"I\";\n}\n\nmjx-c.mjx-c2194::before {\n padding: 0.511em 1em 0.011em 0;\n content: \"\\2194\";\n}\n\nmjx-c.mjx-c1D44A.TEX-I::before {\n padding: 0.683em 1.048em 0.022em 0;\n content: \"W\";\n}\n\nmjx-c.mjx-c37::before {\n padding: 0.676em 0.5em 0.022em 0;\n content: \"7\";\n}\n\nmjx-c.mjx-c1D434.TEX-I::before {\n padding: 0.716em 0.75em 0 0;\n content: \"A\";\n}\n\nmjx-c.mjx-c2212::before {\n padding: 0.583em 0.778em 0.082em 0;\n content: \"\\2212\";\n}\n\nmjx-c.mjx-c54::before {\n padding: 0.677em 0.722em 0 0;\n content: \"T\";\n}\n\nmjx-c.mjx-c1D43D.TEX-I::before {\n padding: 0.683em 0.633em 0.022em 0;\n content: \"J\";\n}\n\nmjx-c.mjx-c1D459.TEX-I::before {\n padding: 0.694em 0.298em 0.011em 0;\n content: \"l\";\n}\n\nmjx-c.mjx-c1D436.TEX-I::before {\n padding: 0.705em 0.76em 0.022em 0;\n content: \"C\";\n}\n\nmjx-c.mjx-c28::before {\n padding: 0.75em 0.389em 0.25em 0;\n content: \"(\";\n}\n\nmjx-c.mjx-c29::before {\n padding: 0.75em 0.389em 0.25em 0;\n content: \")\";\n}\n\nmjx-c.mjx-c36::before {\n padding: 0.666em 0.5em 0.022em 0;\n content: \"6\";\n}\n\nmjx-c.mjx-c1D461.TEX-I::before {\n padding: 0.626em 0.361em 0.011em 0;\n content: \"t\";\n}\n\nmjx-c.mjx-c39::before {\n padding: 0.666em 0.5em 0.022em 0;\n content: \"9\";\n}\n\nmjx-c.mjx-c73::before {\n padding: 0.448em 0.394em 0.011em 0;\n content: \"s\";\n}\n\nmjx-c.mjx-c2248::before {\n padding: 0.483em 0.778em 0 0;\n content: \"\\2248\";\n}\n\nmjx-c.mjx-c6D::before {\n padding: 0.442em 0.833em 0 0;\n content: \"m\";\n}\n\nmjx-c.mjx-c6E::before {\n padding: 0.442em 0.556em 0 0;\n content: \"n\";\n}\n\n\nmjx-container[jax=\"CHTML\"] {\n line-height: 0;\n}\n\nmjx-container [space=\"1\"] {\n margin-left: .111em;\n}\n\nmjx-container [space=\"2\"] {\n margin-left: .167em;\n}\n\nmjx-container [space=\"3\"] {\n margin-left: .222em;\n}\n\nmjx-container [space=\"4\"] {\n margin-left: .278em;\n}\n\nmjx-container [space=\"5\"] {\n margin-left: .333em;\n}\n\nmjx-container [rspace=\"1\"] {\n margin-right: .111em;\n}\n\nmjx-container [rspace=\"2\"] {\n margin-right: .167em;\n}\n\nmjx-container [rspace=\"3\"] {\n margin-right: .222em;\n}\n\nmjx-container [rspace=\"4\"] {\n margin-right: .278em;\n}\n\nmjx-container [rspace=\"5\"] {\n margin-right: .333em;\n}\n\nmjx-container [size=\"s\"] {\n font-size: 70.7%;\n}\n\nmjx-container [size=\"ss\"] {\n font-size: 50%;\n}\n\nmjx-container [size=\"Tn\"] {\n font-size: 60%;\n}\n\nmjx-container [size=\"sm\"] {\n font-size: 85%;\n}\n\nmjx-container [size=\"lg\"] {\n font-size: 120%;\n}\n\nmjx-container [size=\"Lg\"] {\n font-size: 144%;\n}\n\nmjx-container [size=\"LG\"] {\n font-size: 173%;\n}\n\nmjx-container [size=\"hg\"] {\n font-size: 207%;\n}\n\nmjx-container [size=\"HG\"] {\n font-size: 249%;\n}\n\nmjx-container [width=\"full\"] {\n width: 100%;\n}\n\nmjx-box {\n display: inline-block;\n}\n\nmjx-block {\n display: block;\n}\n\nmjx-itable {\n display: inline-table;\n}\n\nmjx-row {\n display: table-row;\n}\n\nmjx-row > * {\n display: table-cell;\n}\n\nmjx-mtext {\n display: inline-block;\n}\n\nmjx-mstyle {\n display: inline-block;\n}\n\nmjx-merror {\n display: inline-block;\n color: red;\n background-color: yellow;\n}\n\nmjx-mphantom {\n visibility: hidden;\n}\n\n_::-webkit-full-page-media, _:future, :root mjx-container {\n will-change: opacity;\n}\n\nmjx-math {\n display: inline-block;\n text-align: left;\n line-height: 0;\n text-indent: 0;\n font-style: normal;\n font-weight: normal;\n font-size: 100%;\n font-size-adjust: none;\n letter-spacing: normal;\n border-collapse: collapse;\n word-wrap: normal;\n word-spacing: normal;\n white-space: nowrap;\n direction: ltr;\n padding: 1px 0;\n}\n\nmjx-container[jax=\"CHTML\"][display=\"true\"] {\n display: block;\n text-align: center;\n margin: 1em 0;\n}\n\nmjx-container[jax=\"CHTML\"][display=\"true\"][width=\"full\"] {\n display: flex;\n}\n\nmjx-container[jax=\"CHTML\"][display=\"true\"] mjx-math {\n padding: 0;\n}\n\nmjx-container[jax=\"CHTML\"][justify=\"left\"] {\n text-align: left;\n}\n\nmjx-container[jax=\"CHTML\"][justify=\"right\"] {\n text-align: right;\n}\n\nmjx-semantics {\n display: inline-block;\n text-align: left;\n}\n\nmjx-msub {\n display: inline-block;\n text-align: left;\n}\n\nmjx-mi {\n display: inline-block;\n text-align: left;\n}\n\nmjx-c {\n display: inline-block;\n}\n\nmjx-utext {\n display: inline-block;\n padding: .75em 0 .2em 0;\n}\n\nmjx-mn {\n display: inline-block;\n text-align: left;\n}\n\nmjx-mrow {\n display: inline-block;\n text-align: left;\n}\n\nmjx-mo {\n display: inline-block;\n text-align: left;\n}\n\nmjx-stretchy-h {\n display: inline-table;\n width: 100%;\n}\n\nmjx-stretchy-h > * {\n display: table-cell;\n width: 0;\n}\n\nmjx-stretchy-h > * > mjx-c {\n display: inline-block;\n transform: scalex(1.0000001);\n}\n\nmjx-stretchy-h > * > mjx-c::before {\n display: inline-block;\n width: initial;\n}\n\nmjx-stretchy-h > mjx-ext {\n /* IE */ overflow: hidden;\n /* others */ overflow: clip visible;\n width: 100%;\n}\n\nmjx-stretchy-h > mjx-ext > mjx-c::before {\n transform: scalex(500);\n}\n\nmjx-stretchy-h > mjx-ext > mjx-c {\n width: 0;\n}\n\nmjx-stretchy-h > mjx-beg > mjx-c {\n margin-right: -.1em;\n}\n\nmjx-stretchy-h > mjx-end > mjx-c {\n margin-left: -.1em;\n}\n\nmjx-stretchy-v {\n display: inline-block;\n}\n\nmjx-stretchy-v > * {\n display: block;\n}\n\nmjx-stretchy-v > mjx-beg {\n height: 0;\n}\n\nmjx-stretchy-v > mjx-end > mjx-c {\n display: block;\n}\n\nmjx-stretchy-v > * > mjx-c {\n transform: scaley(1.0000001);\n transform-origin: left center;\n overflow: hidden;\n}\n\nmjx-stretchy-v > mjx-ext {\n display: block;\n height: 100%;\n box-sizing: border-box;\n border: 0px solid transparent;\n /* IE */ overflow: hidden;\n /* others */ overflow: visible clip;\n}\n\nmjx-stretchy-v > mjx-ext > mjx-c::before {\n width: initial;\n box-sizing: border-box;\n}\n\nmjx-stretchy-v > mjx-ext > mjx-c {\n transform: scaleY(500) translateY(.075em);\n overflow: visible;\n}\n\nmjx-mark {\n display: inline-block;\n height: 0px;\n}\n\nmjx-mover {\n display: inline-block;\n text-align: left;\n}\n\nmjx-mover:not([limits=\"false\"]) {\n padding-top: .1em;\n}\n\nmjx-mover:not([limits=\"false\"]) > * {\n display: block;\n text-align: left;\n}\n\nmjx-c::before {\n display: block;\n width: 0;\n}\n\n.MJX-TEX {\n font-family: MJXZERO, MJXTEX;\n}\n\n.TEX-B {\n font-family: MJXZERO, MJXTEX-B;\n}\n\n.TEX-I {\n font-family: MJXZERO, MJXTEX-I;\n}\n\n.TEX-MI {\n font-family: MJXZERO, MJXTEX-MI;\n}\n\n.TEX-BI {\n font-family: MJXZERO, MJXTEX-BI;\n}\n\n.TEX-S1 {\n font-family: MJXZERO, MJXTEX-S1;\n}\n\n.TEX-S2 {\n font-family: MJXZERO, MJXTEX-S2;\n}\n\n.TEX-S3 {\n font-family: MJXZERO, MJXTEX-S3;\n}\n\n.TEX-S4 {\n font-family: MJXZERO, MJXTEX-S4;\n}\n\n.TEX-A {\n font-family: MJXZERO, MJXTEX-A;\n}\n\n.TEX-C {\n font-family: MJXZERO, MJXTEX-C;\n}\n\n.TEX-CB {\n font-family: MJXZERO, MJXTEX-CB;\n}\n\n.TEX-FR {\n font-family: MJXZERO, MJXTEX-FR;\n}\n\n.TEX-FRB {\n font-family: MJXZERO, MJXTEX-FRB;\n}\n\n.TEX-SS {\n font-family: MJXZERO, MJXTEX-SS;\n}\n\n.TEX-SSB {\n font-family: MJXZERO, MJXTEX-SSB;\n}\n\n.TEX-SSI {\n font-family: MJXZERO, MJXTEX-SSI;\n}\n\n.TEX-SC {\n font-family: MJXZERO, MJXTEX-SC;\n}\n\n.TEX-T {\n font-family: MJXZERO, MJXTEX-T;\n}\n\n.TEX-V {\n font-family: MJXZERO, MJXTEX-V;\n}\n\n.TEX-VB {\n font-family: MJXZERO, MJXTEX-VB;\n}\n\nmjx-stretchy-v mjx-c, mjx-stretchy-h mjx-c {\n font-family: MJXZERO, MJXTEX-S1, MJXTEX-S4, MJXTEX, MJXTEX-A ! important;\n}\n\n@font-face /* 0 */ {\n font-family: MJXZERO;\n src: url(\"https://cdn.jsdelivr.net/npm/mathjax@3/es5/output/chtml/fonts/woff-v2/MathJax_Zero.woff\") format(\"woff\");\n}\n\n@font-face /* 1 */ {\n font-family: MJXTEX;\n src: url(\"https://cdn.jsdelivr.net/npm/mathjax@3/es5/output/chtml/fonts/woff-v2/MathJax_Main-Regular.woff\") format(\"woff\");\n}\n\n@font-face /* 2 */ {\n font-family: MJXTEX-B;\n src: url(\"https://cdn.jsdelivr.net/npm/mathjax@3/es5/output/chtml/fonts/woff-v2/MathJax_Main-Bold.woff\") format(\"woff\");\n}\n\n@font-face /* 3 */ {\n font-family: MJXTEX-I;\n src: url(\"https://cdn.jsdelivr.net/npm/mathjax@3/es5/output/chtml/fonts/woff-v2/MathJax_Math-Italic.woff\") format(\"woff\");\n}\n\n@font-face /* 4 */ {\n font-family: MJXTEX-MI;\n src: url(\"https://cdn.jsdelivr.net/npm/mathjax@3/es5/output/chtml/fonts/woff-v2/MathJax_Main-Italic.woff\") format(\"woff\");\n}\n\n@font-face /* 5 */ {\n font-family: MJXTEX-BI;\n src: url(\"https://cdn.jsdelivr.net/npm/mathjax@3/es5/output/chtml/fonts/woff-v2/MathJax_Math-BoldItalic.woff\") format(\"woff\");\n}\n\n@font-face /* 6 */ {\n font-family: MJXTEX-S1;\n src: url(\"https://cdn.jsdelivr.net/npm/mathjax@3/es5/output/chtml/fonts/woff-v2/MathJax_Size1-Regular.woff\") format(\"woff\");\n}\n\n@font-face /* 7 */ {\n font-family: MJXTEX-S2;\n src: url(\"https://cdn.jsdelivr.net/npm/mathjax@3/es5/output/chtml/fonts/woff-v2/MathJax_Size2-Regular.woff\") format(\"woff\");\n}\n\n@font-face /* 8 */ {\n font-family: MJXTEX-S3;\n src: url(\"https://cdn.jsdelivr.net/npm/mathjax@3/es5/output/chtml/fonts/woff-v2/MathJax_Size3-Regular.woff\") format(\"woff\");\n}\n\n@font-face /* 9 */ {\n font-family: MJXTEX-S4;\n src: url(\"https://cdn.jsdelivr.net/npm/mathjax@3/es5/output/chtml/fonts/woff-v2/MathJax_Size4-Regular.woff\") format(\"woff\");\n}\n\n@font-face /* 10 */ {\n font-family: MJXTEX-A;\n src: url(\"https://cdn.jsdelivr.net/npm/mathjax@3/es5/output/chtml/fonts/woff-v2/MathJax_AMS-Regular.woff\") format(\"woff\");\n}\n\n@font-face /* 11 */ {\n font-family: MJXTEX-C;\n src: url(\"https://cdn.jsdelivr.net/npm/mathjax@3/es5/output/chtml/fonts/woff-v2/MathJax_Calligraphic-Regular.woff\") format(\"woff\");\n}\n\n@font-face /* 12 */ {\n font-family: MJXTEX-CB;\n src: url(\"https://cdn.jsdelivr.net/npm/mathjax@3/es5/output/chtml/fonts/woff-v2/MathJax_Calligraphic-Bold.woff\") format(\"woff\");\n}\n\n@font-face /* 13 */ {\n font-family: MJXTEX-FR;\n src: url(\"https://cdn.jsdelivr.net/npm/mathjax@3/es5/output/chtml/fonts/woff-v2/MathJax_Fraktur-Regular.woff\") format(\"woff\");\n}\n\n@font-face /* 14 */ {\n font-family: MJXTEX-FRB;\n src: url(\"https://cdn.jsdelivr.net/npm/mathjax@3/es5/output/chtml/fonts/woff-v2/MathJax_Fraktur-Bold.woff\") format(\"woff\");\n}\n\n@font-face /* 15 */ {\n font-family: MJXTEX-SS;\n src: url(\"https://cdn.jsdelivr.net/npm/mathjax@3/es5/output/chtml/fonts/woff-v2/MathJax_SansSerif-Regular.woff\") format(\"woff\");\n}\n\n@font-face /* 16 */ {\n font-family: MJXTEX-SSB;\n src: url(\"https://cdn.jsdelivr.net/npm/mathjax@3/es5/output/chtml/fonts/woff-v2/MathJax_SansSerif-Bold.woff\") format(\"woff\");\n}\n\n@font-face /* 17 */ {\n font-family: MJXTEX-SSI;\n src: url(\"https://cdn.jsdelivr.net/npm/mathjax@3/es5/output/chtml/fonts/woff-v2/MathJax_SansSerif-Italic.woff\") format(\"woff\");\n}\n\n@font-face /* 18 */ {\n font-family: MJXTEX-SC;\n src: url(\"https://cdn.jsdelivr.net/npm/mathjax@3/es5/output/chtml/fonts/woff-v2/MathJax_Script-Regular.woff\") format(\"woff\");\n}\n\n@font-face /* 19 */ {\n font-family: MJXTEX-T;\n src: url(\"https://cdn.jsdelivr.net/npm/mathjax@3/es5/output/chtml/fonts/woff-v2/MathJax_Typewriter-Regular.woff\") format(\"woff\");\n}\n\n@font-face /* 20 */ {\n font-family: MJXTEX-V;\n src: url(\"https://cdn.jsdelivr.net/npm/mathjax@3/es5/output/chtml/fonts/woff-v2/MathJax_Vector-Regular.woff\") format(\"woff\");\n}\n\n@font-face /* 21 */ {\n font-family: MJXTEX-VB;\n src: url(\"https://cdn.jsdelivr.net/npm/mathjax@3/es5/output/chtml/fonts/woff-v2/MathJax_Vector-Bold.woff\") format(\"woff\");\n}\n\nmjx-c.mjx-c43::before {\n padding: 0.705em 0.722em 0.021em 0;\n content: \"C\";\n}\n\nmjx-c.mjx-c61::before {\n padding: 0.448em 0.5em 0.011em 0;\n content: \"a\";\n}\n\nmjx-c.mjx-c4F::before {\n padding: 0.705em 0.778em 0.022em 0;\n content: \"O\";\n}\n\nmjx-c.mjx-c33::before {\n padding: 0.665em 0.5em 0.022em 0;\n content: \"3\";\n}\n\nmjx-c.mjx-c4E::before {\n padding: 0.683em 0.75em 0 0;\n content: \"N\";\n}\n\nmjx-c.mjx-c32::before {\n padding: 0.666em 0.5em 0 0;\n content: \"2\";\n}\n\nmjx-c.mjx-c53::before {\n padding: 0.705em 0.556em 0.022em 0;\n content: \"S\";\n}\n\nmjx-c.mjx-c69::before {\n padding: 0.669em 0.278em 0 0;\n content: \"i\";\n}\n\nmjx-c.mjx-cA0::before {\n padding: 0 0.25em 0 0;\n content: \"\\A0\";\n}\n\nmjx-c.mjx-c2B::before {\n padding: 0.583em 0.778em 0.082em 0;\n content: \"+\";\n}\n\nmjx-c.mjx-c1D447.TEX-I::before {\n padding: 0.677em 0.704em 0 0;\n content: \"T\";\n}\n\nmjx-c.mjx-c2192::before {\n padding: 0.511em 1em 0.011em 0;\n content: \"\\2192\";\n}\n","actionData","errors"]

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