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Author: Tinku Tara

Question-198939

Question Number 198939 by Mingma last updated on 26/Oct/23 Answered by witcher3 last updated on 26/Oct/23 $$\varphi:\mathrm{2}\mathbb{N}\rightarrow\mathbb{Z} \\ $$$$\varphi\left(\mathrm{2n}\right)=\begin{cases}{\mathrm{k}\:\mathrm{if}\:\mathrm{n}=\mathrm{2}\left(\mathrm{2k}+\mathrm{1}\right)}\\{−\mathrm{k}\:\mathrm{if}\:\mathrm{n}=\mathrm{2}.\left(\mathrm{2k}\right)}\end{cases} \\ $$$$\varphi\left(\mathrm{m}\right)=\varphi\left(\mathrm{n}\right)\:\Leftrightarrow\mathrm{m}=\mathrm{n} \\ $$$$\varphi\:\mathrm{injective} \\ $$$$\mathrm{if}\:\:\mathrm{n}\in\mathbb{Z}\:\mathrm{if}\:\mathrm{n}\geqslant\mathrm{0}\:…

Question-198988

Question Number 198988 by Safojon last updated on 26/Oct/23 Answered by witcher3 last updated on 26/Oct/23 $$\mathrm{f}\left(\mathrm{x}\right)−\mathrm{g}\left(\mathrm{x}\right)=\mathrm{x}\underset{\mathrm{k}=\mathrm{0}} {\overset{\mathrm{1010}} {\sum}}\left(\mathrm{x}^{\mathrm{2k}} \right)>\mathrm{0} \\ $$$$\mathrm{u}_{\mathrm{n}} =\mathrm{f}^{\left(\mathrm{n}\right)} \left(\frac{\mathrm{1}}{\mathrm{2023}}\right),\mathrm{v}_{\mathrm{n}} =\mathrm{g}^{\left(\mathrm{n}\right)}…

Calcul-determinant-3-16-24-33-1-5-7-9-5-27-36-55-7-38-51-78-

Question Number 198989 by Rodier97 last updated on 26/Oct/23 $$ \\ $$$$ \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Calcul}\:: \\ $$$$\begin{vmatrix}{\mathrm{3}}&{\mathrm{16}}&{\mathrm{24}}&{\mathrm{33}}\\{\mathrm{1}}&{\mathrm{5}}&{\mathrm{7}}&{\mathrm{9}}\\{\mathrm{5}}&{\mathrm{27}}&{\mathrm{36}}&{\mathrm{55}}\\{\mathrm{7}}&{\mathrm{38}}&{\mathrm{51}}&{\mathrm{78}}\end{vmatrix} \\ $$$$ \\ $$$$ \\ $$$$ \\…

Question-198845

Question Number 198845 by sonukgindia last updated on 25/Oct/23 Answered by som(math1967) last updated on 25/Oct/23 $${I}=\int_{−\mathrm{1}} ^{\mathrm{1}} \frac{\left(\mathrm{1}−\mathrm{1}−{x}\right)^{\mathrm{2}} {dx}}{\mathrm{1}+{e}^{\mathrm{1}−\mathrm{1}−{x}} } \\ $$$$=\int_{−\mathrm{1}} ^{\mathrm{1}} \frac{{x}^{\mathrm{2}}…

Question-198901

Question Number 198901 by sonukgindia last updated on 25/Oct/23 Answered by witcher3 last updated on 25/Oct/23 $$\int\frac{\left(\mathrm{cos}\left(\frac{\mathrm{x}}{\mathrm{2}}\right)−\mathrm{sin}\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\right)^{\mathrm{2}} }{\mathrm{6sin}\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\mathrm{cos}\left(\frac{\mathrm{x}}{\mathrm{2}}\right)+\mathrm{10cos}^{\mathrm{2}} \left(\frac{\mathrm{x}}{\mathrm{2}}\right)\mathrm{e}^{\mathrm{x}} }\mathrm{dx}=\mathrm{g}\left(\mathrm{x}\right) \\ $$$$=\int\frac{\left(\mathrm{1}−\mathrm{tg}\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\right)^{\mathrm{2}} }{\mathrm{6tg}\left(\frac{\mathrm{x}}{\mathrm{2}}\right)+\mathrm{10e}^{\mathrm{x}} }\mathrm{dx} \\…