Question Number 217159 by ArshadS last updated on 03/Mar/25 $${Solve}\:{for}\:{x}: \\ $$$$\frac{{x}+\mathrm{3}}{{x}−\mathrm{2}}+\frac{\mathrm{2}{x}−\mathrm{5}}{{x}+\mathrm{4}}=\frac{\mathrm{4}{x}+\mathrm{1}}{{x}^{\mathrm{2}} +\mathrm{2}{x}−\mathrm{8}} \\ $$ Answered by som(math1967) last updated on 03/Mar/25 $$\:\frac{{x}+\mathrm{3}}{{x}−\mathrm{2}}+\frac{\mathrm{2}{x}−\mathrm{5}}{{x}+\mathrm{4}}=\frac{\mathrm{4}{x}+\mathrm{1}}{\left({x}−\mathrm{2}\right)\left({x}+\mathrm{4}\right)} \\ $$$$\Rightarrow\frac{{x}+\mathrm{3}}{{x}−\mathrm{2}}\:−\frac{\mathrm{4}{x}+\mathrm{1}}{\left({x}−\mathrm{2}\right)\left({x}+\mathrm{4}\right)}+\frac{\mathrm{2}{x}−\mathrm{5}}{{x}+\mathrm{4}}=\mathrm{0}…
Question Number 217132 by ArshadS last updated on 02/Mar/25 $$ \\ $$$$\mathrm{Find}\:\mathrm{all}\:\mathrm{integers}\:\:\mathrm{n}>\:\mathrm{1}\:\mathrm{such}\:\mathrm{that} \\ $$$$\:\:\mathrm{n}\:\:\mathrm{divides}\:\:\mathrm{2}^{\mathrm{n}−\mathrm{1}} \:+\:\mathrm{3}^{\mathrm{n}−\mathrm{1}} . \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 217129 by ArshadS last updated on 02/Mar/25 $${prove}\:{that}\:{if}\:{an}\:{integer}\:{n}\:{is}\:{not}\:{divisible}\:{by}\:\mathrm{2}\:{or}\:\mathrm{3} \\ $$$$\:{then}\:{n}^{\mathrm{2}} \equiv\mathrm{1}\left({mod}\:\mathrm{24}\right) \\ $$ Commented by A5T last updated on 02/Mar/25 $$\mathrm{This}\:\mathrm{is}\:\mathrm{not}\:\mathrm{necessarily}\:\mathrm{true}.\: \\ $$$$\mathrm{n}\:\mathrm{could}\:\mathrm{also}\:\mathrm{be}\:\equiv\:\mathrm{5},\mathrm{7},\mathrm{11},\mathrm{13},\mathrm{17},\mathrm{19},\mathrm{23}\:\left(\mathrm{mod}\:\mathrm{24}\right)…
Question Number 217146 by a.lgnaoui last updated on 02/Mar/25 $$\mathrm{determiner}\:\mathrm{le}\:\mathrm{cote}\:\mathrm{du}\:\mathrm{care}\:\boldsymbol{\mathrm{ABCD}} \\ $$$$\mathrm{inscrit}\:\mathrm{dans}\:\mathrm{l}\:\mathrm{elipse}\:\left\{\left(−\mathrm{3},+\mathrm{3}\right):\left(−\mathrm{8},+\mathrm{8}\right)\right\} \\ $$ Commented by a.lgnaoui last updated on 02/Mar/25 Answered by mr W…
Question Number 217130 by ArshadS last updated on 02/Mar/25 $$ \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{for}\:\mathrm{every}\:\mathrm{integer}\:\:\mathrm{n}\geqslant\mathrm{2}\:\:\mathrm{the}\:\mathrm{number}\:\:\mathrm{n}^{\mathrm{4}} +\:\mathrm{4}^{{n}} \:\:\mathrm{is} \\ $$$$\mathrm{c}{o}\mathrm{mposite}. \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
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Question Number 217140 by peter frank last updated on 02/Mar/25 if a, b, c are three digits, abc and bca are two numbers. where abc…
Question Number 217137 by Tinku Tara last updated on 02/Mar/25 $$ \\ $$$$\mathrm{Please}\:\mathrm{do}\:\mathrm{not}\:\mathrm{post}\:\mathrm{totally}\:\mathrm{meaningless} \\ $$$$\mathrm{questions}/\mathrm{answers}. \\ $$$$\mathrm{If}\:\mathrm{you}\:\mathrm{dont}\:\mathrm{know}\:\mathrm{the}\:\mathrm{answer} \\ $$$$\mathrm{leave}\:\mathrm{it}\:\mathrm{unanswered}.\:\mathrm{If}\:\mathrm{some}\:\mathrm{has} \\ $$$$\mathrm{ideas}\:\mathrm{they}\:\mathrm{will}\:\mathrm{post}\:\mathrm{either}\:\mathrm{partially} \\ $$$$\mathrm{or}\:\mathrm{full}\:\mathrm{answers}. \\ $$$$\mathrm{Do}\:\mathrm{not}\:\mathrm{post}\:\mathrm{meaningless}\:\mathrm{answers}\:\mathrm{that}…
Question Number 217101 by MathematicalUser2357 last updated on 01/Mar/25 $$\mathrm{is}\:\mathrm{this}\:\mathrm{right}\:\mathrm{when}\:\left({a}+{bi}\right)^{{c}+{di}} =\mid{a}+{bi}\mid^{{c}+{di}} {e}^{{i}\left({c}+{di}\right)\mathrm{arg}\left({a}+{bi}\right)} ? \\ $$$$\mathrm{I}\:\mathrm{had}\:\mathrm{let}\:\mathrm{arg}\left({a}+{bi}\right)=\begin{cases}{\mathrm{tan}^{−\mathrm{1}} \left(\frac{{b}}{{a}}\right)}&{{a}\geqslant\mathrm{0}\:\mathrm{and}\:{b}\geqslant\mathrm{0}}\\{\pi−\mathrm{tan}^{−\mathrm{1}} \left(−\frac{{b}}{{a}}\right)}&{{a}<\mathrm{0}\:\mathrm{and}\:{b}\geqslant\mathrm{0}}\\{−\left(\pi−\mathrm{tan}^{−\mathrm{1}} \left(\frac{{b}}{{a}}\right)\right)}&{{a}<\mathrm{0}\:\mathrm{and}\:{b}<\mathrm{0}}\\{−\mathrm{tan}^{−\mathrm{1}} \left(\frac{{b}}{{a}}\right)}&{{a}\geqslant\mathrm{0}\:\mathrm{and}\:{b}<\mathrm{0}}\end{cases}\:\mathrm{before}\:\mathrm{I}\:\mathrm{solved}\:\mathrm{it} \\ $$$$\left({a}+{bi}\right)^{{c}+{di}} =\mid{a}+{bi}\mid^{{c}+{di}} {e}^{{i}\left({c}+{di}\right)\mathrm{arg}\left({a}+{bi}\right)} \\ $$$$=\mid{a}+{bi}\mid^{{c}}…
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Question Number 217122 by efronzo1 last updated on 01/Mar/25 $$\:\:\:\:\:\int\:\frac{\sqrt{\mathrm{cos}\:\mathrm{2x}}}{\mathrm{cos}\:\mathrm{x}}\:\mathrm{dx}\:=? \\ $$ Answered by Frix last updated on 01/Mar/25 $$\int\frac{\sqrt{\mathrm{cos}\:\mathrm{2}{x}}}{\mathrm{cos}\:{x}}{dx}=\int\frac{\sqrt{−\mathrm{1}+\mathrm{2cos}^{\mathrm{2}} \:{x}}}{\mathrm{cos}\:{x}}{dx}\:\overset{\left[{t}=\sqrt{\mathrm{2}}\mathrm{sin}\:{x}\right]} {=} \\ $$$$=\sqrt{\mathrm{2}}\int\frac{\sqrt{\mathrm{1}−{t}^{\mathrm{2}} }}{\mathrm{2}−{t}^{\mathrm{2}}…