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Category: Arithmetic

Question-211871

Question Number 211871 by Spillover last updated on 22/Sep/24 Answered by Ghisom last updated on 23/Sep/24 $$=−\underset{\mathrm{0}} {\overset{\mathrm{arccos}\:\frac{\sqrt{\mathrm{6}}}{\mathrm{3}}} {\int}}\frac{\mathrm{2}+\mathrm{tan}\:{x}}{\mathrm{3}−\mathrm{2cos}^{\mathrm{2}} \:{x}\:−\mathrm{sin}^{\mathrm{2}} \:{x}}{dx}= \\ $$$$\:\:\:\:\:\left[{t}=\sqrt{\mathrm{2}}\mathrm{tan}\:{x}\right] \\ $$$$=−\frac{\mathrm{1}}{\mathrm{2}}\underset{\mathrm{0}}…

Question-211873

Question Number 211873 by Spillover last updated on 22/Sep/24 Answered by IbtisamAdnan last updated on 23/Sep/24 $$\:\:\:\boldsymbol{\mathrm{lim}}_{\boldsymbol{\mathrm{x}}\rightarrow\mathrm{4}} \frac{\left(\boldsymbol{\mathrm{cos}\alpha}\right)^{\mathrm{x}} −\left(\mathrm{sin}\alpha\right)^{\mathrm{x}} −\mathrm{cos}\:\mathrm{2}\alpha}{\mathrm{x}−\mathrm{4}} \\ $$$$\boldsymbol{\mathrm{lim}}_{\boldsymbol{\mathrm{x}}\rightarrow\mathrm{4}} \:\frac{\left(\boldsymbol{\mathrm{cos}\alpha}\right)^{\mathrm{x}} .\:\mathrm{ln}\:\mathrm{cos}\alpha\:\:−\:\left(\mathrm{sin}\:\alpha\right)^{\mathrm{x}} .\mathrm{ln}\:\mathrm{sin}\alpha}{\mathrm{1}}\left[\mathrm{L}\:\mathrm{hospital}\:\mathrm{rule}\right]…

Question-211636

Question Number 211636 by BaliramKumar last updated on 15/Sep/24 Answered by Rasheed.Sindhi last updated on 15/Sep/24 $$\blacktriangleright{A}={x}^{\mathrm{8}{k}+\mathrm{3}} +{x}^{\mathrm{8}{k}+\mathrm{6}} +{x}^{\mathrm{8}{k}+\mathrm{9}} +{x}^{\mathrm{8}{k}+\mathrm{12}} \\ $$$$\:\:\:\:\:\:\:\:={x}^{\mathrm{8}{k}+\mathrm{3}} \left(\mathrm{1}+{x}^{\mathrm{3}} +{x}^{\mathrm{6}} +{x}^{\mathrm{9}}…

Question-211105

Question Number 211105 by peter frank last updated on 28/Aug/24 Answered by mm1342 last updated on 28/Aug/24 $${z}={z}_{\mathrm{1}} {z}_{\mathrm{2}} ={cos}\frac{\mathrm{12}\pi}{\mathrm{5}}+{isin}\frac{\mathrm{12}\pi}{\mathrm{5}} \\ $$$$={cos}\frac{\mathrm{2}\pi}{\mathrm{5}}+{isin}\frac{\mathrm{2}\pi}{\mathrm{5}}={e}^{\frac{\mathrm{2}\pi}{\mathrm{5}}{i}\:} \Rightarrow{z}^{\mathrm{5}} ={e}^{\mathrm{2}\pi{i}} =\mathrm{1}…

Question-211020

Question Number 211020 by peter frank last updated on 26/Aug/24 Answered by mahdipoor last updated on 26/Aug/24 $${dV}=\left(\pi{y}^{\mathrm{2}} \right)\left({dx}\right)=\pi\frac{\left({x}−{a}\right)^{\mathrm{2}} \left({x}−{b}\right)^{\mathrm{2}} }{{c}^{\mathrm{2}} }{dx} \\ $$$${V}=\int{dV}=\frac{\pi}{{c}^{\mathrm{2}} }\int_{{a}}…

Question-211019

Question Number 211019 by peter frank last updated on 26/Aug/24 Answered by som(math1967) last updated on 26/Aug/24 $$\:{let}\:{x}={sin}\alpha\:\:{y}={sin}\beta \\ $$$$\:{cos}\alpha+{cos}\beta={a}\left({sin}\alpha−{sin}\beta\right) \\ $$$$\Rightarrow\frac{\mathrm{2}{cos}\frac{\alpha+\beta}{\mathrm{2}}{cos}\frac{\alpha−\beta}{\mathrm{2}}}{\mathrm{2}{sin}\frac{\alpha−\beta}{\mathrm{2}}{cos}\frac{\alpha+\beta}{\mathrm{2}}}={a} \\ $$$$\Rightarrow{cot}\frac{\alpha−\beta}{\mathrm{2}}={a} \\…