Question Number 67482 by Rasheed.Sindhi last updated on 27/Aug/19 $${I}\:{have}\:{tried}\:{to}\:{solve}\:{Q}#\mathrm{67299} \\ $$$${Please}\:{see}\:{and}\:{give}\:{critical}\:{remarks} \\ $$ Commented by mr W last updated on 28/Aug/19 $${your}\:{solution}\:{is}\:{correct}\:{sir}.\:{but}\:{i}'{m} \\ $$$${not}\:{sure}\:{if}\:{the}\:{solution}\:{is}\:{unique}.…
Question Number 133016 by metamorfose last updated on 18/Feb/21 $$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left({tan}\left({x}\right)\right)^{\frac{\mathrm{1}}{{n}}} {dx}\:… \\ $$ Answered by Ar Brandon last updated on 18/Feb/21 $$\mathcal{I}=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}}…
Question Number 67481 by Mohamed Amine Bouguezzoul last updated on 27/Aug/19 $${p}\:{is}\:{a}\:{prime}\:{number}\:{such}\:{that}\:\left(\mathrm{1}+{p}\right)^{{p}} \equiv\mathrm{2}\left[\mathrm{7}\right] \\ $$$${find}\:{all}\:{k}\:{such}\:{that}\:{p}\equiv{k}\left[\mathrm{42}\right] \\ $$ Commented by Rasheed.Sindhi last updated on 30/Aug/19 $$\boldsymbol{{Some}}\:{values}\:{of}\:{k}…
Question Number 1942 by Yozzi last updated on 25/Oct/15 $${Let}\:{N}\:{be}\:{a}\:{positive}\:{integer}\:{with}\:{prime} \\ $$$${factorisation}\: \\ $$$$\:\:{N}={p}_{\mathrm{1}} ^{{m}_{\mathrm{1}} } {p}_{\mathrm{2}} ^{{m}_{\mathrm{2}} } {p}_{\mathrm{3}} ^{{m}_{\mathrm{3}} } ×…×{p}_{{n}−\mathrm{1}} ^{{m}_{{n}−\mathrm{1}} }…
Question Number 133009 by shaker last updated on 18/Feb/21 Commented by liberty last updated on 18/Feb/21 $$?\:=\:\mathrm{1} \\ $$ Answered by Rasheed.Sindhi last updated on…
Question Number 1937 by Rasheed Soomro last updated on 25/Oct/15 $$\bullet{Is}\:\:\:'\Leftrightarrow'\:\:{necessary}\:{and}\:{suficient}\:{for}\:{two} \\ $$$${inequalities}\:{to}\:{be}\:{equivalent}? \\ $$$$\bullet{If}\:\:\boldsymbol{\mathrm{a}}>\boldsymbol{\mathrm{b}}\:: \\ $$$${Are}\:\:\boldsymbol{\mathrm{A}}>\boldsymbol{\mathrm{B}}\:{and}\:\boldsymbol{\mathrm{A}}+\boldsymbol{\mathrm{a}}\:>\:\boldsymbol{\mathrm{B}}+\boldsymbol{\mathrm{b}}\:{equivalent}? \\ $$ Answered by 123456 last updated on…
Question Number 133010 by rs4089 last updated on 18/Feb/21 $${for}\:{the}\:{extremum}\:{values}\:{of}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} \:{subject}\:{to}\:{the}\:{constraints} \\ $$$${ax}^{\mathrm{2}} +{by}^{\mathrm{2}} +{cz}^{\mathrm{2}} =\mathrm{1}\:{and}\:{lx}+{my}+{nz}=\mathrm{0}\:. \\ $$$${show}\:{that}\:{the}\:{stationary}\:{points}\:{setisfy}\:{the}\:{realation} \\ $$$$\frac{{l}^{\mathrm{2}} }{\mathrm{1}+{a}\lambda_{\mathrm{1}} }+\frac{{m}^{\mathrm{2}} }{\mathrm{1}+{b}\lambda_{\mathrm{1}}…
Question Number 1936 by 123456 last updated on 25/Oct/15 $${f}^{\mathrm{2}} \left({x}\right)−{f}\left({x}^{\mathrm{2}} \right)={a}\:\left[\mathrm{G}.\mathrm{Q1902}\right] \\ $$$${f}\left({x}\right)=? \\ $$ Commented by prakash jain last updated on 25/Oct/15 $$\mathrm{I}\:\mathrm{don}'\mathrm{t}\:\mathrm{think}\:\mathrm{that}\:\mathrm{a}\:\mathrm{solution}\:\mathrm{exists}\:\mathrm{except}…
Question Number 133004 by liberty last updated on 18/Feb/21 $$\mathrm{If}\:\int\:\frac{\mathrm{tan}\:\mathrm{x}}{\mathrm{1}+\mathrm{tan}\:\mathrm{x}+\mathrm{tan}\:^{\mathrm{2}} \mathrm{x}}\:\mathrm{dx}\:=\:\mathrm{x}−\frac{{k}}{\:\sqrt{{A}}}\:\mathrm{tan}^{−\mathrm{1}} \left(\frac{{k}\:\mathrm{tan}\:{x}+\mathrm{1}}{\:\sqrt{{A}}}\right)+\mathrm{C} \\ $$$$\mathrm{where}\:\mathrm{C}\:\mathrm{is}\:\mathrm{constant}\:\mathrm{of}\:\mathrm{integration}. \\ $$$$\mathrm{then}\:\mathrm{the}\:\mathrm{ordered}\:\mathrm{pair}\:\left({k},\mathrm{A}\right)\:\mathrm{is}\: \\ $$$$\mathrm{equal}\:\mathrm{to}\: \\ $$ Answered by EDWIN88 last updated…
Question Number 67471 by AnjanDey last updated on 27/Aug/19 $$\boldsymbol{{Evaluate}}:\int\sqrt{{x}\sqrt{{x}+\mathrm{1}}}\:{dx} \\ $$ Commented by MJS last updated on 28/Aug/19 $$\int\sqrt{{x}\sqrt{{x}+\mathrm{1}}}{dx}= \\ $$$$\:\:\:\:\:\left[{t}=\sqrt{{x}}\:\rightarrow\:{dx}=\mathrm{2}\sqrt{{x}}{dt}\right] \\ $$$$=\mathrm{2}\int{t}^{\mathrm{2}} \left({t}^{\mathrm{2}}…