Category: Algebra

Question Number 211932 by a.lgnaoui last updated on 25/Sep/24 $$\mathrm{determiner}\mathbf{R}1\mathbf{R}2\mathrm{et}\mathbf{R}3$$$$\mathrm{segment}\mathrm{de}\mathrm{longueur}\mathbf{a}\mathbf{est}\mathbf{tangent}\mathbf{aux}$$$$\mathbf{cercles}1\mathbf{et}2.$$$$\mathbf{MN}//\mathbf{EF};\mathbf{EF}=\mathbf{a};\mathrm{OM}=\mathrm{ON}=\frac{3\mathbf{a}}{2}.$$$$\left(\mathrm{length}\mathbf{a}\mathrm{is}\mathrm{tangent}\mathrm{to}\mathrm{cirles}\mathrm{C}1\left(\mathrm{radius}\mathrm{R}1\right)\mathrm{and}\mathrm{circldC}2\left(\mathrm{radius}\mathrm{R}2\right)\right).$$ Commented by a.lgnaoui last updated…

Question Number 211917 by Spillover last updated on 24/Sep/24 Answered by BHOOPENDRA last updated on 24/Sep/24 $$Use{L}^{\prime }{Hopital}^{\prime }sRule$$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{mn}\left(\mathrm{1}+{nx}\right)^{{m}−\mathrm{1}} −{mn}\left(\mathrm{1}+{mx}\right)^{{n}−\mathrm{1}} }{\left(\mathrm{1}+{mx}\right)^{\frac{\mathrm{1}}{{m}}−\mathrm{1}} −\left(\mathrm{1}+{nx}\right)^{\frac{\mathrm{1}}{{n}}−\mathrm{1}} } \…

Question Number 211896 by Spillover last updated on 23/Sep/24 Answered by BHOOPENDRA last updated on 23/Sep/24 $$=-\frac{1}{2}(Re\underset{z=0}{s}\left(f\left(z\right)\right)+Res\underset{z=-1}{}\left(f\left(z\right)\right))$$$$=-\frac{1}{2}(2n-\frac{2}{3}(2n^2+1))$$$$=\frac{\mathrm{2}{n}^{\mathrm{2}} −\mathrm{3}{n}+\mathrm{1}}{\mathrm{3}}…

Question Number 211885 by Spillover last updated on 23/Sep/24 Answered by Frix last updated on 23/Sep/24 $$\int \frac{x^2}{{(x\sin x+\cos x)}^2}dx=\frac{y}{x\sin x+\cos x}+C$$$$\frac{d}{dx}\left[\frac{y}{x\sin x+\cos x}\right]=\frac{x^2}{{(x\sin x+\cos x)}^2}$$$$\frac{{y}'\left({x}\mathrm{sin}\:{x}\:+\mathrm{cos}\:{x}\right)−{yx}\mathrm{cos}\:{x}}{\left({x}\mathrm{sin}\:{x}\:+\mathrm{cos}\:{x}\right)^{\mathrm{2}}…

Question Number 211881 by Spillover last updated on 23/Sep/24 Answered by som(math1967) last updated on 23/Sep/24 $$\alpha +\beta +\gamma =1,\alpha \beta +\beta \gamma +\gamma \alpha =-2$$$$\alpha \beta \gamma =-1$$$$\:\alpha^{\mathrm{2}} +\beta^{\mathrm{2}} +\gamma^{\mathrm{2}} =\mathrm{1}+\mathrm{4}=\mathrm{5} \…