Category: Algebra

Question Number 206082 by hardmath last updated on 06/Apr/24 $$\mathrm{If}\cos \mathit{\alpha }=\sin \mathit{\alpha }+\frac{1}{\sqrt{3}}$$$$\mathrm{Find}\sin 2\mathit{\alpha }=?$$ Answered by MATHEMATICSAM last updated on 29/Apr/24 $$\cos \alpha =\sin \alpha +\frac{1}{\sqrt{3}}$$$$\Rightarrow\:\mathrm{cos}\alpha\:−\:\mathrm{sin}\alpha\:=\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}} \…

Question Number 205988 by cortano12 last updated on 04/Apr/24 $$Givenf(x+1)=2^{f\left(x\right)}.f\left(1\right)$$$$andf\left(1\right)=16$$$$thenf\left(2016\right)=?$$ Answered by Berbere last updated on 04/Apr/24 $${f}\left(\mathrm{2}\right)=\mathrm{2}^{\mathrm{20}}…

Question Number 205941 by cortano12 last updated on 03/Apr/24 $$2+\frac{2}{2-\frac{2}{2+\frac{2}{2-\frac{2}{2+\frac{2}{3}}}}}=?$$$$$$ Answered by mr W last updated on 03/Apr/24 $$2+\frac{2}{2-\frac{2}{2+\frac{2}{2-\frac{2}{2+\frac{2}{3}}}}}$$$$=\mathrm{2}+\frac{\mathrm{2}}{\mathrm{2}−\frac{\mathrm{2}}{\mathrm{2}+\frac{\mathrm{2}}{\mathrm{2}−\frac{\mathrm{3}}{\mathrm{4}}}}}…

Question Number 205914 by mr W last updated on 02/Apr/24 $$ifa+b+c+d+e+f=10and$$$$a^2+b^2+c^2+d^2+e^2+f^2=25,find$$$$a_{min}and{f}_{max}.$$…

Question Number 205885 by hardmath last updated on 01/Apr/24 $$\mathrm{Find}:\underset{\mathbf{n}\to \mathrm{\infty }}{\lim }\sqrt[\mathbf{n}]{}\left(\begin{array}{c}2\mathrm{n}\\ \mathrm{n}\end{array}\right)=?$$ Answered by MM42 last updated on 03/Apr/24 $$\left(\begin{array}{c}2n\\ n\end{array}\right)=\frac{\left(2n\right)(2n-1)(2n-2)\dots (n+3)(n+2)(n+1)}{n(n-1)(n-2)\dots 3\times 2\times 1}$$$$a=\sqrt[n]{}\left(\begin{array}{c}2n\\ n\end{array}\right)$$$$\Rightarrow{lna}=\frac{\mathrm{1}}{{n}}\left[{ln}\left(\mathrm{2}\right)+{ln}\left(\mathrm{2}+\frac{\mathrm{1}}{{n}−\mathrm{1}}\right)+{ln}\left(\mathrm{2}+\frac{\mathrm{2}}{{n}−\mathrm{2}}\right)+…+{ln}\left(\mathrm{2}+\frac{{n}−\mathrm{3}}{\mathrm{3}}\right)\left(\mathrm{2}+\frac{{n}−\mathrm{2}}{\mathrm{2}}\right)\left(\mathrm{2}+\frac{{n}−\mathrm{1}}{\mathrm{1}}\right)\right.…