Question Number 67992 by MJS last updated on 03/Sep/19 $$\left(\mathrm{1}\right)\:{z}={a}+{b}\mathrm{i} \\ $$$$\left(\mathrm{2}\right)\:{z}={r}\mathrm{e}^{\mathrm{i}\theta} \\ $$$$\mathrm{express}\:\mathrm{the}\:\mathrm{values}\:\mathrm{of} \\ $$$$\left(\mathrm{a}\right)\:\mathrm{real}\:\left({z}^{{z}} \right)\:\:\:\:\:\left[\mathrm{real}\:\mathrm{part}\right] \\ $$$$\left(\mathrm{b}\right)\:\mathrm{imag}\:\left({z}^{{z}} \right)\:\:\:\:\:\left[\mathrm{imaginary}\:\mathrm{part}\right] \\ $$$$\left(\mathrm{c}\right)\:\mathrm{abs}\:\left({z}^{{z}} \right)\:\:\:\:\:\left[\mathrm{absolute}\:\mathrm{value}\right] \\ $$$$\left(\mathrm{d}\right)\:\mathrm{arg}\:\left({z}^{{z}}…
Question Number 2454 by Yozzi last updated on 20/Nov/15 $${Find}\:{smallest}\:{a}>\mathrm{1}\:{for}\:{which} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{{a}+{sinx}}{{a}+{siny}}\leqslant{e}^{{y}−{x}} \\ $$$${for}\:\forall\:{x}\leqslant{y}. \\ $$ Commented by Rasheed Soomro last updated on 21/Nov/15 $${Find}\:{smallest}\:{a}>\mathrm{1}\:{for}\:{which}…
Question Number 2441 by Yozzi last updated on 20/Nov/15 $${Prove}\:{or}\:{disprove}\:{that}\:,\:{for}\:{even}\: \\ $$$${positive}\:{n}, \\ $$$$\mathrm{2}×\underset{{k}=\mathrm{1}} {\overset{\frac{{n}}{\mathrm{2}}−\mathrm{1}} {\sum}}\left(−\mathrm{1}\right)^{{k}} \begin{pmatrix}{{n}}\\{{k}}\end{pmatrix}+\left(−\mathrm{1}\right)^{\left({n}/\mathrm{2}\right)} \frac{{n}!}{\left(\left({n}/\mathrm{2}\right)!\right)^{\mathrm{2}} }=−\mathrm{2} \\ $$ Commented by Rasheed Soomro…
Question Number 133469 by Eric002 last updated on 22/Feb/21 $${find}\:{x}\:{in}\:{terms}\:{of}\:{a} \\ $$$$\frac{\mathrm{1}+{x}−\sqrt{\mathrm{2}{x}+{x}^{\mathrm{2}} }}{\mathrm{1}+{x}+\sqrt{\mathrm{2}{x}+{x}^{\mathrm{2}} }}={a}^{\mathrm{3}} \frac{\sqrt{\mathrm{2}+{x}}+\sqrt{{x}}}{\:\sqrt{\mathrm{2}+{x}}−\sqrt{{x}}} \\ $$ Answered by EDWIN88 last updated on 22/Feb/21 $$\frac{\left[\left(\mathrm{1}+\mathrm{x}\right)−\sqrt{\mathrm{2x}+\mathrm{x}^{\mathrm{2}}…
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Question Number 133423 by mathlove last updated on 22/Feb/21 Commented by liki last updated on 24/Feb/21 $$\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{x}\:\mathrm{and}\:\mathrm{y}? \\ $$ Commented by MJS_new last updated on…
Question Number 67881 by mr W last updated on 01/Sep/19 $${find}\:{all}\:{x},{y}\:\in{R}\:{such}\:{that} \\ $$$$\left({x}+{yi}\right)^{\mathrm{2019}} ={x}−{yi} \\ $$ Answered by mind is power last updated on 01/Sep/19…
Question Number 2341 by Rasheed Soomro last updated on 17/Nov/15 $${For}\:{what}\:{conditions} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\left({i}\right)\:{x}^{{y}^{{z}} } <\left({x}^{{y}} \right)^{{z}} \:\:? \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\left({ii}\right)\:{y}^{{x}^{{z}} } <\left({x}^{{y}} \right)^{{z}} \:? \\ $$…
Question Number 2304 by Ashis last updated on 14/Nov/15 $${if}\:\:{x}^{\mathrm{2}} +\mathrm{5}{x}−\mathrm{9}=\:\mathrm{0}\:\:{then}\:{x}=? \\ $$ Answered by prakash jain last updated on 14/Nov/15 $${x}^{\mathrm{2}} +\mathrm{5}{x}−\mathrm{9} \\ $$$${x}=\frac{−\mathrm{5}\pm\sqrt{\left(−\mathrm{5}\right)^{\mathrm{2}}…
Question Number 67826 by peter frank last updated on 31/Aug/19 Commented by gunawan last updated on 01/Sep/19 $${x}=\mathrm{1}+{a}+{a}^{\mathrm{2}} +{a}^{\mathrm{3}} +.. \\ $$$${x}=\frac{\mathrm{1}}{\mathrm{1}−{a}}\:\Rightarrow\mathrm{1}−\:{a}=\frac{\mathrm{1}}{{x}}\Rightarrow\:{a}=\mathrm{1}−\frac{\mathrm{1}}{{x}}=\frac{{x}−\mathrm{1}}{{x}} \\ $$$${y}=\mathrm{1}+{b}+{b}^{\mathrm{2}} +{b}^{\mathrm{3}}…