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Question-73663

Question Number 73663 by ajfour last updated on 14/Nov/19 Answered by ajfour last updated on 15/Nov/19 $${See}\:{Q}.\mathrm{73673} \\ $$$${V}=\mathrm{2}\int\left(\left(\frac{\rho^{\mathrm{2}} \left(\mathrm{2}\phi\right)}{\mathrm{2}}×\left(−{ds}\right)\right)\right. \\ $$$$\:\rho={s}\mathrm{sin}\:\alpha={s}\mathrm{sin}\:\frac{\pi}{\mathrm{6}}\:=\:\frac{{s}}{\mathrm{2}} \\ $$$$\:{s}=\frac{{a}\sqrt{\mathrm{3}}−{r}\left(\mathrm{1}−\mathrm{cos}\:\theta\right)}{\mathrm{cos}\:\alpha}=\frac{\mathrm{2}{a}}{\mathrm{3}}\left(\mathrm{2}+\mathrm{cos}\:\theta\right) \\…

m-1-32-3m-2-n-1-10-sin-2npi-11-icos-2npi-11-m-A-4-1-i-B-12-1-i-C-12-1-i-D-48-1-i-

Question Number 139193 by EnterUsername last updated on 23/Apr/21 $$\underset{{m}=\mathrm{1}} {\overset{\mathrm{32}} {\sum}}\left(\mathrm{3}{m}+\mathrm{2}\right)\left(\underset{{n}=\mathrm{1}} {\overset{\mathrm{10}} {\sum}}\left(\mathrm{sin}\left(\frac{\mathrm{2}{n}\pi}{\mathrm{11}}\right)−\mathrm{icos}\left(\frac{\mathrm{2}{n}\pi}{\mathrm{11}}\right)\right)\right)^{{m}} = \\ $$$$\left(\mathrm{A}\right)\:\mathrm{4}\left(\mathrm{1}−{i}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{B}\right)\:\mathrm{12}\left(\mathrm{1}+{i}\right) \\ $$$$\left(\mathrm{C}\right)\:\mathrm{12}\left(\mathrm{1}−{i}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{D}\right)\:\mathrm{48}\left(\mathrm{1}−{i}\right) \\ $$ Answered by qaz last…

If-z-1-and-z-2-are-complex-nth-roots-of-unity-which-sub-tend-right-angle-at-the-origin-then-n-must-be-of-the-form-A-4K-1-B-4K-2-C-4K-3-D-4K-

Question Number 139192 by EnterUsername last updated on 23/Apr/21 $$\mathrm{If}\:{z}_{\mathrm{1}\:} \:\mathrm{and}\:{z}_{\mathrm{2}} \:\mathrm{are}\:\mathrm{complex}\:{n}\mathrm{th}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{unity}\:\mathrm{which}\:\mathrm{sub}- \\ $$$$\mathrm{tend}\:\mathrm{right}\:\mathrm{angle}\:\mathrm{at}\:\mathrm{the}\:\mathrm{origin},\:\mathrm{then}\:{n}\:\mathrm{must}\:\mathrm{be}\:\mathrm{of}\:\mathrm{the}\:\mathrm{form} \\ $$$$\left(\mathrm{A}\right)\:\mathrm{4K}+\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{B}\right)\:\mathrm{4K}+\mathrm{2} \\ $$$$\left(\mathrm{C}\right)\:\mathrm{4K}+\mathrm{3}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{D}\right)\:\mathrm{4K} \\ $$ Answered by mr W last…

a-b-R-ax-b-1-x-2-1-x-R-prove-a-2-b-1-

Question Number 139190 by mathdanisur last updated on 23/Apr/21 $${a};{b}\in\mathbb{R}\:,\:\frac{\mid{ax}+{b}\mid}{\mathrm{1}+{x}^{\mathrm{2}} }\:\leqslant\:\mathrm{1}\:,\:\forall{x}\in\mathbb{R} \\ $$$${prove}:\:\:\mid{a}\mid\leqslant\mathrm{2}\:;\:\mid{b}\mid\leqslant\mathrm{1} \\ $$ Answered by mitica last updated on 24/Apr/21 $${x}=\mathrm{0}\Rightarrow\mid{b}\mid\leqslant\mathrm{1} \\ $$$${x}=\mathrm{1}\Rightarrow\mid{a}+{b}\mid\leqslant\mathrm{2}…