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Author: Tinku Tara

0-9999-1-or-0-

Question Number 2384 by 123456 last updated on 18/Nov/15 $$\lfloor\mathrm{0},\mathrm{9999}….\rfloor=\mathrm{1}\:\mathrm{or}\:\mathrm{0}? \\ $$ Answered by Filup last updated on 19/Nov/15 $$\mathrm{0}.\overset{−} {\mathrm{9}}=\mathrm{1}\:\left(\mathrm{through}\:\mathrm{algerbraic}\:\mathrm{menipulation}\right) \\ $$$$\therefore\lfloor\mathrm{0}.\overset{−} {\mathrm{9}}\rfloor=\lfloor\mathrm{1}\rfloor=\mathrm{1} \\…

Question-67918

Question Number 67918 by aliesam last updated on 02/Sep/19 Answered by $@ty@m123 last updated on 02/Sep/19 $${Let}\:\alpha+\beta={x} \\ $$$$\mathrm{tan}\:{x}=\frac{\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{3}}}{\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}×\frac{\mathrm{1}}{\mathrm{3}}}=\mathrm{1}\Rightarrow{x}=\mathrm{45}^{\mathrm{o}} \:\:…\left(\mathrm{1}\right) \\ $$$$\mathrm{tan}\:\beta=\frac{\mathrm{1}}{\mathrm{3}}\Rightarrow\mathrm{sin}\:\beta=\frac{\mathrm{1}}{\:\sqrt{\mathrm{10}}}\:\&\:\mathrm{cos}\:\beta=\frac{\mathrm{3}}{\:\sqrt{\mathrm{10}}}\:..\left(\mathrm{2}\right) \\ $$$$\therefore\:\mathrm{cos}\:\left(\mathrm{12}\alpha+\mathrm{13}\beta\right)=\mathrm{cos}\:\left(\mathrm{12}{x}+\beta\right) \\…

Question-133452

Question Number 133452 by bagjagunawan last updated on 22/Feb/21 Answered by mnjuly1970 last updated on 22/Feb/21 $$\boldsymbol{\phi}\overset{{x}=\mathrm{2}{y}−\mathrm{1}} {=}\int_{\frac{\mathrm{1}}{\mathrm{2}}\:} ^{\:\mathrm{1}} \frac{{ln}\left(\mathrm{2}−\mathrm{2}{y}\right){ln}\left({ln}\left(\mathrm{2}{y}\right)\right)}{\mathrm{2}{y}}\left(\mathrm{2}\right){dy} \\ $$$$=\int_{\frac{\mathrm{1}}{\mathrm{2}}} ^{\:\mathrm{1}} \left({ln}\left(\mathrm{2}\right)+{ln}\left(\mathrm{1}−{y}\right)\right)\left({ln}\left(\mathrm{2}\right)+{ln}\left({y}\right)\right)\frac{{dy}}{{y}} \\…

Of-the-numbers-1-2-3-6000-how-many-are-not-multiples-of-2-3-or-5-

Question Number 2381 by Yozzi last updated on 18/Nov/15 $${Of}\:{the}\:{numbers}\:\mathrm{1},\:\mathrm{2},\:\mathrm{3},\:…\:,\:\mathrm{6000}, \\ $$$${how}\:{many}\:{are}\:{not}\:{multiples}\:{of}\:\mathrm{2},\:\mathrm{3}\:{or}\:\mathrm{5}? \\ $$ Commented by 123456 last updated on 18/Nov/15 $$\mathrm{N}=\mathrm{N}_{\mathrm{2}} +\mathrm{N}_{\mathrm{3}} +\mathrm{N}_{\mathrm{5}} −\mathrm{N}_{\mathrm{2},\mathrm{3}}…

Question-133454

Question Number 133454 by rs4089 last updated on 22/Feb/21 Answered by TheSupreme last updated on 22/Feb/21 $$\Omega=\left\{\left({x},{y}\right)\mid\:{x}>\mathrm{0}\:;\:\mathrm{0}<{y}<{x}\right\} \\ $$$$\Omega=\left\{\left({x},{y}\right)\mid\:{y}>\mathrm{0},\:{x}>{y}\right\} \\ $$$${I}=\int_{\mathrm{0}} ^{\infty} \int_{\mathrm{0}} ^{{x}} {e}^{−{xy}}…