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Author: Tinku Tara

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Question Number 203192 by MrGHK last updated on 12/Jan/24 $$\boldsymbol{\mathrm{find}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{last}}\:\boldsymbol{\mathrm{four}}\:\boldsymbol{\mathrm{digits}}\:\boldsymbol{\mathrm{of}}\:\mathrm{2022}^{\mathrm{2023}} +\mathrm{2023}^{\mathrm{2022}} \\ $$ Answered by AST last updated on 12/Jan/24 $${x}=\mathrm{2022}^{\mathrm{2023}} +\mathrm{2023}^{\mathrm{2022}} \overset{\mathrm{16}} {\equiv}\mathrm{6}^{\mathrm{2023}} +\mathrm{7}^{\mathrm{2022}}…

Question-203219

Question Number 203219 by hardmath last updated on 12/Jan/24 Answered by witcher3 last updated on 15/Jan/24 $$\mathrm{evident} \\ $$$$\left(\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{d}\right)^{\mathrm{2}} =\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} +\mathrm{c}^{\mathrm{2}} +\mathrm{d}^{\mathrm{2}} +\mathrm{2}\left(\mathrm{ab}+\mathrm{bc}+\mathrm{cd}+\mathrm{da}\right) \\…

Question-203187

Question Number 203187 by Amidip last updated on 12/Jan/24 Answered by mr W last updated on 12/Jan/24 $${supposed}:\:{AB}\bot{AC},\:{AD}\bot{BC} \\ $$$$\frac{\mathrm{45}}{{x}}=\frac{{x}+\mathrm{48}}{\mathrm{45}}\: \\ $$$$\Rightarrow{x}^{\mathrm{2}} +\mathrm{48}{x}−\mathrm{45}^{\mathrm{2}} =\mathrm{0} \\…

Question-203211

Question Number 203211 by ajfour last updated on 12/Jan/24 Answered by MM42 last updated on 12/Jan/24 $${p}^{\mathrm{2}} =\mathrm{2}−\mathrm{2}{cosa} \\ $$$${q}^{\mathrm{2}} =\mathrm{2}−\mathrm{2}{sina} \\ $$$$\Rightarrow\left({p}^{\mathrm{2}} −\mathrm{2}\right)^{\mathrm{2}} +\left({q}^{\mathrm{2}}…

Question-203206

Question Number 203206 by hardmath last updated on 12/Jan/24 Answered by mr W last updated on 12/Jan/24 $$\underset{{k}=\mathrm{1}} {\overset{\mathrm{100}} {\sum}}\mid{i}−{k}\mid \\ $$$$=\underset{{k}=\mathrm{1}} {\overset{{i}} {\sum}}\mid{i}−{k}\mid+\underset{{k}={i}+\mathrm{1}} {\overset{\mathrm{100}}…