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Author: Tinku Tara

Question-215229

Question Number 215229 by Mingma last updated on 01/Jan/25 Answered by devdutt last updated on 01/Jan/25 $$\mathrm{1}.\:{f}\left(\frac{{k}}{{n}}\right)\:=\:\frac{{n}^{\mathrm{2025}} }{{n}^{\mathrm{2025}} +{k}^{\mathrm{2025}} } \\ $$$$ \\ $$$$\mathrm{2}.\:{f}\left(\frac{{k}}{{n}}\right)+{f}\left(\frac{{n}}{{k}}\right)\:=\:\mathrm{1},\: \\…

Question-215231

Question Number 215231 by Mingma last updated on 01/Jan/25 Answered by A5T last updated on 01/Jan/25 $$\mathrm{2024}=\mathrm{8}×\mathrm{253}=\mathrm{8}×\mathrm{11}×\mathrm{23} \\ $$$$\mathrm{2025}=\mathrm{45}^{\mathrm{2}} =\mathrm{3}^{\mathrm{4}} ×\mathrm{5}^{\mathrm{2}} ;\:\:\:\:\mathrm{2026}=\mathrm{2}×\mathrm{1013} \\ $$$$\Rightarrow\mathrm{smallest}\:\mathrm{perfect}\:\mathrm{square} \\…

Question-215256

Question Number 215256 by ajfour last updated on 01/Jan/25 Commented by ajfour last updated on 01/Jan/25 $${After}\:{this}\:{elastic}\:{collision}\:{on}\:{a}\:{flat} \\ $$$${frictionless}\:{surface}\:{Find}\:{after}\:{what} \\ $$$${time}\:{at}\:{what}\:{y}\:{coordinate}\:{does}\:{point} \\ $$$${A}\:{cross}\:{the}\:{y}-{axis}. \\ $$…

20-25-2-2025-only-3-other-4-digit-numbers-00-01-2-0001-30-25-2-3025-98-01-2-9801-

Question Number 215227 by Ghisom last updated on 01/Jan/25 $$\left(\mathrm{20}+\mathrm{25}\right)^{\mathrm{2}} =\mathrm{2025} \\ $$$$\mathrm{only}\:\mathrm{3}\:\mathrm{other}\:\mathrm{4}\:\mathrm{digit}\:\mathrm{numbers}: \\ $$$$\left(\mathrm{00}+\mathrm{01}\right)^{\mathrm{2}} =\mathrm{0001} \\ $$$$\left(\mathrm{30}+\mathrm{25}\right)^{\mathrm{2}} =\mathrm{3025} \\ $$$$\left(\mathrm{98}+\mathrm{01}\right)^{\mathrm{2}} =\mathrm{9801} \\ $$ Answered…

a-b-c-are-pythagorean-triples-For-the-following-values-determinant-a-b-c-5-12-13-a-2-b-c-5-2-12-13-25-determinant-a-b-c-45-1012-1013-a-2-b-c-45-2-1012

Question Number 215255 by Rasheed.Sindhi last updated on 01/Jan/25 $${a},{b},{c}\:{are}\:{pythagorean}\:{triples} \\ $$$${For}\:{the}\:{following}\:{values}: \\ $$$$\begin{array}{|c|c|}{{a}}&\hline{{b}}&\hline{\:{c}}\\{\mathrm{5}}&\hline{\mathrm{12}}&\hline{\mathrm{13}}\\\hline\end{array}\:\:\underset{\Rightarrow\mathrm{5}^{\mathrm{2}} =\mathrm{12}+\mathrm{13}=\mathrm{25}} {{a}^{\mathrm{2}} ={b}+{c}\:\:} \\ $$$$\begin{array}{|c|c|}{{a}}&\hline{{b}}&\hline{\:{c}}\\{\mathrm{45}}&\hline{\mathrm{1012}}&\hline{\mathrm{1013}}\\\hline\end{array}\underset{\:\:\:\:\:\:\:\Rightarrow\mathrm{45}^{\mathrm{2}} =\mathrm{1012}+\mathrm{1013}=\mathrm{2025}} {{a}^{\mathrm{2}} ={b}+{c}} \\ $$$$\Rightarrow\mathrm{45}^{\mathrm{2}} =\mathrm{1012}+\mathrm{1013}=\mathrm{2025}…

10-7-9-4-21-8-

Question Number 215248 by sudipyt44 last updated on 01/Jan/25 $$\frac{\mathrm{10}}{\mathrm{7}}\boldsymbol{\div}\frac{\mathrm{9}}{\mathrm{4}}×\frac{\mathrm{21}}{\mathrm{8}} \\ $$ Commented by A5T last updated on 01/Jan/25 $$\left(\mathrm{a}\boldsymbol{\div}\mathrm{b}\right)×\mathrm{c}=\frac{\mathrm{ac}}{\mathrm{b}} \\ $$$$\mathrm{a}\boldsymbol{\div}\left(\mathrm{b}×\mathrm{c}\right)=\frac{\mathrm{a}}{\mathrm{bc}} \\ $$ Answered…

Question-215232

Question Number 215232 by Mingma last updated on 01/Jan/25 Answered by devdutt last updated on 01/Jan/25 $$\mathrm{ln}\:\mathrm{3}\sqrt{\mathrm{5}}\:+\:\mathrm{ln}\:{x}\:\mathrm{log}\:_{\mathrm{45}} {x}\:=\:\mathrm{2ln}\:{x}\: \\ $$$$\Rightarrow\:\mathrm{ln}\:\mathrm{3}\sqrt{\mathrm{5}}\:+\:\frac{\mathrm{ln}\:^{\mathrm{2}} {x}}{\mathrm{ln}\:\mathrm{45}}\:=\:\mathrm{2ln}\:{x}\: \\ $$$$\Rightarrow\:\mathrm{ln}\:^{\mathrm{2}} {x}\:−\:\mathrm{2ln}\:\mathrm{45}\:\mathrm{ln}\:{x}\:+\:\mathrm{ln}\:\mathrm{3}\sqrt{\mathrm{5}}\:\mathrm{ln}\:\mathrm{45}\:=\:\mathrm{0} \\…