Question Number 145002 by mathdanisur last updated on 01/Jul/21 $${Find}\:{the}\:{last}\:{digit}\:{of}\:{the}\:{number}: \\ $$$$\mathrm{1}^{\mathrm{1989}} +\mathrm{2}^{\mathrm{1989}} +\mathrm{3}^{\mathrm{1989}} +…+\mathrm{1989}^{\mathrm{1989}} \\ $$ Commented by mr W last updated on 01/Jul/21…
Question Number 144989 by mathdanisur last updated on 01/Jul/21 $${lcm}\left(\mathrm{2}{a};\mathrm{3}{a}\right)={lcm}\left(\mathrm{45};\mathrm{100}\right)\Rightarrow{a}=? \\ $$ Answered by Rasheed.Sindhi last updated on 01/Jul/21 $${lcm}\left(\mathrm{2}{a};\mathrm{3}{a}\right)={lcm}\left(\mathrm{45};\mathrm{100}\right)\Rightarrow{a}=? \\ $$$${lcm}\left(\mathrm{2}{a};\mathrm{3}{a}\right)=\mathrm{6}{a}={lcm}\left(\mathrm{45};\mathrm{100}\right)=\mathrm{900} \\ $$$${a}=\mathrm{900}/\mathrm{6}=\mathrm{150} \\…
Question Number 144981 by mathdanisur last updated on 01/Jul/21 $$\underset{{k}=\mathrm{1}} {\overset{\mathrm{35}} {\sum}}\:\frac{\sqrt{{k}}}{{k}\:+\:\sqrt{{k}^{\mathrm{2}} \:+\:{k}}}\:=\:? \\ $$ Answered by liberty last updated on 01/Jul/21 $$\:\frac{\sqrt{\mathrm{k}}}{\mathrm{k}+\sqrt{\mathrm{k}^{\mathrm{2}} +\mathrm{k}}}\:=\:\frac{\sqrt{\mathrm{k}}\:\left(\mathrm{k}−\sqrt{\mathrm{k}^{\mathrm{2}} +\mathrm{k}}\right)}{−\mathrm{k}}…
Question Number 144961 by mathdanisur last updated on 30/Jun/21 $${x}\in\mathbb{Z}^{+} \\ $$$$\mathrm{15}^{\mathrm{48}\boldsymbol{{a}}+\mathrm{1}} \:\equiv\:{x}\:\left({mod}\:\mathrm{17}\right)\:\:{find}\:\:{x}=?\: \\ $$ Answered by Rasheed.Sindhi last updated on 01/Jul/21 $$\mathrm{15}^{\mathrm{8}} \equiv{x}\left({mod}\mathrm{17}\right) \\…
Question Number 79424 by peter frank last updated on 25/Jan/20 $${Derive}\:{the}\:{width}\:{of}\:{the} \\ $$$${diffraction}\:{pattern}\:{for} \\ $$$${the}\:{case}\:{of} \\ $$$$\left({i}\right){single}\:{slits} \\ $$$$\left({ii}\right){double}\:{slits} \\ $$ Terms of Service Privacy…
Question Number 144959 by mathdanisur last updated on 30/Jun/21 Answered by Ar Brandon last updated on 01/Jul/21 $$\mathrm{sin72}°=\mathrm{cos18}°=\frac{\sqrt{\mathrm{10}+\mathrm{2}\sqrt{\mathrm{5}}}}{\mathrm{4}} \\ $$ Commented by Ar Brandon last…
Question Number 144951 by loveineq last updated on 30/Jun/21 $$\mathrm{Let}\:{a},{b}\:>\:\mathrm{0}\:\mathrm{and}\:{a}+{b}+\mathrm{1}\:=\:\mathrm{3}{ab}.\:\mathrm{Prove}\:\mathrm{that} \\ $$$$\left(\mathrm{1}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{{a}^{\mathrm{2}} }{{a}+\mathrm{1}}+\frac{{b}^{\mathrm{2}} }{{b}+\mathrm{1}}\:\geqslant\:\frac{{a}}{{a}^{\mathrm{2}} +\mathrm{1}}+\frac{{b}}{{b}^{\mathrm{2}} +\mathrm{1}} \\ $$$$\left(\mathrm{2}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{{a}^{\mathrm{2}} }{{b}+\mathrm{1}}+\frac{{b}^{\mathrm{2}} }{{a}+\mathrm{1}}\:\geqslant\:\frac{{a}}{{b}^{\mathrm{2}} +\mathrm{1}}+\frac{{b}}{{a}^{\mathrm{2}} +\mathrm{1}} \\ $$ Commented…
Question Number 79417 by mr W last updated on 25/Jan/20 $${For}\:{x},{y}\in\mathbb{R}\:{find}\:{the}\:{minimum}\:{and} \\ $$$${maximum}\:{of}\:\mathrm{2}{x}^{\mathrm{2}} −\mathrm{3}{x}+\mathrm{4}{y} \\ $$$${if}\:{x}^{\mathrm{2}} +\mathrm{2}{y}^{\mathrm{2}} −{xy}−\mathrm{5}{x}−\mathrm{7}{y}−\mathrm{30}=\mathrm{0}. \\ $$ Commented by john santu last…
Question Number 144947 by mathdanisur last updated on 30/Jun/21 $$\underset{{k}=\mathrm{1}} {\overset{\mathrm{12}} {\prod}}\mathrm{2}\centerdot{sin}\left(\frac{\pi{k}}{\mathrm{24}}\right)\:=\:? \\ $$ Answered by Olaf_Thorendsen last updated on 01/Jul/21 $$\mathrm{sin2}\theta\:=\:\mathrm{2cos}\theta\mathrm{sin}\theta \\ $$$$\mathrm{2cos}\theta\:=\:\frac{\mathrm{sin2}\theta}{\mathrm{sin}\theta} \\…
Question Number 144936 by mathdanisur last updated on 30/Jun/21 Answered by mindispower last updated on 30/Jun/21 $${sec}^{\mathrm{4}} \left({x}\right)+{csc}^{\mathrm{4}} \left({x}\right)\geqslant\frac{\mathrm{2}}{{cos}^{\mathrm{2}} \left({x}\right){sin}^{\mathrm{2}} \left({x}\right)}=\frac{\mathrm{8}}{\left({sin}\left(\mathrm{2}{x}\right)\right)^{\mathrm{2}} } \\ $$$$\geqslant\mathrm{8} \\…