Question Number 148525 by nadovic last updated on 28/Jul/21 $$\:\mathrm{Express}\:\:\frac{\sqrt{\mathrm{15}}+\sqrt{\mathrm{35}}+\sqrt{\mathrm{21}}+\mathrm{5}}{\:\sqrt{\mathrm{3}}+\mathrm{2}\sqrt{\mathrm{5}}+\sqrt{\mathrm{7}}}\:\mathrm{in}\:\mathrm{the}\:\mathrm{form} \\ $$$$\:{a}\sqrt{\mathrm{3}\:}+\:{b}\sqrt{\mathrm{7}}. \\ $$ Answered by Ar Brandon last updated on 28/Jul/21 $$\frac{\sqrt{\mathrm{15}}+\sqrt{\mathrm{35}}+\sqrt{\mathrm{21}}+\mathrm{5}}{\:\sqrt{\mathrm{3}}+\mathrm{2}\sqrt{\mathrm{5}}+\sqrt{\mathrm{7}}} \\ $$$$=\frac{\sqrt{\mathrm{3}}\centerdot\sqrt{\mathrm{5}}+\sqrt{\mathrm{7}}\centerdot\sqrt{\mathrm{5}}+\sqrt{\mathrm{3}}\centerdot\sqrt{\mathrm{7}}+\sqrt{\mathrm{5}}\overset{\mathrm{2}}…
Question Number 17454 by Tinkutara last updated on 06/Jul/17 $$\mathrm{Find}\:\mathrm{all}\:\mathrm{integers}\:{n}\:\mathrm{such}\:\mathrm{that} \\ $$$$\left({n}^{\mathrm{2}} \:−\:{n}\:−\:\mathrm{1}\right)^{{n}\:+\:\mathrm{2}} \:=\:\mathrm{1} \\ $$ Answered by mrW1 last updated on 06/Jul/17 $$\mathrm{n}^{\mathrm{2}} −\mathrm{n}−\mathrm{1}=\mathrm{1}\:\mathrm{if}\:\mathrm{n}\:\mathrm{is}\:\mathrm{odd}\:\mathrm{or}\:\mathrm{even}\:\:\left(\mathrm{i}\right)…
Question Number 148522 by mathdanisur last updated on 28/Jul/21 Answered by Rasheed.Sindhi last updated on 29/Jul/21 $${P}\left({x}\right)=\mathrm{16}{x}^{\mathrm{2}} +\mathrm{24}{x}+\mathrm{9} \\ $$$${Q}\left({x}\right)={x}^{\mathrm{3}} −\mathrm{3}{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{2} \\ $$$${If}\:{x}=−\mathrm{1}.\mathrm{25} \\…
Question Number 148513 by mathdanisur last updated on 28/Jul/21 $$\mathrm{6}\:+\:{log}_{\mathrm{2}} \:{sin}\mathrm{15}°\:-\:{log}_{\frac{\mathrm{1}}{\mathrm{2}}} {sin}\mathrm{75}°\:=\:? \\ $$ Answered by liberty last updated on 29/Jul/21 $$\chi\:=\:\mathrm{6}\:+\mathrm{log}\:_{\mathrm{2}} \mathrm{sin}\:\mathrm{15}°+\mathrm{log}\:_{\mathrm{2}} \mathrm{sin}\:\mathrm{75}° \\…
Question Number 148515 by mathdanisur last updated on 28/Jul/21 $${Find}\:{the}\:{sum}\:{of}\:{the}\:{roots}\:{of}\:{the}\:{equation}: \\ $$$${x}^{\mathrm{2}} \:-\:\mathrm{2}{x}\:-\:\mathrm{3}\:\mid{x}\:-\:\mathrm{1}\mid\:+\:\mathrm{3}\:=\:\mathrm{0} \\ $$ Answered by dumitrel last updated on 28/Jul/21 $$\mid{x}−\mathrm{1}\mid^{\mathrm{2}} −\mathrm{3}\mid{x}−\mathrm{1}\mid+\mathrm{2}=\mathrm{0}\Rightarrow\mid{x}−\mathrm{1}\mid\in\left\{\mathrm{1},\mathrm{2}\right\} \\…
Question Number 148505 by mathdanisur last updated on 28/Jul/21 Answered by dumitrel last updated on 28/Jul/21 $${O}\left(\mathrm{0},\mathrm{0}\right);{A}\left({x},{y}\right);{B}\left(\mathrm{4};\mathrm{3}\right) \\ $$$${OB}=\sqrt{\mathrm{4}^{\mathrm{2}} +\mathrm{3}^{\mathrm{2}} }=\mathrm{5} \\ $$$$\Rightarrow{OA}+{AB}={OB}\Rightarrow{A}\in\left({OB}\right)\Rightarrow{x},{y}>\mathrm{0} \\ $$$$\frac{{x}}{\mathrm{4}}=\frac{{y}}{\mathrm{3}}\Rightarrow\mathrm{4}{y}=\mathrm{3}{x}\Rightarrow\mathrm{3}{x}^{\mathrm{2}}…
Question Number 82952 by TawaTawa1 last updated on 26/Feb/20 $$\mathrm{Show}\:\mathrm{that}:\:\:\:\:\:\:\mathrm{y}\:\:+\:\:\sqrt{\mathrm{y}^{\mathrm{2}} \:−\:\mathrm{1}}\:\:\:\geqslant\:\:\mathrm{1}\:\:\:\:\:\mathrm{and}\:\:\:\:\mathrm{0}\:\:<\:\:\mathrm{y}\:\:−\:\:\sqrt{\mathrm{y}^{\mathrm{2}} \:−\:\mathrm{1}}\:\:\leqslant\:\:\mathrm{1} \\ $$$$\mathrm{if}\:\:\mathrm{y}\:\:\geqslant\:\mathrm{1} \\ $$ Answered by MJS last updated on 26/Feb/20 $${t}^{\mathrm{2}} ={y}^{\mathrm{2}}…
Question Number 148482 by mathdanisur last updated on 28/Jul/21 Commented by hknkrc46 last updated on 28/Jul/21 $$\bigstar\:\frac{\mathrm{1}}{\mathrm{78}}\:+\:\frac{\mathrm{2}}{\mathrm{79}}\:+\:\frac{\mathrm{3}}{\mathrm{80}}\:−\:\mathrm{3} \\ $$$$=\:\left(\frac{\mathrm{1}}{\mathrm{78}}\:−\:\mathrm{1}\right)\:+\:\left(\frac{\mathrm{2}}{\mathrm{79}}\:−\:\mathrm{1}\right)\:+\:\left(\frac{\mathrm{3}}{\mathrm{80}}\:−\:\mathrm{1}\right) \\ $$$$=\:\frac{−\mathrm{77}}{\mathrm{78}}\:+\:\frac{−\mathrm{77}}{\mathrm{79}}\:+\:\frac{−\mathrm{77}}{\mathrm{80}} \\ $$$$=\:−\mathrm{77}\left(\frac{\mathrm{1}}{\mathrm{78}}\:+\:\frac{\mathrm{1}}{\mathrm{79}}\:+\:\frac{\mathrm{1}}{\mathrm{80}}\right) \\ $$$$\bigstar\:\frac{−\mathrm{77}\left(\frac{\mathrm{1}}{\mathrm{78}}\:+\:\frac{\mathrm{1}}{\mathrm{79}}\:+\:\frac{\mathrm{1}}{\mathrm{80}}\right)}{\frac{\mathrm{1}}{\mathrm{78}}\:+\:\frac{\mathrm{1}}{\mathrm{79}}\:+\:\frac{\mathrm{1}}{\mathrm{80}}}\:=\:−\mathrm{77}…
Question Number 148453 by mathdanisur last updated on 28/Jul/21 $$\frac{\left({n}\:+\:\mathrm{1}\right)!}{{n}!}\:=\:\mathrm{38}\:\:\Rightarrow\:\:{n}=? \\ $$ Answered by puissant last updated on 28/Jul/21 $$\Rightarrow\frac{\left(\mathrm{n}+\mathrm{1}\right)×\mathrm{n}!}{\mathrm{n}!}=\mathrm{38} \\ $$$$\Rightarrow\mathrm{n}+\mathrm{1}=\mathrm{38} \\ $$$$\Rightarrow\mathrm{n}=\mathrm{37}.. \\…
Question Number 148452 by mathdanisur last updated on 28/Jul/21 $$\underset{\:\mathrm{1}} {\overset{\:\mathrm{4}} {\int}}\mathrm{2}{sin}^{\mathrm{2}} {x}\:{dx}\:+\:\underset{\:\mathrm{1}} {\overset{\:\mathrm{4}} {\int}}\left(\mathrm{1}+{cos}\mathrm{2}{x}\right){dx}\:=\:? \\ $$ Answered by puissant last updated on 28/Jul/21 $$=\int_{\mathrm{1}}…