Question Number 211255 by ajfour last updated on 02/Sep/24 $$\sqrt{{a}+\sqrt{{b}−{x}}+\sqrt{{b}−\sqrt{{a}+{x}}}}=\mathrm{2}{x} \\ $$$${solve}\:{for}\:{x}.\:\:\:\: \\ $$ Commented by Ghisom last updated on 03/Sep/24 $$\sqrt{{a}+\sqrt{{b}−{x}}+\sqrt{{b}−\sqrt{{a}+{x}}}}=\mathrm{2}{x} \\ $$$${x}=\left({b}−\left(\mathrm{4}{x}^{\mathrm{2}} −\sqrt{{b}−{x}}−{a}\right)^{\mathrm{2}}…
Question Number 211245 by Ghisom last updated on 01/Sep/24 $$\mathrm{prove}: \\ $$$$\underset{\mathrm{0}} {\overset{\infty} {\int}}\frac{{t}^{\alpha−\mathrm{1}} }{{t}^{\pi} +\mathrm{1}}{dt}=\frac{\mathrm{1}}{\mathrm{sin}\:\alpha} \\ $$ Answered by Frix last updated on 02/Sep/24…
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Question Number 211241 by efronzo1 last updated on 01/Sep/24 $$\:\:\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{cos}\:\mathrm{x}^{\mathrm{2}} −\mathrm{cos}\:\left(\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}\:\right)}{\mathrm{x}^{\mathrm{6}} }\:=? \\ $$ Commented by Frix last updated on 01/Sep/24 $$\mathrm{Using}\:\mathrm{Taylor}\:\mathrm{polynomials}\:\mathrm{I}\:\mathrm{get}\:−\frac{\mathrm{1}}{\mathrm{3}} \\…
Question Number 211250 by RojaTaniya last updated on 01/Sep/24 $${Find}\:{the}\:{number}\:{of}\:\mathrm{4}\:{digit}\:{numbers} \\ $$$$\:{so}\:{that}\:{when}\:{decomposed}\:{into}\:{prime} \\ $$$$\:{factors},\:{have}\:{the}\:{sum}\:{of}\:{prime}\:{factors} \\ $$$$\:{equal}\:{to}\:{the}\:{sum}\:{of}\:{the}\:{exponents}? \\ $$ Answered by mr W last updated on…
Question Number 211251 by hardmath last updated on 01/Sep/24 Commented by a.lgnaoui last updated on 04/Sep/24 Commented by a.lgnaoui last updated on 05/Sep/24 $$\frac{\mathrm{sin}\:\mathrm{C}}{\mathrm{MN}}=\frac{\mathrm{sin}\:\mathrm{M}}{\mathrm{BC}−\mathrm{AB}}=\frac{\mathrm{sin}\:\mathrm{N}}{\mathrm{CD}−\mathrm{AD}}\left(\mathrm{1}\right) \\…
Question Number 211235 by behi834171 last updated on 01/Sep/24 Commented by behi834171 last updated on 01/Sep/24 $$\angle\boldsymbol{{ABC}};\:{is}\:{given}. \\ $$$$\boldsymbol{{D}};{is}\:{a}\:{point}\:{as}\:{it}\:{showen},{and} \\ $$$$\boldsymbol{{DG}}\:\:\&\:\boldsymbol{{DF}};\:{are}\:{distance}\:{of}:\boldsymbol{{D}},\:{from}: \\ $$$$\boldsymbol{{AB}}\:\&\:\boldsymbol{{BC}},{such}\:{that}:\frac{\boldsymbol{{DG}}}{\boldsymbol{{DF}}}=\boldsymbol{{k}}\left(\boldsymbol{{constant}}\right) \\ $$$$\boldsymbol{{find}}\:\:\boldsymbol{{the}}\:\:\boldsymbol{{locus}}\:\boldsymbol{{of}}:\:\boldsymbol{{D}}\:.…
Question Number 211222 by RojaTaniya last updated on 31/Aug/24 $${How}\:{many}\:{integers}\:{are}\:{there} \\ $$$$\:\:\:{such}\:{that},\:\:\mathrm{0}\leqslant{n}\leqslant\mathrm{720}\:{and}\: \\ $$$$\:\:\:{n}^{\mathrm{2}} \equiv\mathrm{1}\left({mod}\right)\mathrm{720}? \\ $$ Answered by mahdipoor last updated on 31/Aug/24 $$\Rightarrow{n}^{\mathrm{2}}…
Question Number 211207 by Etimbuk last updated on 31/Aug/24 Answered by mr W last updated on 03/Sep/24 $${e}_{\mathrm{1}} ={a}+{b}+{c}+{d} \\ $$$${e}_{\mathrm{2}} ={ab}+{bc}+{cd}+{da}+{ac}+{bd} \\ $$$${e}_{\mathrm{3}} ={abc}+{bcd}+{cda}+{dab}…