Question Number 217377 by ArshadS last updated on 12/Mar/25 $$\mathrm{Let}\:\mathrm{x},\mathrm{y},\mathrm{z}\:\mathrm{be}\:\mathrm{real}\:\mathrm{numbers}\:\mathrm{satisfying}\:\mathrm{the}\:\mathrm{equations} \\ $$$$\mathrm{x}\:+\:\mathrm{y}\:+\:\mathrm{z}=\:\mathrm{7} \\ $$$$\mathrm{xy}\:+\:\mathrm{yz}\:+\:\mathrm{zx}=\mathrm{10} \\ $$$$\mathrm{xyz}=\mathrm{6} \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$$\mathrm{x}^{\mathrm{3}} \:+\:\mathrm{y}^{\mathrm{3}} \:+\:\mathrm{z}^{\mathrm{3}} \\ $$ Answered…
Question Number 217359 by ArshadS last updated on 11/Mar/25 $$\mathrm{Find}\:\mathrm{all}\:\mathrm{three}-\mathrm{digit}\:\mathrm{numbers}\:\mathrm{such}\:\mathrm{that}\:\mathrm{when}\:\mathrm{the}\:\mathrm{number}\:\mathrm{is} \\ $$$$\mathrm{divided}\:\mathrm{by}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{its}\:\mathrm{digits}\:\mathrm{the}\:\mathrm{quotient}\:\mathrm{is}\:\mathrm{7}\:\mathrm{and}\:\mathrm{the}\: \\ $$$$\mathrm{remainder}\:\mathrm{is}\:\mathrm{5}. \\ $$ Answered by Rasheed.Sindhi last updated on 12/Mar/25 $$\mathrm{100}{h}+\mathrm{10}{t}+{u}=\mathrm{7}\left({h}+{t}+{u}\right)+\mathrm{5} \\…
Question Number 217326 by Rasheed.Sindhi last updated on 10/Mar/25 $${Find}\:{three}\:{consecutive}\:{integers}\: \\ $$$${such}\:{that}\:{the}\:{sum}\:{of}\:{their}\:{squares} \\ $$$$\:{is}\:\mathrm{50}. \\ $$ Answered by Hanuda354 last updated on 10/Mar/25 $$\mathrm{Let}\:{n},\:{n}+\mathrm{1}\:\:\mathrm{and}\:\:{n}+\mathrm{2}\:\:\mathrm{are}\:\:\mathrm{required}\:\:\mathrm{numbers}. \\…
Question Number 217321 by ArshadS last updated on 10/Mar/25 $$\mathrm{Find}\:\mathrm{three}\:\mathrm{consecutive}\:\mathrm{integers}\:\mathrm{such}\:\mathrm{that}\:\mathrm{the}\:\mathrm{product}\:\mathrm{of}\:\mathrm{the}\: \\ $$$$\mathrm{first}\:\mathrm{two}\:\mathrm{is}\:\mathrm{16}\:\mathrm{more}\:\mathrm{than}\:\mathrm{twice}\:\mathrm{the}\:\mathrm{third}\:\mathrm{integer}.\:\mathrm{Provide}\:\mathrm{all}\: \\ $$$$\mathrm{possible}\:\mathrm{solutions}. \\ $$ Answered by Rasheed.Sindhi last updated on 10/Mar/25 $${let}\:{x}−\mathrm{1},{x},{x}+\mathrm{1}\:{are}\:{required}\:{numbers} \\…
Question Number 217300 by peter frank last updated on 09/Mar/25 Answered by Marzuk last updated on 09/Mar/25 $$\:\:\:\:\:\:\:\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }\:−\:\mathrm{2}{a}\frac{{dy}}{{dx}}\:+\:\left({a}^{\mathrm{2}} \:+\:{b}^{\mathrm{2}} \right){y}\:=\:\mathrm{0} \\ $$$${or},\frac{{dy}'}{{dx}}\:−\:\mathrm{2}{a}\frac{{d}}{{dx}}\:\left[{e}^{{ax}} {sin}\left({bx}\right)\:\right]\:+\:{a}^{\mathrm{2}}…
Question Number 217299 by peter frank last updated on 09/Mar/25 Answered by mr W last updated on 10/Mar/25 $${x}={r}\:\mathrm{cos}\:\theta \\ $$$${y}={r}\:\mathrm{sin}\:\theta \\ $$$$\left({r}\:\mathrm{cos}\:\theta−\mathrm{1}\right)^{\mathrm{2}} +\left({r}\:\mathrm{sin}\:\theta\right)^{\mathrm{2}} =\mathrm{16}…
Question Number 217291 by ArshadS last updated on 08/Mar/25 $$\mathrm{Solve}: \\ $$$$\frac{{x}+\mathrm{2}}{{x}−\mathrm{3}}−\frac{{x}−\mathrm{1}}{{x}+\mathrm{4}}=\frac{\mathrm{10}}{{x}^{\mathrm{2}} +{x}−\mathrm{12}} \\ $$ Answered by Hanuda354 last updated on 08/Mar/25 $$\frac{\left({x}^{\mathrm{2}} +\mathrm{6}{x}+\mathrm{8}\right)−\left({x}^{\mathrm{2}} −\mathrm{4}{x}+\mathrm{3}\right)}{{x}^{\mathrm{2}}…
Question Number 217262 by ArshadS last updated on 07/Mar/25 $${Solve}: \\ $$$$\frac{{x}^{\mathrm{2}} +\mathrm{3}{x}}{{x}^{\mathrm{3}} −\mathrm{4}{x}}−\frac{\mathrm{2}}{{x}^{\mathrm{2}} +\mathrm{2}{x}}=\frac{\mathrm{1}}{{x}−\mathrm{2}} \\ $$ Answered by ibrahimmatematic last updated on 07/Mar/25 $$\frac{{x}\left({x}+\mathrm{3}\right)}{{x}\left({x}−\mathrm{2}\right)\left({x}+\mathrm{2}\right)}−\frac{\mathrm{2}}{{x}\left({x}+\mathrm{2}\right)}=\frac{\mathrm{1}}{{x}−\mathrm{2}}…
Question Number 217225 by hardmath last updated on 06/Mar/25 $$\mathrm{Find}:\:\:\:\:\:\int\:−\:\frac{{d}\mathrm{x}}{\:\sqrt{\mathrm{24x}\:−\:\mathrm{16x}^{\mathrm{2}} \:−\:\mathrm{8}}}\:=\:? \\ $$ Answered by Frix last updated on 06/Mar/25 $$−\int\frac{{dx}}{\:\sqrt{−\mathrm{16}{x}^{\mathrm{2}} +\mathrm{24}{x}−\mathrm{8}}}\:\overset{\left[{t}=\mathrm{sin}^{−\mathrm{1}} \:\left(\mathrm{4}{x}−\mathrm{3}\right)\right]} {=} \\…
Question Number 217197 by hardmath last updated on 05/Mar/25 $$\mathrm{Find}: \\ $$$$\sqrt[{\mathrm{5}}]{−\mathrm{32}}\:\:+\:\:\sqrt[{\mathrm{8}}]{\left(−\mathrm{3}\right)^{\mathrm{8}} }\:\:=\:\:? \\ $$ Answered by Rasheed.Sindhi last updated on 05/Mar/25 $$\sqrt[{\mathrm{5}}]{−\mathrm{32}}\:\:+\:\:\sqrt[{\mathrm{8}}]{\left(−\mathrm{3}\right)^{\mathrm{8}} }\: \\…