Question Number 63532 by Rio Michael last updated on 05/Jul/19 $${prove}\:{that}\:{there}\:{exist}\:{unique}\:{intergers}\:{p}\:{and}\:{s}\:{sucb}\:{that} \\ $$$${a}\:=\:{bp}\:+\:{s}\:{with}\:−\frac{\mid{b}\mid}{\mathrm{2}}<\:{s}\:\leqslant\frac{\mid{b}\mid}{\mathrm{2}} \\ $$$${hence}\:{find}\:{p}\:{and}\:{s}\:{given}\:{that}\:{a}=\mathrm{49}\:{and}\:{b}=\mathrm{26} \\ $$ Answered by MJS last updated on 05/Jul/19 $${a}={bp}+{s}\:\Rightarrow\:{s}={a}−{bp}…
Question Number 63517 by Rio Michael last updated on 05/Jul/19 $$\mathrm{Given}\:\mathrm{that}\:\:\mid{z}−\mathrm{6}\mid=\mathrm{2}\mid{z}+\mathrm{6}−\mathrm{9}{i}\mid, \\ $$$$\left.\mathrm{a}\right)\:\mathrm{Use}\:\mathrm{algebra}\:\mathrm{to}\:\mathrm{show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{locus}\:\mathrm{of}\:{z}\:\mathrm{is}\:\mathrm{a}\:\mathrm{circle}, \\ $$$$\mathrm{stating}\:\mathrm{its}\:\mathrm{center}\:\mathrm{and}\:\mathrm{its}\:\mathrm{radius}. \\ $$$$\left.\mathrm{b}\right)\:\mathrm{sketch}\:\mathrm{the}\:\mathrm{locus}\:{z}\:\mathrm{on}\:\mathrm{an}\:\mathrm{argand}\:\mathrm{diagram}. \\ $$ Answered by MJS last updated on…
Question Number 129047 by Dwaipayan Shikari last updated on 12/Jan/21 $$\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}\left(\frac{\mathrm{1}}{\mathrm{2}.\mathrm{1}!}\right)^{\mathrm{2}} +\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{2}} }\left(\frac{\mathrm{1}.\mathrm{3}}{\mathrm{2}^{\mathrm{2}} .\mathrm{2}!}\right)^{\mathrm{2}} +\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{3}} }\left(\frac{\mathrm{1}.\mathrm{3}.\mathrm{5}}{\mathrm{2}^{\mathrm{3}} .\mathrm{3}!}\right)^{\mathrm{2}} +…=\frac{\sqrt{\pi}}{\Gamma^{\mathrm{2}} \left(\frac{\mathrm{3}}{\mathrm{4}}\right)} \\ $$ Answered by mindispower last…
Question Number 63507 by mathmax by abdo last updated on 05/Jul/19 $${let}\:{U}_{{n}} =\int_{\frac{\mathrm{1}}{{n}}} ^{\mathrm{1}} \left(\sqrt{{x}^{\mathrm{2}} +{x}+\mathrm{1}}\:−\sqrt{{x}^{\mathrm{2}} −{x}+\mathrm{1}}\right){dx}\:\:\:\left({n}>\mathrm{0}\right) \\ $$$$\left.\mathrm{1}\right){calculate}\:{lim}_{{n}\rightarrow+\infty} \:\:{U}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:\:{find}\:{nature}\:{of}\:\:\Sigma\:{U}_{{n}} \\ $$ Answered…
Question Number 63499 by Rio Michael last updated on 04/Jul/19 $${given}\:{that}\:\:\:{a}\mid{b},\:{show}\:{that}\:−{a}\mid{b}. \\ $$ Answered by MJS last updated on 04/Jul/19 $${a}\mid{b}\:\Rightarrow\:\frac{{b}}{{a}}={c}\:\mathrm{with}\:{c}\in\mathbb{Z} \\ $$$$\Rightarrow\:−\frac{{b}}{{a}}=\left(−\mathrm{1}\right)\frac{{b}}{{a}}=\left(−\mathrm{1}\right){c}=−{c} \\ $$$${c}\in\mathbb{Z}\:\Leftrightarrow\:−{c}\in\mathbb{Z}…
Question Number 129018 by BHOOPENDRA last updated on 12/Jan/21 $$\frac{\left(\sqrt{{s}}−\mathrm{1}\right)^{\mathrm{2}} }{{s}^{\mathrm{2}} }\:{find}\:{the}\:{inverse}\:{laplace}\:{transformtion} \\ $$ Answered by Dwaipayan Shikari last updated on 12/Jan/21 $$\mathscr{L}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{{s}}−\frac{\mathrm{2}}{\:{s}^{\frac{\mathrm{3}}{\mathrm{2}}} }+\frac{\mathrm{1}}{{s}^{\mathrm{2}}…
Question Number 129010 by BHOOPENDRA last updated on 12/Jan/21 Commented by BHOOPENDRA last updated on 12/Jan/21 $${find}\:{the}\:{inverse}\:{laplace}\:{transformation}? \\ $$ Answered by Dwaipayan Shikari last updated…
Question Number 63447 by minh2001 last updated on 04/Jul/19 $${How}\:{to}\:{calculate}\:\lceil\left({n}\right)\: \\ $$$${using}\:{gamma}\:{function} \\ $$$$\forall{n}\in{R} \\ $$ Commented by minh2001 last updated on 04/Jul/19 $${I}\:{can}'{t}\:{remember}\:{it}\:{until}\: \\…
Question Number 63428 by Rio Michael last updated on 04/Jul/19 $${It}\:{is}\:{given}\:{that}\:{S}_{{n}} =\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}\left(\mathrm{3}{r}^{\mathrm{2}\:} −\mathrm{3}{r}−\mathrm{1}\right).\:{Use}\:{the}\:{the}\:{formulae} \\ $$$${of}\:\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}{r}^{\mathrm{2}\:\:} {and}\:\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}{r}\:\:{to}\:{show}\:{that}\:{S}_{{n}} ={n}^{\mathrm{3}} . \\…
Question Number 63424 by Rio Michael last updated on 04/Jul/19 $${A}\:{colony}\:{of}\:{bacteria}\:{if}\:{left}\:{undisturbed}\:{will}\:{grow}\:{at}\:{a}\:{rate} \\ $$$${proportional}\:{to}\:{the}\:{number}\:{of}\:{bacteria},\:{P}\:{present}\:{at}\:{time},{t}. \\ $$$${However},{a}\:{toxic}\:{substance}\:{is}\:{being}\:{added}\:{slowly}\:{such}\:{that} \\ $$$${at}\:{time}\:{t},\:{the}\:{bacteria}\:{also}\:{die}\:{at}\:{the}\:{rate}\:\mu{Pt}\:{where}\:\mu\:{is} \\ $$$${a}\:{positive}\:{constant}. \\ $$$$\left({a}\right)\:\:{Show}\:{that}\:{at}\:{time}\:{t}\:{the}\:{rate}\:{of}\:{growth}\:{of}\:{the}\:{bacteria}\:{in} \\ $$$${the}\:{colony}\:{is}\:{governed}\:{by}\:{the}\:{differential}\:{equation} \\ $$$$\:\frac{{dP}}{{dt}}=\:\left({k}−\mu{t}\right){p}\:{where}\:{k}\:{is}\:{apositive}\:{constant}.…