Question Number 147585 by mathdanisur last updated on 22/Jul/21 Answered by Rasheed.Sindhi last updated on 22/Jul/21 $$\:\:\:{z}={w}+{f}\::\:{whole}\:{number}\:\&\:{fraction}<\mathrm{1} \\ $$$$\:\:\:\:{w}+{f}−\mathrm{2021}{f}=\mathrm{2021} \\ $$$$\:\:\:\:{w}=\mathrm{2021}{f}−{f}+\mathrm{2021} \\ $$$$\:\:\:\:{w}=\mathrm{2020}{f}+\mathrm{2021} \\ $$$${Let}\:{f}=\frac{{p}}{{q}}\:;\:{p}\:\&\:{q}\:{coprime}\:{p}<{q}…
Question Number 16514 by Sai dadon. last updated on 23/Jun/17 $${Help}\:{plz} \\ $$$${Given}\:\mathrm{2}{d}^{\mathrm{2}} {x}/{dt}^{\mathrm{2}} +\mathrm{2}{dx}/{dt}+{x}={ksint} \\ $$$${where}\:{k}\:{is}\:{constant}\:{show}\:{that}\:{if}\:{x}={n} \\ $$$${and}\:{dx}/{dt}=\mathrm{0}\:{when}\:{t}=\mathrm{0}\:{then}\: \\ $$$$\left.{x}={e}^{−\mathrm{1}/\mathrm{2}{t}} \left\{\left({n}+\mathrm{2}/\mathrm{5}{k}\right){cos}\left({t}/\mathrm{2}\right)+\right)+\left({n}+\mathrm{4}/\mathrm{5}{k}\right){sin}\left({t}/\mathrm{2}\right)\right\} \\ $$$$−\mathrm{1}/\mathrm{5}{ksint}−\mathrm{2}/\mathrm{5}{kcost}. \\…
Question Number 147587 by mathdanisur last updated on 22/Jul/21 Commented by ajfour last updated on 22/Jul/21 $${what}\:{if}\:{we}\:{replace}\:{the}\:{r}.{h}.{s}.\:{in} \\ $$$${this}\:{question}\:{by}\:\:\mathrm{2}{x}\:? \\ $$ Answered by iloveisrael last…
Question Number 147581 by EDWIN88 last updated on 22/Jul/21 $$\:\:\mathrm{2sin}\:\mathrm{2x}\:−\mathrm{4sin}\:^{\mathrm{2}} \mathrm{x}\:=\:\mathrm{7cos}\:\mathrm{2x}\: \\ $$$$\:\frac{\pi}{\mathrm{2}}<\mathrm{x}<\pi\:\Rightarrow\:\mathrm{sin}\:\mathrm{2x}\:=? \\ $$ Answered by iloveisrael last updated on 22/Jul/21 $$\:\mathrm{If}\:\mathrm{2sin}\:\mathrm{2x}−\mathrm{4sin}\:^{\mathrm{2}} \mathrm{x}\:=\:\mathrm{7cos}\:\mathrm{2x}\: \\…
Question Number 147582 by mathdanisur last updated on 22/Jul/21 $${if}\:\:{x};{y};{z}>\mathrm{0}\:\:{prove}\:{that} \\ $$$$\frac{{x}^{\mathrm{3}} +{y}^{\mathrm{3}} +{z}^{\mathrm{3}} }{{xyz}}\:\geqslant\:\mathrm{2}\left(\frac{{x}}{{y}+{z}}\:+\:\frac{{y}}{{z}+{x}}\:+\:\frac{{z}}{{x}+{y}}\right) \\ $$ Answered by mitica last updated on 22/Jul/21 $${m}_{{h}}…
Question Number 82042 by ajfour last updated on 17/Feb/20 Commented by ajfour last updated on 17/Feb/20 $${Find}\:{ratio}\:{of}\:{side}\:{lengths}\:{of} \\ $$$${outer}\:{equilateral}\:\bigtriangleup{KLM}\:{to} \\ $$$${that}\:{of}\:{inner}\:{equilateral}\:\bigtriangleup{PQR}. \\ $$ Commented by…
Question Number 147576 by qaz last updated on 22/Jul/21 $$\left(\mathrm{1}\right)::\:\:\:\:\:\underset{\mathrm{i}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\underset{\mathrm{j}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\mid\mathrm{i}−\mathrm{j}\mid=? \\ $$$$\left(\mathrm{2}\right)::\:\:\:\:\:\underset{\mathrm{i}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\underset{\mathrm{j}=\mathrm{i}} {\overset{\mathrm{n}} {\sum}}\frac{\mathrm{1}}{\mathrm{j}}=? \\ $$$$\left(\mathrm{3}\right)::\:\:\:\:\:\:\underset{\mathrm{i}=\mathrm{1}} {\overset{\mathrm{n}^{\mathrm{2}} } {\sum}}\left[\sqrt{\mathrm{i}}\right]=?…
Question Number 16506 by Tinkutara last updated on 23/Jun/17 $$\mathrm{A}\:\mathrm{train}\:\mathrm{1}\:\mathrm{moves}\:\mathrm{from}\:\mathrm{east}\:\mathrm{to}\:\mathrm{west} \\ $$$$\left(\mathrm{clockwise}\right)\:\mathrm{and}\:\mathrm{another}\:\mathrm{train}\:\mathrm{2}\:\mathrm{moves} \\ $$$$\mathrm{from}\:\mathrm{west}\:\mathrm{to}\:\mathrm{east}\:\left(\mathrm{anticlockwise}\right)\:\mathrm{on} \\ $$$$\mathrm{the}\:\mathrm{equator}\:\mathrm{with}\:\mathrm{equal}\:\mathrm{speeds}\:\mathrm{relative} \\ $$$$\mathrm{to}\:\mathrm{ground}.\:\mathrm{The}\:\mathrm{ratio}\:\mathrm{of}\:\mathrm{their} \\ $$$$\mathrm{centripetal}\:\mathrm{acceleration}\:\frac{{a}_{\mathrm{1}} }{{a}_{\mathrm{2}} }\:\mathrm{relative}\:\mathrm{to} \\ $$$$\mathrm{centre}\:\mathrm{of}\:\mathrm{earth}\:\mathrm{is} \\…
Question Number 16505 by Tinkutara last updated on 23/Jun/17 $$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{maximum}\:\mathrm{magnitude}\:\mathrm{of} \\ $$$$\mathrm{change}\:\mathrm{in}\:\mathrm{velocity}\:\mathrm{of}\:\mathrm{a}\:\mathrm{motorcycle} \\ $$$$\mathrm{moving}\:\mathrm{with}\:\mathrm{a}\:\mathrm{uniform}\:\mathrm{speed}\:{v}_{\mathrm{0}} \:\mathrm{in}\:\mathrm{a} \\ $$$$\mathrm{circular}\:\mathrm{path}\:\mathrm{of}\:\mathrm{length}\:{l}\:=\:\frac{\pi}{\mathrm{3}}{R}\:\mathrm{and} \\ $$$$\mathrm{radius}\:\mathrm{R}?\:\mathrm{Treat}\:\mathrm{motorcycle}\:\mathrm{as}\:\mathrm{a} \\ $$$$\mathrm{particle} \\ $$$$\left(\mathrm{1}\right)\:\mid\Delta\overset{\rightarrow} {{v}}\mid\:=\:\frac{{v}_{\mathrm{0}} }{\mathrm{2}}…
Question Number 82041 by M±th+et£s last updated on 17/Feb/20 $${show}\:{that} \\ $$$$\pi^{{ie}} +\frac{\mathrm{1}}{\mathrm{2}}=\mathrm{0} \\ $$ Commented by mr W last updated on 17/Feb/20 $${certainly}\:{it}'{s}\:{not}\:{your}\:{fault}\:{that}\:{it}\:{is} \\…