Question Number 152879 by SANOGO last updated on 02/Sep/21 $$\underset{{x}\rightarrow+{oo}} {\mathrm{lim}}\frac{\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}{k}^{\alpha} \:\:\:}{\left({n}+\mathrm{1}\right)^{\alpha} \:\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\left({n}\alpha+\mathrm{1}\right)}=\frac{\mathrm{1}}{\mathrm{6}{o}}\:\:.\alpha=? \\ $$ Commented by talminator2856791 last updated on…
Question Number 87340 by john santu last updated on 04/Apr/20 $$\int\:\frac{\mathrm{cos}\:\mathrm{x}}{\left(\mathrm{5}+\mathrm{4cos}\:\mathrm{x}\right)^{\mathrm{2}} }\:\mathrm{dx}\:= \\ $$ Commented by john santu last updated on 04/Apr/20 $$\mathrm{dear}\:\mathrm{mr}\:\mathrm{Mjs}.\:\mathrm{whether}\:\mathrm{methods}\: \\ $$$$\mathrm{besides}\:\mathrm{weierstrass}\:\mathrm{substitution}?…
Question Number 21802 by Tinkutara last updated on 04/Oct/17 $$\mathrm{Five}\:\mathrm{balls}\:\mathrm{are}\:\mathrm{to}\:\mathrm{be}\:\mathrm{placed}\:\mathrm{in}\:\mathrm{three} \\ $$$$\mathrm{boxes}.\:\mathrm{Each}\:\mathrm{can}\:\mathrm{hold}\:\mathrm{all}\:\mathrm{the}\:\mathrm{five}\:\mathrm{balls}. \\ $$$$\mathrm{In}\:\mathrm{how}\:\mathrm{many}\:\mathrm{different}\:\mathrm{ways}\:\mathrm{can}\:\mathrm{we} \\ $$$$\mathrm{place}\:\mathrm{the}\:\mathrm{balls}\:\mathrm{so}\:\mathrm{that}\:\mathrm{no}\:\mathrm{box}\:\mathrm{remains} \\ $$$$\mathrm{empty},\:\mathrm{if}\:\mathrm{balls}\:\mathrm{are}\:\mathrm{different}\:\mathrm{but}\:\mathrm{boxes} \\ $$$$\mathrm{are}\:\mathrm{identical}? \\ $$ Commented by mrW1…
Question Number 21801 by Tinkutara last updated on 04/Oct/17 $$\mathrm{There}\:\mathrm{are}\:{n}\:\mathrm{straight}\:\mathrm{lines}\:\mathrm{in}\:\mathrm{a}\:\mathrm{plane},\:\mathrm{no} \\ $$$$\mathrm{two}\:\mathrm{of}\:\mathrm{which}\:\mathrm{are}\:\mathrm{parallel}\:\mathrm{and}\:\mathrm{no}\:\mathrm{three} \\ $$$$\mathrm{pass}\:\mathrm{through}\:\mathrm{the}\:\mathrm{same}\:\mathrm{point}.\:\mathrm{Their} \\ $$$$\mathrm{point}\:\mathrm{of}\:\mathrm{intersection}\:\mathrm{are}\:\mathrm{joined}.\:\mathrm{Then} \\ $$$$\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{fresh}\:\mathrm{lines}\:\mathrm{thus}\:\mathrm{obtained} \\ $$$$\mathrm{is} \\ $$ Terms of Service…
Question Number 152874 by mnjuly1970 last updated on 02/Sep/21 $$ \\ $$$$\:\:\:{solve}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{I}\::=\:\int_{\mathrm{0}} ^{\:\infty} \left(\frac{\:{tanh}\:\left({x}\right)\:}{{x}}\:\right)^{\:\mathrm{2}} {dx}\:=\:? \\ $$$$\:\:{m}.{n}. \\ $$ Answered by Ar Brandon…
Question Number 21800 by Tinkutara last updated on 04/Oct/17 $$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{integers}\:\mathrm{which}\:\mathrm{lie} \\ $$$$\mathrm{between}\:\mathrm{1}\:\mathrm{and}\:\mathrm{10}^{\mathrm{6}} \:\mathrm{and}\:\mathrm{which}\:\mathrm{have}\:\mathrm{sum} \\ $$$$\mathrm{of}\:\mathrm{digits}\:\mathrm{equal}\:\mathrm{to}\:\mathrm{12}\:\mathrm{is} \\ $$ Commented by mrW1 last updated on 27/Dec/17 $$\mathrm{For}\:\mathrm{example}\:\mathrm{to}\:\mathrm{find}\:\mathrm{such}\:\mathrm{numbers}…
Question Number 21799 by Tinkutara last updated on 04/Oct/17 $$\mathrm{There}\:\mathrm{are}\:{n}\:\mathrm{white}\:\mathrm{and}\:{n}\:\mathrm{red}\:\mathrm{balls} \\ $$$$\mathrm{marked}\:\mathrm{1},\:\mathrm{2},\:\mathrm{3},\:….{n}.\:\mathrm{The}\:\mathrm{number}\:\mathrm{of} \\ $$$$\mathrm{ways}\:\mathrm{we}\:\mathrm{can}\:\mathrm{arrange}\:\mathrm{these}\:\mathrm{balls}\:\mathrm{in}\:\mathrm{a} \\ $$$$\mathrm{row}\:\mathrm{so}\:\mathrm{that}\:\mathrm{neighbouring}\:\mathrm{balls}\:\mathrm{are}\:\mathrm{of} \\ $$$$\mathrm{different}\:\mathrm{colours}\:\mathrm{is} \\ $$ Commented by mrW1 last updated…
Question Number 87335 by naka3546 last updated on 04/Apr/20 Answered by TANMAY PANACEA. last updated on 04/Apr/20 $$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{{sin}^{{a}} {x}}{{sin}^{{a}} {x}+{cos}^{{a}} {x}}{dx}=\frac{{I}\pi}{\mathrm{4}} \\ $$$${using}\:\int_{\mathrm{0}}…
Question Number 21795 by hi147 last updated on 04/Oct/17 $${help} \\ $$$${x}\in{N} \\ $$$${determine}\:{x}\:{where}\:\mathrm{7}\:{divise}\:\mathrm{2}^{{x}} +\mathrm{3}^{{x}} \\ $$$$ \\ $$ Commented by Rasheed.Sindhi last updated on…
Question Number 21793 by Tinkutara last updated on 04/Oct/17 $$\mathrm{A}\:\mathrm{plank}\:\mathrm{of}\:\mathrm{mass}\:\mathrm{10}\:\mathrm{kg}\:\mathrm{rests}\:\mathrm{on}\:\mathrm{a}\:\mathrm{smooth} \\ $$$$\mathrm{horizontal}\:\mathrm{surface}.\:\mathrm{Two}\:\mathrm{blocks}\:\mathrm{A}\:\mathrm{and} \\ $$$$\mathrm{B}\:\mathrm{of}\:\mathrm{masses}\:\mathrm{m}_{\mathrm{A}} \:=\:\mathrm{2}\:\mathrm{kg}\:\mathrm{and}\:\mathrm{m}_{\mathrm{B}} \:=\:\mathrm{1}\:\mathrm{kg} \\ $$$$\mathrm{lies}\:\mathrm{at}\:\mathrm{a}\:\mathrm{distance}\:\mathrm{of}\:\mathrm{3}\:\mathrm{m}\:\mathrm{on}\:\mathrm{the}\:\mathrm{plank}. \\ $$$$\mathrm{The}\:\mathrm{friction}\:\mathrm{coefficient}\:\mathrm{between}\:\mathrm{the} \\ $$$$\mathrm{blocks}\:\mathrm{and}\:\mathrm{plank}\:\mathrm{are}\:\mu_{\mathrm{A}} \:=\:\mathrm{0}.\mathrm{3}\:\mathrm{and}\:\mu_{\mathrm{B}} \:= \\…