Question Number 1387 by navajyoti.tamuli.tamuli@gmail. last updated on 27/Jul/15 $${Q}.\:{is}\:{there}\:{any}\:{angle}\:{in}\:{a}\:{circle}? \\ $$ Commented by Rasheed Soomro last updated on 28/Jul/15 $${Two}\:{answers}\:{can}\:{be}\:{given}.{Each}\:{has}\:\:{its}\:\:{own}\:{reasoning}. \\ $$$$\left({i}\right)\:{There}\:{are}\:{infinity}\:{number}\:{of}\:{angles}. \\ $$$${Consider}\:{a}\:{polygon}\:{of}\:{n}\:{angles}\:{inscribed}\:{in}\:{a}\:{circle}:…
Question Number 132459 by mnjuly1970 last updated on 14/Feb/21 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:….\:{nice}\:\:\:{calculus}\:…. \\ $$$$\:\:\:\:\:\:\:\:\:\boldsymbol{\phi}=\int_{\mathrm{0}} ^{\:\infty} \frac{{sin}\left({x}^{\mathrm{2}} \right){ln}\left({x}\right)}{{x}^{\frac{\mathrm{3}}{\mathrm{2}}} }{dx}= \\ $$$$\:\:{solution}: \\ $$$$\boldsymbol{\phi}\overset{{x}^{\mathrm{2}} ={t}} {=}\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\:\infty} \frac{{sin}\left({t}\right){ln}\left(\sqrt{{t}}\:\right)}{{t}^{\frac{\mathrm{3}}{\mathrm{4}}} }\:\frac{{dt}}{{t}^{\frac{\mathrm{1}}{\mathrm{2}}}…
Question Number 1379 by 314159 last updated on 27/Jul/15 $${Find}\:{all}\:{positive}\:{integers}\:{n}\:{such}\:{that}\:\: \\ $$$${n}\leqslant\mathrm{2014}\:{and}\:\mathrm{3}^{{n}−\mathrm{1}} .{n}\:\:{will}\:{be}\:{a}\:{perfect}\:{square}\:{integer}. \\ $$ Commented by 123456 last updated on 26/Jul/15 $${m}\equiv\mathrm{0},\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4}\left(\mathrm{mod}\:\mathrm{5}\right) \\ $$$${n}={m}^{\mathrm{2}}…
Question Number 1378 by Rasheed Soomro last updated on 28/Jul/15 $$\:\:\:\:\:{Four}\:{sides}\:{m}\overline {\boldsymbol{\mathrm{AB}}}\:,\:{m}\overline {\boldsymbol{\mathrm{BC}}}\:,\:{m}\overline {\boldsymbol{\mathrm{CD}}}\:\:{and}\:\:{m}\overline {\boldsymbol{\mathrm{DA}}}\:{of}\:{a}\:\boldsymbol{\mathrm{quadrilateral}}\: \\ $$$$\boldsymbol{\mathrm{ABCD}}\:\:{have}\:{measurement}\:\boldsymbol{{a}}\:,\:\boldsymbol{{b}}\:,\:\boldsymbol{{c}}\:\:{and}\:\boldsymbol{{d}}\:{units}\:{respectively}. \\ $$$$\:\:\:\:\:{Let}\:{the}\:{sum}\:{of}\:{any}\:{adjacent}\:{sides}\:{is}\:{not}\:{equal}\:{to}\:{the}\:{sum}\:{of} \\ $$$${remaining}\:{adjacent}\:{sides}\:\:{and}\:{measurement}\:{of}\:{all}\:{the}\:{sides}\: \\ $$$${is}\:{positive}\:{and}\:{real}. \\ $$$$\:\:\:\:\:\:{What}\:{could}\:{be}\:{the}\:{possible}\:{minimum}\:{and}\:{maximum}\:{values}…
Question Number 66910 by naka3546 last updated on 20/Aug/19 $${P}\:\:=\:\:\underset{{n}=\mathrm{1}} {\overset{\mathrm{999}} {\sum}}\:\frac{\mathrm{1}}{\left(\mathrm{2}{n}−\mathrm{1}\right)\left(\mathrm{2}{n}\right)} \\ $$$${Q}\:\:=\:\:\underset{{n}=\mathrm{1}} {\overset{\mathrm{999}} {\sum}}\:\frac{\mathrm{1}}{\left(\mathrm{999}+{n}\right)\left(\mathrm{1999}−{n}\right)} \\ $$$$\frac{{P}}{{Q}}\:\:=\:\:? \\ $$ Commented by naka3546 last updated…
Question Number 1375 by 123456 last updated on 26/Jul/15 $$\mathrm{lets}\:{f}:\mathbb{R}\rightarrow\mathbb{R}\:\mathrm{and}\:{g}:\mathbb{R}\rightarrow\mathbb{R}\:\mathrm{two}\:\mathrm{continuous}\:\mathrm{and}\:\mathrm{differentiable}\:\mathrm{functions} \\ $$$$\mathrm{suppose}\:\mathrm{that}\:{g}\left(\mathrm{0}\right)=\mathrm{0},\:\mathrm{then}\:\mathrm{compute} \\ $$$${h}\left({x}\right)=\underset{\Delta{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{g}\left({f}\left({x}+\Delta{x}\right)−{f}\left({x}\right)\right)}{\Delta{x}} \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 132444 by liberty last updated on 14/Feb/21 Answered by bemath last updated on 14/Feb/21 Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 66909 by naka3546 last updated on 20/Aug/19 $$\int\:\:\frac{\sqrt{{x}^{\mathrm{2019}} +\mathrm{2019}}\:\:+\:\:\sqrt[{\mathrm{3}}]{{x}^{\mathrm{2018}} +\mathrm{2018}}}{\:\sqrt[{\mathrm{4}}]{{x}^{\mathrm{2017}} +\mathrm{2017}}\:\:+\:\:\sqrt[{\mathrm{5}}]{{x}^{\mathrm{2016}} +\mathrm{2016}}}\:\:{dx}\:\:=\:\:? \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 66906 by Kunal12588 last updated on 20/Aug/19 $${Intergrate}\:{I}=\int\:\frac{{t}^{\mathrm{2}} }{\mathrm{1}+{t}^{\mathrm{4}} }\:{dt} \\ $$ Commented by Prithwish sen last updated on 20/Aug/19 $$\frac{\mathrm{1}}{\mathrm{2i}}\int\frac{\mathrm{1}}{\mathrm{1}−\mathrm{it}^{\mathrm{2}} }\:−\:\frac{\mathrm{1}}{\mathrm{1}+\mathrm{it}^{\mathrm{2}} }\:\mathrm{dt}=\frac{\mathrm{1}}{\mathrm{2}}\left[−\mathrm{sin}^{−\mathrm{1}}…
Question Number 132440 by liberty last updated on 14/Feb/21 $$\mathrm{A}\:\mathrm{vessel}\:\mathrm{containing}\:\mathrm{water}\:\mathrm{has}\:\mathrm{the}\: \\ $$$$\mathrm{shape}\:\mathrm{of}\:\mathrm{and}\:\mathrm{inverted}\:\mathrm{right}\:\mathrm{circular} \\ $$$$\mathrm{cone}\:\mathrm{with}\:\mathrm{base}\:\mathrm{radius}\:\mathrm{2m}\:\mathrm{and}\:\mathrm{height}\:\mathrm{5m} \\ $$$$\mathrm{The}\:\mathrm{water}\:\mathrm{flows}\:\mathrm{from}\:\mathrm{the}\:\mathrm{apex} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{cone}\:\mathrm{at}\:\mathrm{a}\:\mathrm{constant}\:\mathrm{rate}\: \\ $$$$\mathrm{of}\:\mathrm{0}.\mathrm{2}\:\mathrm{m}^{\mathrm{3}} /\mathrm{min}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{rate}\:\mathrm{at} \\ $$$$\mathrm{which}\:\mathrm{the}\:\mathrm{water}\:\mathrm{level}\:\mathrm{is}\:\mathrm{dropping} \\ $$$$\mathrm{when}\:\mathrm{the}\:\mathrm{depth}\:\mathrm{of}\:\mathrm{the}\:\mathrm{water}\:\mathrm{is}…