Question Number 119831 by talminator2856791 last updated on 27/Oct/20 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\mathrm{evaluate}:\:\:\:{I}\:\:=\:\int_{\mathrm{1}} ^{\:\infty} \left(\frac{\mathrm{1}}{{x}}\right)^{{x}} {dx} \\ $$$$\: \\ $$$$\: \\ $$…
Question Number 54278 by ajfour last updated on 01/Feb/19 $${If}\:{f}\:{be}\:{n}\rightarrow{n}\:\:\:\left({n}\in\mathbb{N}\right)\:{and}\:{is}\:{increasing}, \\ $$$${f}\left({f}\left({n}\right)\right)=\mathrm{3}{n};\:{find}\:{f}\left(\mathrm{2}\right). \\ $$$${options}:\:\:\mathrm{1},\:\mathrm{2},\:\mathrm{3},\mathrm{4}\:. \\ $$ Commented by mr W last updated on 02/Feb/19 $${if}\:{f}\left({f}\left({x}\right)\right)=\mathrm{3}{x}…
Question Number 119807 by Ar Brandon last updated on 27/Oct/20 $$\mathrm{Let}\:{f}\:\mathrm{be}\:\mathrm{a}\:\mathrm{real}-\mathrm{valued}\:\mathrm{function}\:\mathrm{defined}\:\mathrm{on}\:\mathrm{the}\:\mathrm{inte}- \\ $$$$\mathrm{rval}\:\left[−\mathrm{1},\:\mathrm{1}\right].\:\mathrm{If}\:\mathrm{the}\:\mathrm{area}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equilateral}\:\mathrm{triangle}\:\mathrm{with} \\ $$$$\left(\mathrm{0},\:\mathrm{0}\right)\:\mathrm{and}\:\left(\mathrm{x},\:{f}\left(\mathrm{x}\right)\right)\:\mathrm{as}\:\mathrm{two}\:\mathrm{vertices}\:\mathrm{is}\:\sqrt{\mathrm{3}}/\mathrm{4},\:\mathrm{then}\:{f}\left(\mathrm{x}\right) \\ $$$$\mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$$$\left(\mathrm{A}\right)\:\sqrt{\mathrm{1}−\mathrm{x}^{\mathrm{2}} }\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{B}\right)\:\sqrt{\mathrm{1}+\mathrm{x}^{\mathrm{2}} } \\ $$$$\left(\mathrm{C}\right)\:−\sqrt{\mathrm{1}−\mathrm{x}^{\mathrm{2}} }\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{D}\right)\:−\sqrt{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }…
Question Number 54271 by naveen200601 last updated on 01/Feb/19 Commented by Meritguide1234 last updated on 02/Feb/19 $$\mathrm{post}\:\mathrm{many}\:\mathrm{problem}\:\mathrm{on}\:\mathrm{this}\:\mathrm{type} \\ $$ Answered by tanmay.chaudhury50@gmail.com last updated on…
Question Number 185337 by Shrinava last updated on 20/Jan/23 Answered by qaz last updated on 20/Jan/23 $$\underset{{k}=\mathrm{2}} {\overset{{n}} {\sum}}\frac{{k}+\sqrt{{k}}}{{k}}\sim\int_{\mathrm{2}} ^{{n}} \frac{{x}+\sqrt{{x}}}{{x}}{dx}\sim{n} \\ $$$$\underset{{k}=\mathrm{2}} {\overset{{n}} {\sum}}\frac{{k}+\sqrt[{\mathrm{3}}]{{k}}}{{k}}\sim\int_{\mathrm{2}}…
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Question Number 119801 by Ar Brandon last updated on 27/Oct/20 $$\mathrm{If}\:{f}:\mathbb{R}\rightarrow\mathbb{R}\:\mathrm{is}\:\mathrm{a}\:\mathrm{function}\:\mathrm{such}\:\mathrm{that}\:{f}\left(\mathrm{0}\right)=\mathrm{1}\:\mathrm{and}\:{f}\left(\mathrm{x}+{f}\left(\mathrm{y}\right)\right)= \\ $$$${f}\left(\mathrm{x}\right)+\mathrm{y}\:\mathrm{for}\:\mathrm{all}\:\mathrm{x},\:\mathrm{y}\in\mathbb{R},\:\mathrm{then} \\ $$$$\left(\mathrm{A}\right)\:\mathrm{1}\:\mathrm{is}\:\mathrm{a}\:\mathrm{period}\:\mathrm{of}\:{f} \\ $$$$\left(\mathrm{B}\right)\:{f}\left({n}\right)=\mathrm{1}\:\mathrm{for}\:\mathrm{all}\:\mathrm{integers}\:{n} \\ $$$$\left(\mathrm{C}\right)\:{f}\left({n}\right)={n}\:\mathrm{for}\:\mathrm{all}\:\mathrm{integers}\:{n} \\ $$$$\left(\mathrm{D}\right)\:{f}\left(−\mathrm{1}\right)=\mathrm{0} \\ $$ Terms of…
Question Number 119800 by Ar Brandon last updated on 27/Oct/20 $$\mathrm{Examples}\:\mathrm{of}\:\mathrm{functions}\:\mathrm{such}\:\mathrm{that} \\ $$$${f}\left(\mathrm{x}+\mathrm{y}\right)={f}\left(\mathrm{x}\right)+{f}\left(\mathrm{y}\right)\:\mathrm{for}\:\mathrm{all}\:\mathrm{x},\mathrm{y}\in\mathbb{R} \\ $$ Commented by mr W last updated on 27/Oct/20 $${f}\left({x}\right)={ax} \\…
Question Number 119797 by Ar Brandon last updated on 27/Oct/20 $$\mathrm{Q1} \\ $$$$\mathrm{Let}\:{M}_{\mathrm{2}} \:\mathrm{be}\:\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{square}\:\mathrm{matrices}\:\mathrm{of}\:\mathrm{order}\:\mathrm{2}\:\mathrm{over} \\ $$$$\mathrm{the}\:\mathrm{real}\:\mathrm{number}\:\mathrm{system}\:\mathrm{and} \\ $$$$\:\:\:\:\:\mathcal{R}=\left\{\left({A},{B}\right)\in{M}_{\mathrm{2}} ×{M}_{\mathrm{2}} \mid{A}={P}^{\:\mathrm{T}} {BP}\:\:\mathrm{for}\:\mathrm{some}\right. \\ $$$$\left.\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{non}-\mathrm{singular}\:{P}\:\in{M}\right\} \\ $$$$\mathrm{Then}\:\mathcal{R}\:\mathrm{is}…
Question Number 119795 by bobhans last updated on 27/Oct/20 $${Solve}\:{in}\:{real}\:{numbers}\:{the}\:{system}\:{of} \\ $$$${equations}\:\begin{cases}{\left(\mathrm{3}{x}+{y}\right)\left({x}+\mathrm{3}{y}\right)\sqrt{{xy}}\:=\mathrm{14}}\\{\left({x}+{y}\right)\left({x}^{\mathrm{2}} +\mathrm{14}{xy}+{y}^{\mathrm{2}} \right)=\:\mathrm{36}}\end{cases}\: \\ $$ Answered by 1549442205PVT last updated on 27/Oct/20 $$\mathrm{Put}\:\mathrm{x}=\mathrm{ky}\:\mathrm{we}\:\mathrm{get} \\…