Question Number 11609 by Nayon last updated on 29/Mar/17 $$ \\ $$$$ \\ $$$${Evaluate}\:\int{x}^{{x}} {dx} \\ $$ Commented by Nayon last updated on 29/Mar/17 $${pls}\:{ans}\:{mrw}\mathrm{1}…
Question Number 11608 by agni5 last updated on 29/Mar/17 $$\mathrm{If}\:\mathrm{f}\left(\mathrm{x}\right)=\mathrm{xtan}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{x}}\right)\:,\:\:\:\:\:\:\:\mathrm{x}\neq\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{0}\:,\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{x}=\mathrm{0} \\ $$$$\mathrm{show}\:\mathrm{that}\:\mathrm{f}\:\mathrm{is}\:\mathrm{countinous}\:\mathrm{but}\:\mathrm{not}\:\mathrm{differentiable} \\ $$$$\mathrm{at}\:\mathrm{x}=\mathrm{0}. \\ $$ Answered by mrW1 last updated on…
Question Number 11607 by Joel576 last updated on 29/Mar/17 $$\mathrm{A}\:\mathrm{parabola}\:\mathrm{with}\:\mathrm{equation}\:{y}\:=\:\frac{{x}^{\mathrm{2}} }{{k}}\:−\:\mathrm{5}\:\mathrm{intersects} \\ $$$$\mathrm{a}\:\mathrm{circle}\:\mathrm{with}\:\mathrm{equation}\:{x}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} \:=\:\mathrm{25}\:\mathrm{at}\:\mathrm{exactly}\:\mathrm{3}\:\mathrm{points}\:{A},\:{B},\:{C} \\ $$$$\mathrm{Determine}\:\mathrm{all}\:\mathrm{such}\:\mathrm{positive}\:\mathrm{integers}\:{k}\:\mathrm{for}\:\mathrm{which} \\ $$$$\mathrm{the}\:\mathrm{area}\:\mathrm{of}\:\Delta{ABC}\:\mathrm{is}\:\mathrm{an}\:\mathrm{integer} \\ $$ Answered by mrW1 last…
Question Number 77143 by Boyka last updated on 03/Jan/20 Commented by mr W last updated on 04/Jan/20 $${Master}\:{sir}: \\ $$$${if}\:{you}\:{don}'{t}\:{understand}\:{the}\:{answer} \\ $$$${i}\:{have}\:{given}\:{in}\:{your}\:{earlier}\:{post}, \\ $$$${you}\:{can}\:{tell}\:{me}\:{what}\:{you}\:{don}'{t} \\…
Question Number 11606 by Joel576 last updated on 29/Mar/17 $$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{smallest}\:\mathrm{positive}\:\mathrm{integer}\:{x}\:\mathrm{for} \\ $$$$\mathrm{which}\:\frac{\mathrm{1}}{\mathrm{32}}\:=\:\frac{{x}}{\mathrm{10}^{{y}} }\:\:\mathrm{for}\:\mathrm{some}\:\mathrm{positive}\:\mathrm{integer}\:{y}\:? \\ $$ Answered by mrW1 last updated on 29/Mar/17 $$\mathrm{32}{x}=\mathrm{10}^{{y}} =\left(\mathrm{2}×\mathrm{5}\right)^{{y}} =\mathrm{2}^{{y}}…
Question Number 142679 by Gbenga last updated on 03/Jun/21 $${how}\:{to}\:{get}\:{exert}\:{value}\:{of}\:{a}\:{lambert}\:{w}\:{function}\:{question}\:{without}\:{wolfram}\:{alpha} \\ $$ Commented by Dwaipayan Shikari last updated on 03/Jun/21 $${W}\left({x}\right)=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(−{n}\right)^{{n}−\mathrm{1}} }{{n}!}{x}^{{n}} \\…
Question Number 142674 by Snail last updated on 03/Jun/21 $${If}\:{f}\left({x}\right)=\frac{{x}−\mathrm{3}}{{x}+\mathrm{1}}\:{and}\:{h}\left({x}\right)={f}\left({f}\left({f}\left({f}\left(..\mathrm{2022}\:{times}\left({f}\left({x}\right)\right)\right)\right)\right)\right) \\ $$$${then}\:\:\:{h}\left(\mathrm{1}\right)\:=? \\ $$ Answered by iloveisrael last updated on 04/Jun/21 $${f}\left({f}\left({x}\right)\right)=\frac{\frac{{x}−\mathrm{3}}{{x}+\mathrm{1}}−\mathrm{3}}{\frac{{x}−\mathrm{3}}{{x}+\mathrm{1}}+\mathrm{1}}\:=\:\frac{{x}−\mathrm{3}−\mathrm{3}{x}−\mathrm{3}}{{x}−\mathrm{3}+{x}+\mathrm{1}} \\ $$$${f}\left({f}\left({x}\right)\right)=\:\frac{−\mathrm{2}{x}−\mathrm{6}}{\mathrm{2}{x}−\mathrm{2}}\:=\:\frac{−{x}−\mathrm{3}}{{x}−\mathrm{1}}={f}^{−\mathrm{1}} \left({x}\right)…
Question Number 11600 by tawa last updated on 28/Mar/17 $$\mathrm{A}\:\mathrm{number}\:\mathrm{is}\:\mathrm{selected}\:\mathrm{at}\:\mathrm{random}\:\mathrm{from}\:\mathrm{the}\:\mathrm{integers}\:\mathrm{1}\:\mathrm{to}\:\mathrm{100}\:\mathrm{exclusive},\: \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{chance}\:\mathrm{of}\:\mathrm{choosing}\:\mathrm{at}\:\mathrm{random}\:\mathrm{a}\:\mathrm{two}\:\mathrm{digit}\:\mathrm{number}\:\mathrm{which}\:\mathrm{is} \\ $$$$\left(\mathrm{a}\right)\:\mathrm{Divisible}\:\mathrm{by}\:\mathrm{4} \\ $$$$\left(\mathrm{b}\right)\:\mathrm{The}\:\mathrm{multiple}\:\mathrm{of}\:\mathrm{5} \\ $$ Answered by mrW1 last updated on 29/Mar/17…
Question Number 11599 by tawa last updated on 28/Mar/17 $$\mathrm{A}\:\mathrm{number}\:\mathrm{is}\:\mathrm{selected}\:\mathrm{at}\:\mathrm{random}\:\mathrm{from}\:\mathrm{the}\:\mathrm{integers}\:\mathrm{10}\:\mathrm{to}\:\mathrm{35}\:\mathrm{exclussive},\:\mathrm{such} \\ $$$$\mathrm{that}\:\mathrm{the}\:\mathrm{chances}\:\mathrm{take}\:\mathrm{the}\:\mathrm{subsets}\:\mathrm{of}\:\mathrm{P},\:\mathrm{Q}\:\mathrm{and}\:\mathrm{R}\:\mathrm{respectively},\:\mathrm{where}\:\mathrm{P}\:\mathrm{is} \\ $$$$\mathrm{even}\:,\:\mathrm{Q}\:\mathrm{is}\:\mathrm{even}\:\mathrm{or}\:\mathrm{prime}\:\mathrm{number},\:\mathrm{and}\:\mathrm{R}\:\mathrm{is}\:\mathrm{odd}\:\mathrm{prime}\:\mathrm{numbers}.\:\mathrm{What}\:\mathrm{is} \\ $$$$\mathrm{the}\:\mathrm{probability}\: \\ $$$$\left(\mathrm{a}\right)\:\mathrm{That}\:\mathrm{exacly}\:\mathrm{one}\:\mathrm{of}\:\mathrm{the}\:\mathrm{events}\:\mathrm{of}\:\mathrm{the}\:\mathrm{subset}\:\mathrm{is}\:\mathrm{choosen}\:? \\ $$$$\left(\mathrm{b}\right)\:\mathrm{Of}\:\mathrm{choosen}\:\mathrm{at}\:\mathrm{most}\:\mathrm{two}\:\mathrm{of}\:\mathrm{the}\:\mathrm{subset}\:\mathrm{of}\:\mathrm{the}\:\mathrm{event}. \\ $$ Answered by mrW1…
Question Number 142668 by mathdanisur last updated on 03/Jun/21 $$\underset{{n}\rightarrow\infty} {{lim}}\left(\frac{{n}+\mathrm{6}}{{n}}\right)^{\frac{\mathrm{6}}{{n}}} =\:? \\ $$ Commented by iloveisrael last updated on 04/Jun/21 $$\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\left(\frac{{n}+\mathrm{6}}{{n}}\right)^{\frac{\mathrm{6}}{{n}}} \:= \\…