Question Number 1752 by Rasheed Ahmad last updated on 14/Sep/15 $${Prove}\:{that}: \\ $$$$\left(−{x}\right)\left(−{y}\right)={xy} \\ $$ Commented by 123456 last updated on 15/Sep/15 $${ax}+{bx}=\left({a}+{b}\right){x} \\ $$$$−{x}=−\mathrm{1}×{x}…
Question Number 1748 by Rasheed Ahmad last updated on 14/Sep/15 $${Why}\:\:\:{a}^{\mathrm{0}} =\mathrm{1}\:{and}\:\:{a}^{−\mathrm{3}} =\frac{\mathrm{1}}{{a}^{\mathrm{3}} } \\ $$ Answered by 123456 last updated on 14/Sep/15 $$\frac{{a}^{{x}} }{{a}^{{y}}…
Question Number 1738 by Rasheed Sindhi last updated on 08/Sep/15 $${Let}\:\boldsymbol{\mathrm{a}}\:{belongs}\:{to}\:{an}\:{interval}\:\boldsymbol{\mathrm{A}},\boldsymbol{\mathrm{k}}\:{is}\:{aconstant}\:{such}\:{that}\: \\ $$$$\boldsymbol{\mathrm{k}}\in\mathbb{R}^{+} \:{and}\:\boldsymbol{\mathrm{k}}<\boldsymbol{\mathrm{a}}\:. \\ $$$${Find}\:{out}\:\boldsymbol{\mathrm{A}}\:{in}\:{case}: \\ $$$$\:\:\:\:\:\:\:\:\left(\:\boldsymbol{\mathrm{a}}−\boldsymbol{\mathrm{k}}\right)^{\boldsymbol{\mathrm{a}}+\boldsymbol{\mathrm{k}}} \geqslant\left(\boldsymbol{\mathrm{a}}+\boldsymbol{\mathrm{k}}\right)^{\boldsymbol{\mathrm{a}}−\boldsymbol{\mathrm{k}}} \\ $$ Terms of Service Privacy…
Question Number 1729 by Rasheed Ahmad last updated on 04/Sep/15 $${Prove}/{disprove}/{prove}\:{for}\:{an} \\ $$$${interval}\:{as}\:{the}\:{case}\:{may}\:{be}: \\ $$$$\left({x}!\right)^{\frac{\mathrm{1}}{{x}}} \:\overset{?} {<}\:\left\{\left({x}+\mathrm{1}\right)!\right\}^{\frac{\mathrm{1}}{{x}+\mathrm{1}}} \:\:,\:{x}\in\mathbb{N}\:\left[{x}\neq\mathrm{0}\right] \\ $$$$\left({Generalization}\:{of}\:{Q}\:\mathrm{1700}\right) \\ $$ Commented by 123456…
Question Number 1734 by Rasheed Ahmad last updated on 07/Sep/15 $$\boldsymbol{\mathrm{a}}\:\:{belongs}\:{to}\:{an}\:{interval}\:\boldsymbol{\mathrm{A}}\:{and} \\ $$$$\boldsymbol{\mathrm{k}}\in\mathbb{R}^{+} \:\:{is}\:{a}\:{constant}.\:{Determine} \\ $$$$\boldsymbol{\mathrm{A}}\:{in}\:{case} \\ $$$$\:\:\:\:\:\:\:\:\:\:\mid\boldsymbol{\mathrm{a}}−\boldsymbol{\mathrm{k}}\mid^{\mid\:\boldsymbol{\mathrm{a}}+\boldsymbol{\mathrm{k}}\:\mid} \geqslant\:\mid\boldsymbol{\mathrm{a}}+\boldsymbol{\mathrm{k}}\mid^{\mid\boldsymbol{\mathrm{a}}−\boldsymbol{\mathrm{k}}\mid} \: \\ $$$$ \\ $$ Terms…
Question Number 1724 by Hasan Mohamed last updated on 03/Sep/15 $$ \\ $$ Answered by 123456 last updated on 04/Sep/15 $${f}\left({x}\right)={x}!=\Gamma\left({x}+\mathrm{1}\right) \\ $$$${f}'\left({x}\right)={x}!\psi\left({x}+\mathrm{1}\right) \\ $$$$\psi\left({x}\right)=\frac{{d}}{{dx}}\mathrm{ln}\:\Gamma\left({x}\right)…
Question Number 1716 by Rasheed Soomro last updated on 02/Sep/15 $${Determine}\:{interval}\:\boldsymbol{\mathrm{A}}\:{of}\:{real}\:{numbers}\:{for}\:{which} \\ $$$${a}^{{a}+\mathrm{1}} \geqslant\left({a}+\mathrm{1}\right)^{{a}} \:\:\:\:\:\:{whenever}\:{a}\in\boldsymbol{\mathrm{A}} \\ $$ Commented by Rasheed Ahmad last updated on 03/Sep/15…
Question Number 67244 by Learner-123 last updated on 24/Aug/19 $${Which}\:{of}\:{the}\:{series}\:{converge}\:{and}\: \\ $$$${which}\:{diverge}?\:{Check}\:{by}\:{the}\:{limit} \\ $$$${comparison}\:{test}. \\ $$$$\left.\mathrm{1}\right)\:\underset{{n}=\mathrm{2}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}+{n}\:{ln}\left({n}\right)}{{n}^{\mathrm{2}} +\mathrm{5}} \\ $$$$\left.\mathrm{2}\right)\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{{ln}\left({n}\right)}{{n}^{\frac{\mathrm{3}}{\mathrm{2}}} } \\…
Question Number 67208 by mr W last updated on 24/Aug/19 $${Find}\:{the}\:{times}\:{in}\:{a}\:{day}\:{when} \\ $$$${the}\:{hour}'{s},\:{minute}'{s}\:{and}\:{second}'{s} \\ $$$${hand}\:{of}\:{a}\:{clock}\:{occupy}\:{the}\:{same} \\ $$$${angular}\:{position}. \\ $$$$\left[{old}\:{question}\:{reposted}\right] \\ $$ Commented by Kunal12588 last…
Question Number 67167 by behi83417@gmail.com last updated on 23/Aug/19 $$\boldsymbol{\mathrm{solve}}\:\boldsymbol{\mathrm{for}}\:\boldsymbol{\mathrm{real}}\:\:\boldsymbol{\mathrm{x}}\:\boldsymbol{\mathrm{and}}\:\:\boldsymbol{\mathrm{y}}:\left[\mathrm{a},\mathrm{b}\in\mathrm{R}\right] \\ $$$$\boldsymbol{\mathrm{a}}.\begin{cases}{\boldsymbol{\mathrm{x}}^{\mathrm{3}} +\mathrm{1}=\boldsymbol{\mathrm{y}}^{\mathrm{3}} }\\{\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\mathrm{1}=\boldsymbol{\mathrm{y}}^{\mathrm{2}} }\end{cases}\:\:\:\:\:\:\:\: \\ $$$$\boldsymbol{\mathrm{b}}.\begin{cases}{\boldsymbol{\mathrm{x}}^{\mathrm{3}} +\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\mathrm{1}=\boldsymbol{\mathrm{y}}^{\mathrm{3}} }\\{\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\boldsymbol{\mathrm{x}}+\mathrm{1}=\boldsymbol{\mathrm{y}}^{\mathrm{2}} }\end{cases} \\ $$$$\boldsymbol{\mathrm{c}}.\begin{cases}{\boldsymbol{\mathrm{x}}^{\mathrm{3}}…