Question Number 2441 by Yozzi last updated on 20/Nov/15 $${Prove}\:{or}\:{disprove}\:{that}\:,\:{for}\:{even}\: \\ $$$${positive}\:{n}, \\ $$$$\mathrm{2}×\underset{{k}=\mathrm{1}} {\overset{\frac{{n}}{\mathrm{2}}−\mathrm{1}} {\sum}}\left(−\mathrm{1}\right)^{{k}} \begin{pmatrix}{{n}}\\{{k}}\end{pmatrix}+\left(−\mathrm{1}\right)^{\left({n}/\mathrm{2}\right)} \frac{{n}!}{\left(\left({n}/\mathrm{2}\right)!\right)^{\mathrm{2}} }=−\mathrm{2} \\ $$ Commented by Rasheed Soomro…
Question Number 67977 by behi83417@gmail.com last updated on 02/Sep/19 Commented by behi83417@gmail.com last updated on 02/Sep/19 $$\boldsymbol{\mathrm{AD}}=\boldsymbol{\mathrm{DC}},\angle\boldsymbol{\mathrm{AEB}}=\mathrm{90}^{\bullet} \:\:. \\ $$$$\boldsymbol{\mathrm{find}}:\:\:\:\boldsymbol{\mathrm{ED}}\:\:\boldsymbol{\mathrm{in}}\:\boldsymbol{\mathrm{terms}}\:\boldsymbol{\mathrm{of}}:\boldsymbol{\mathrm{a}},\boldsymbol{\mathrm{b}},\boldsymbol{\mathrm{c}}. \\ $$ Answered by $@ty@m123…
Question Number 67974 by mathmax by abdo last updated on 02/Sep/19 $${let}\:{F}\left({x}\right)\:=\int_{{x}} ^{{x}^{\mathrm{2}} } \:\:\:\frac{{arctan}\left({xt}\right)}{{x}^{\mathrm{2}} \:+{t}^{\mathrm{2}} }{dt}\:\:{calculate}\:{F}\:^{'} \left({x}\right). \\ $$ Commented by mathmax by abdo…
Question Number 133508 by Ahmed1hamouda last updated on 22/Feb/21 Commented by Ahmed1hamouda last updated on 22/Feb/21 $$\boldsymbol{\mathrm{S}}\mathrm{olve}\:\mathrm{the}\:\mathrm{differential}\:\mathrm{equations} \\ $$ Answered by Olaf last updated on…
Question Number 67972 by mathmax by abdo last updated on 02/Sep/19 $${if}\:{F}\left({x}\right)=\int_{{u}\left({x}\right)} ^{{v}\left({x}\right)} {g}\left({x},{t}\right){dt}\:\:\:\:\:{determine}\:{a}\:{expression}\:{for}\:{F}\:^{'} \left({x}\right). \\ $$ Answered by Tanmay chaudhury last updated on 03/Sep/19…
Question Number 67973 by behi83417@gmail.com last updated on 02/Sep/19 Commented by behi83417@gmail.com last updated on 02/Sep/19 $$\mathrm{A}\overset{\bigtriangleup} {\mathrm{B}C},\mathrm{is}\:\mathrm{given}. \\ $$$$\boldsymbol{\mathrm{show}}\:\boldsymbol{\mathrm{that}}: \\ $$$$\mathrm{1}.\:\:\:\:\:\frac{\boldsymbol{\mathrm{BX}}}{\boldsymbol{\mathrm{XC}}}\:×\:\:\frac{\boldsymbol{\mathrm{CY}}}{\boldsymbol{\mathrm{YA}}}\:×\:\frac{\boldsymbol{\mathrm{AZ}}}{\boldsymbol{\mathrm{ZB}}}=\mathrm{1}. \\ $$$$\mathrm{2}.\:\:\:\:\:\frac{\boldsymbol{\mathrm{PX}}}{\boldsymbol{\mathrm{XA}}}\:+\:\:\frac{\boldsymbol{\mathrm{PY}}}{\boldsymbol{\mathrm{YB}}}\:+\:\frac{\boldsymbol{\mathrm{PZ}}}{\boldsymbol{\mathrm{ZC}}}=\mathrm{1}. \\…
Question Number 2435 by Yozzi last updated on 20/Nov/15 $${Prove}\:{that}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Gamma'\left(\mathrm{1}\right)=\int_{\mathrm{0}} ^{\infty} {e}^{−{x}} {lnxdx} \\ $$$${is}\:{a}\:{negative}\:{number}. \\ $$ Commented by prakash jain last updated…
Question Number 2434 by Yozzi last updated on 20/Nov/15 $${Prove}\:{that}\:{for}\:{m}=\mathrm{0},\mathrm{1},\mathrm{2},\mathrm{3},… \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{{x}\rightarrow−{m}} {\mathrm{lim}}\Gamma\left({x}\right)=\infty. \\ $$ Answered by 123456 last updated on 25/Nov/15 $$\Gamma\left({x}+\mathrm{1}\right)={x}\Gamma\left({x}\right) \\ $$$$\mathrm{so}…
Question Number 67969 by behi83417@gmail.com last updated on 02/Sep/19 $$\boldsymbol{\mathrm{Two}}\:\boldsymbol{\mathrm{triangles}}\:\bigtriangleup_{\mathrm{1}} \:\boldsymbol{\mathrm{and}}\:\bigtriangleup_{\mathrm{2}} \:\boldsymbol{\mathrm{are}}\:\boldsymbol{\mathrm{given}},\boldsymbol{\mathrm{such}}\: \\ $$$$\boldsymbol{\mathrm{that}}\:\boldsymbol{\mathrm{length}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{sides}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{triangle}}\:\mathrm{1},\boldsymbol{\mathrm{are}}\: \\ $$$$\boldsymbol{\mathrm{equail}}\:\boldsymbol{\mathrm{to}}\:\boldsymbol{\mathrm{length}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{medians}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{triangle}}\:\mathrm{2}. \\ $$$$\mathrm{1}.\boldsymbol{\mathrm{find}}\:\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{ratio}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{areas}}\:\boldsymbol{\mathrm{of}}\:\:\boldsymbol{\mathrm{triangles}}. \\ $$$$\mathrm{2}.\boldsymbol{\mathrm{given}}\:\boldsymbol{\mathrm{that}}\:\boldsymbol{\mathrm{small}}\:\boldsymbol{\mathrm{side}}\:\boldsymbol{\mathrm{of}}\:\bigtriangleup_{\mathrm{1}} ,\:\boldsymbol{\mathrm{be}}\:\boldsymbol{\mathrm{equail}}\:\boldsymbol{\mathrm{to}}:\sqrt{\mathrm{2}} \\ $$$$\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{one}}\:\boldsymbol{\mathrm{angle}}\:\boldsymbol{\mathrm{be}}:\mathrm{90}^{\bullet} . \\…
Question Number 2432 by prakash jain last updated on 19/Nov/15 $$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{digits}\:\mathrm{of}\:\mathrm{3333}^{\mathrm{4444}} , \\ $$$$\mathrm{Say}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{all}\:\mathrm{digits}\:\mathrm{of}\:\mathrm{3333}^{\mathrm{4444}} \:\mathrm{is}\:\mathrm{A}, \\ $$$$\mathrm{If}\:\mathrm{A}>\mathrm{10}\:\mathrm{then}\:\mathrm{sum}\:\mathrm{all}\:\mathrm{digits}\:\mathrm{of}\:\mathrm{A}. \\ $$$$\mathrm{This}\:\mathrm{process}\:\mathrm{is}\:\mathrm{repeated}\:\mathrm{until}\:\mathrm{a}\:\mathrm{single} \\ $$$$\mathrm{digits}\:\mathrm{sum}\:{x}\:\mathrm{in}\:\mathrm{obtained}. \\ $$$${x}=? \\ $$…