Question Number 159171 by LEKOUMA last updated on 13/Nov/21 $${Prove}\:{by}\:{absurd}\:{that}\:\mathrm{log}\:\mathrm{2}\:{is}\:{the} \\ $$$${number}\:{irrational} \\ $$ Answered by mr W last updated on 14/Nov/21 $${say}\:\mathrm{log}\:\mathrm{2}=\frac{{p}}{{q}}\:{with}\:{p},{q}\in\mathbb{Z} \\ $$$$\mathrm{2}=\mathrm{10}^{\frac{{p}}{{q}}}…
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Question Number 159123 by BHOOPENDRA last updated on 13/Nov/21 Answered by aleks041103 last updated on 14/Nov/21 $${Def}.\:{Linear}\:{transform}\:{L}: \\ $$$$\left.\mathrm{1}\right)\:{L}\left({a}+{b}\right)={L}\left({a}\right)+{L}\left({b}\right) \\ $$$$\left.\mathrm{2}\right){L}\left(\alpha{a}\right)=\alpha{L}\left({a}\right) \\ $$$$ \\ $$$${T}\left(\begin{bmatrix}{{a}}\\{{b}}\\{{c}}\\{{d}}\end{bmatrix}+\begin{bmatrix}{{A}}\\{{B}}\\{{C}}\\{{D}}\end{bmatrix}\right)={T}\left(\begin{bmatrix}{{a}+{A}}\\{{b}+{B}}\\{{c}+{C}}\\{{d}+{D}}\end{bmatrix}\right)=\begin{bmatrix}{{a}+{A}+{b}+{B}}\\{{b}+{B}−{c}−{C}}\\{{a}+{A}+{d}+{D}}\end{bmatrix}=…
Question Number 28006 by sorour87 last updated on 18/Jan/18 $$\int_{\mathrm{0}} ^{\infty} \left(\mathrm{ln}\:{x}\right)^{−\mathrm{3}} {dx} \\ $$ Commented by abdo imad last updated on 20/Jan/18 $${the}\:{ch}\:.{lnx}={t}\:{give}\:\:\:\int_{\mathrm{0}} ^{\infty}…
Question Number 159078 by LEKOUMA last updated on 12/Nov/21 $$\left.\mathrm{1}\right)\:{Prove}\:{by}\:{recurrence}\:{that}\: \\ $$$${for}\:{n}\geqslant\mathrm{28},\:\:\:{n}!\geqslant\mathrm{11}^{{n}} \: \\ $$$$\left.\mathrm{2}\right)\:{On}\:{subtract}\:{the}\:{limit}\:{of}\:{the}\: \\ $$$${suite}\:\left(\frac{{n}!}{\mathrm{10}^{{n}} }\right)\:{when}\:{n}\:{tended}\:{at}\:+\infty \\ $$ Terms of Service Privacy Policy…
Question Number 27997 by ajfour last updated on 18/Jan/18 Commented by mrW2 last updated on 18/Jan/18 $$\omega=\sqrt{\frac{\mathrm{3}{g}}{\mathrm{2}{a}\:\mathrm{cos}\:\alpha}} \\ $$ Commented by ajfour last updated on…
Question Number 27986 by ajfour last updated on 18/Jan/18 $${Prove}\:{that}\:{the}\:{angular}\:{momentum} \\ $$$$\bar {\boldsymbol{{H}}}_{{G}} \:{of}\:{a}\:{rigid}\:{body}\:{about}\:{its}\:{mass} \\ $$$${center}\:{is}\:{given}\:{by}\:: \\ $$$${H}_{{x}} =\bar {{I}}_{{x}} \omega_{{x}} −\bar {{I}}_{{xy}} \omega_{{y}} −\bar…
Question Number 93483 by Rio Michael last updated on 13/May/20 $$\:\mathrm{prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{the}\:\mathrm{normal}\:\mathrm{to}\:\mathrm{the}\:\mathrm{rectangular} \\ $$$$\mathrm{hyperbola}\:{xy}\:=\:{c}^{\mathrm{2}} \:\mathrm{at}\:\mathrm{the}\:\mathrm{point}\:{P}\left({ct},\:{c}/{t}\right)\:\mathrm{is}\:{t}^{\mathrm{3}} {x}\:−{ty}\:=\:{c}\left({t}^{\mathrm{4}} −\mathrm{1}\right). \\ $$$$\mathrm{the}\:\mathrm{normal}\:\mathrm{to}\:{P}\:\:\mathrm{on}\:\mathrm{the}\:\mathrm{hyperbola}\:\mathrm{meets}\:\mathrm{the}\:\mathrm{x}−\mathrm{axis}\:\mathrm{at}\:{Q}\:\mathrm{and}\:\mathrm{the} \\ $$$$\mathrm{tangent}\:\mathrm{to}\:{P}\:\mathrm{meets}\:\mathrm{the}\:\mathrm{yaxis}\:\mathrm{at}\:{R}.\:\mathrm{show}\:\mathrm{that} \\ $$$$\mathrm{the}\:\mathrm{locus}\:\mathrm{of}\:\mathrm{the}\:\mathrm{midpoint}\:\:\mathrm{oc}\:{QR},\:\mathrm{as}\:{P}\:\mathrm{varies}\:\mathrm{is}\:\mathrm{2}{c}^{\mathrm{2}} {xy}\:+\:{y}^{\mathrm{4}} \:=\:{c}^{\mathrm{4}} .…
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Question Number 27946 by das47955@mail.com last updated on 17/Jan/18 $$\boldsymbol{\mathrm{Q}}.\boldsymbol{\mathrm{N}}− \\ $$$$\:\:\:\:\:\:\:\boldsymbol{\mathrm{Find}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{coefficint}}\:\boldsymbol{\mathrm{of}}\:\:\boldsymbol{\mathrm{x}}^{\mathrm{4}} \\ $$$$\boldsymbol{\mathrm{in}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{expansion}}\:\boldsymbol{\mathrm{of}} \\ $$$$\:\:\:\:\:\left(\mathrm{1}+\mathrm{3}\boldsymbol{\mathrm{x}}+\mathrm{10}\boldsymbol{\mathrm{x}}^{\mathrm{2}} \right)\left(\boldsymbol{\mathrm{x}}+\frac{\mathrm{1}}{\boldsymbol{\mathrm{x}}}\right)^{\mathrm{10}} \\ $$$$ \\ $$$$\:\:\:\:\:\:\: \\ $$ Answered by…