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Question Number 154013 by mathdanisur last updated on 13/Sep/21 $$\mathrm{Prove}\:\mathrm{without}\:\mathrm{any}\:\mathrm{software} \\ $$$$\underset{\:\frac{\mathrm{1}}{\mathrm{2}}} {\overset{\:\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}} {\int}}\:\frac{\mathrm{1}}{\mathrm{x}}\:\mathrm{log}\left(\mathrm{1}+\mathrm{2x}^{\mathrm{2}} +\mathrm{x}^{\mathrm{4}} \right)\mathrm{dx}\:<\:\sqrt{\mathrm{7}}\:-\:\sqrt{\mathrm{5}} \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 154012 by mathdanisur last updated on 13/Sep/21 $$\mathrm{Determine}\:\mathrm{all}\:\mathrm{the}\:\mathrm{perfect}\:\mathrm{squares}\:\mathrm{on} \\ $$$$\mathrm{form}\:\:\boldsymbol{\mathrm{p}}^{\boldsymbol{\mathrm{n}}} \:+\:\mathrm{1}\:\:\mathrm{where}\:\:\boldsymbol{\mathrm{p}}\:\:\mathrm{is}\:\mathrm{a}\:\mathrm{prime}\:\mathrm{number} \\ $$$$\mathrm{and}\:\:\boldsymbol{\mathrm{n}}\:\:\mathrm{a}\:\mathrm{positive}\:\mathrm{integer}. \\ $$ Commented by Rasheed.Sindhi last updated on 13/Sep/21 $$\mathrm{For}\:\mathrm{some}\:\mathrm{cases}…
Question Number 88479 by Ar Brandon last updated on 10/Apr/20 $${Consider}\:{the}\:{transformation}\:\boldsymbol{{f}}\:{of}\:{the}\:{plane}\:{with}\:{all}\:{points} \\ $$$$\boldsymbol{{M}}\:{wity}\:{affix}\:\boldsymbol{{z}}\:{mapped}\:{to}\:{the}\:{point}\:\boldsymbol{{M}}\:'\:{with}\:{affix}\:\boldsymbol{{z}}\:' \\ $$$${such}\:{that}\:\boldsymbol{{z}}\:'=−\left(\sqrt{\mathrm{3}}+{i}\right){z}−\mathrm{1}+{i}\left(\mathrm{1}+\sqrt{\mathrm{3}}\right) \\ $$$$\left.\mathrm{1}\right)\:{Given}\:\boldsymbol{{M}}_{\mathrm{0}} \:{the}\:{point}\:\boldsymbol{{z}}_{\mathrm{0}} =\frac{\sqrt{\mathrm{3}}}{\mathrm{4}}+\frac{\mathrm{3}}{\mathrm{4}}{i} \\ $$$${calculate}\:\boldsymbol{{AM}}_{\mathrm{0}} \:{and}\:{deduce}\:{the}\:{angle}\:{in}\:{radians} \\ $$$$\left({Taking}\:\boldsymbol{{A}}\:{as}\:{the}\:{center}\:{of}\:{the}\:{transformation}\right) \\…
Question Number 154015 by mathdanisur last updated on 13/Sep/21 $$\mathrm{If}\:\:\mathrm{x};\mathrm{y};\mathrm{z}>\mathrm{0}\:\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\frac{\mathrm{x}}{\mathrm{x}^{\mathrm{2}} +\mathrm{yz}}\:+\:\frac{\mathrm{y}}{\mathrm{y}^{\mathrm{2}} +\mathrm{zx}}\:+\:\frac{\mathrm{z}}{\mathrm{z}^{\mathrm{2}} +\mathrm{xy}}\:\leqslant\:\frac{\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} +\mathrm{z}^{\mathrm{2}} }{\mathrm{2xyz}} \\ $$ Terms of Service Privacy Policy…
Question Number 22941 by tawa tawa last updated on 24/Oct/17 Commented by tawa tawa last updated on 24/Oct/17 $$\mathrm{please}\:\mathrm{help}.\: \\ $$ Commented by ajfour last…
Question Number 22940 by selestian last updated on 24/Oct/17 Commented by ajfour last updated on 24/Oct/17 $${T}_{{r}} =\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{2}^{{r}−\mathrm{1}} }{\mathrm{1}+\mathrm{2}^{\mathrm{2}{r}−\mathrm{1}} }\right) \\ $$$$\Rightarrow\:\:\mathrm{tan}\:{T}_{{r}} =\frac{\mathrm{2}^{{r}} −\mathrm{2}^{{r}−\mathrm{1}}…
Question Number 22939 by selestian last updated on 24/Oct/17 Answered by ajfour last updated on 24/Oct/17 $$\left({B}\right)\:\mathrm{1}/\mathrm{8} \\ $$$$\frac{{a}}{\mathrm{1}−{r}}=\mathrm{4}\:\:\:\:;\:\:\frac{{a}^{\mathrm{3}} }{\mathrm{1}−{r}^{\mathrm{3}} }=\frac{\mathrm{64}}{\mathrm{7}} \\ $$$$\Rightarrow\:\:\frac{\mathrm{7}{a}^{\mathrm{3}} }{\mathrm{1}−{r}^{\mathrm{3}} }=\frac{{a}^{\mathrm{3}}…