Question Number 180744 by harckinwunmy last updated on 16/Nov/22 Commented by cyathokoza last updated on 17/Nov/22 $$\boldsymbol{{I}}\:\boldsymbol{{TRY}}\:\:\:\boldsymbol{{THIS}}\:\boldsymbol{{THEN}}\:\boldsymbol{{SUBMIT}}.\:\boldsymbol{{THE}}\:\boldsymbol{{ANSWERS}} \\ $$ Commented by Acem last updated on…
Question Number 180728 by CrispyXYZ last updated on 16/Nov/22 $$\mathrm{Given}\:{x},\:{y},\:{z}\in\mathbb{R}^{+} \:\mathrm{such}\:\mathrm{that}\:{x}^{\mathrm{2}} −\mathrm{3}{xy}+\mathrm{4}{y}^{\mathrm{2}} −{z}=\mathrm{0}. \\ $$$$\mathrm{when}\:\frac{{xy}}{{z}}\:\mathrm{reaches}\:\mathrm{its}\:\mathrm{max}\:\mathrm{value},\:\mathrm{find}\:\mathrm{the}\:\mathrm{max}\:\mathrm{value}\:\mathrm{of} \\ $$$$\frac{\mathrm{2}}{{x}}+\frac{\mathrm{1}}{{y}}−\frac{\mathrm{2}}{{z}}. \\ $$ Answered by mr W last updated…
Question Number 49647 by maxmathsup by imad last updated on 08/Dec/18 $${let}\:{p}\left({x}\right)\:={x}^{\mathrm{2}{n}} \:−{x}^{{n}} \:+\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{determine}\:{the}\:{roots}\:{of}\:{p}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{factorize}\:{inside}\:{C}\left[{x}\right]\:{the}\:{polynom}\:{p}\left({x}\right)\:. \\ $$$$\left.\mathrm{3}\right){solve}\:{p}\left({x}\right)=\mathrm{0}\:\:{and}\:{p}\left({x}\right)\:=\mathrm{2} \\ $$ Commented by maxmathsup…
Question Number 49642 by maxmathsup by imad last updated on 08/Dec/18 $${if}\:\:{a}+{b}\:={s}\:{and}\:{a}^{\mathrm{3}} \:+{b}^{\mathrm{3}} \:={t}\:\:{find}\:{a}^{\mathrm{2}} \:+{b}^{\mathrm{2}} \:\:{and}\:{a}^{\mathrm{4}} \:+{b}^{\mathrm{4}} \:{interms}\:{of}\:{s}\:{and}\:{t}\:. \\ $$ Answered by mr W last…
Question Number 49604 by Tawa1 last updated on 08/Dec/18 $$\mathrm{Show}\:\mathrm{that}:\:\:\:\:\:\:\:\:\:\:\:\frac{\left(\mathrm{a}\:+\:\mathrm{b}\right)^{\mathrm{2}} }{\mathrm{2}}\:\:\leqslant\:\:\mathrm{a}^{\mathrm{2}} \:+\:\mathrm{b}^{\mathrm{2}} \\ $$ Answered by afachri last updated on 08/Dec/18 $$\mathrm{let}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left({a}\:−\:{b}\right)^{\mathrm{2}} \:\:\geqslant\:\:\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:{a}^{\mathrm{2}}…
Question Number 115135 by Algoritm last updated on 23/Sep/20 Commented by Dwaipayan Shikari last updated on 23/Sep/20 $$\frac{\mathrm{1}}{\mathrm{51}.\mathrm{52}…\mathrm{100}}=\frac{\mathrm{1}.\mathrm{2}.\mathrm{3}..}{\mathrm{100}!}=\frac{\mathrm{50}!}{\mathrm{100}!} \\ $$$$\mathrm{1}.\mathrm{3}.\mathrm{5}..\mathrm{9}…=\frac{\mathrm{1}.\mathrm{2}.\mathrm{3}.\mathrm{4}…}{\mathrm{2}^{\mathrm{100}} .\mathrm{50}!} \\ $$$$\frac{\mathrm{100}!}{\mathrm{2}^{\mathrm{50}} .\mathrm{50}!}.\frac{\mathrm{50}!}{\mathrm{100}!}=\mathrm{2}^{\mathrm{x}} \\…
Question Number 115136 by Algoritm last updated on 23/Sep/20 Answered by Olaf last updated on 23/Sep/20 $$\mathrm{sin2}\theta\:=\:\mathrm{2sin}\theta\mathrm{cos}\theta \\ $$$$\mathrm{cos}\theta\:=\:\frac{\mathrm{1}}{\mathrm{2}}.\frac{\mathrm{sin2}\theta}{\mathrm{sin}\theta} \\ $$$$\theta\:=\:\frac{{k}\pi}{\mathrm{2}{n}+\mathrm{1}},\:\mathrm{cos}\left(\frac{{k}\pi}{\mathrm{2}{n}+\mathrm{1}}\right)\:=\:\frac{\mathrm{1}}{\mathrm{2}}.\frac{\mathrm{sin}\left(\frac{\mathrm{2}{k}\pi}{\mathrm{2}{n}+\mathrm{1}}\right)}{\mathrm{sin}\left(\frac{{k}\pi}{\mathrm{2}{n}+\mathrm{1}}\right)} \\ $$$$\underset{{k}=\mathrm{1}} {\overset{{n}} {\prod}}\mathrm{cos}\left(\frac{{k}\pi}{\mathrm{2}{n}+\mathrm{1}}\right)\:=\:\frac{\mathrm{1}}{\mathrm{2}^{{n}}…
Question Number 180653 by Shrinava last updated on 14/Nov/22 Answered by aleks041103 last updated on 17/Nov/22 $${A}=\begin{pmatrix}{{a}}&{{b}}\\{−{b}}&{{a}}\end{pmatrix}\:={a}\begin{pmatrix}{\mathrm{1}}&{\mathrm{0}}\\{\mathrm{0}}&{\mathrm{1}}\end{pmatrix}\:+{b}\begin{pmatrix}{\mathrm{0}}&{\mathrm{1}}\\{−\mathrm{1}}&{\mathrm{0}}\end{pmatrix}\:={aE}+{bS} \\ $$$${AB}=\left({aE}+{bS}\right)\left({cE}+{dS}\right)= \\ $$$$={acE}+{bdS}^{\mathrm{2}} +\left({ad}+{bc}\right){S} \\ $$$${S}^{\mathrm{2}} =\begin{pmatrix}{\mathrm{0}}&{\mathrm{1}}\\{−\mathrm{1}}&{\mathrm{0}}\end{pmatrix}\begin{pmatrix}{\mathrm{0}}&{\mathrm{1}}\\{−\mathrm{1}}&{\mathrm{0}}\end{pmatrix}\:=\:\begin{pmatrix}{−\mathrm{1}}&{\mathrm{0}}\\{\mathrm{0}}&{−\mathrm{1}}\end{pmatrix}\:=−{E}…
Question Number 180652 by Shrinava last updated on 14/Nov/22 Answered by aleks041103 last updated on 17/Nov/22 $${let}\:{detX}={x}\Rightarrow{a}=\mathrm{1} \\ $$$${det}\left({A}^{\mathrm{2}} \right)=\left({detA}\right)^{\mathrm{2}} =\mathrm{1}={det}\left({BC}\right)={detBdetC} \\ $$$$\Rightarrow{bc}=\mathrm{1}\Rightarrow{b}=\frac{\mathrm{1}}{{c}} \\ $$$${B}^{\mathrm{2}}…
Question Number 180641 by mr W last updated on 14/Nov/22 $${if}\:{a}+{b}+{c}+{d}+{e}=\mathrm{8}\:{and} \\ $$$${a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} +{d}^{\mathrm{2}} +{e}^{\mathrm{2}} =\mathrm{16},\:{what}\:{is}\:{the} \\ $$$${maximal}\:{value}\:{of}\:{a}\:? \\ $$ Answered by Emrice…