Question Number 118712 by 1549442205PVT last updated on 19/Oct/20 $$\mathrm{Prove}\:\mathrm{the}\:\mathrm{following}\:\mathrm{inequalities}: \\ $$$$\left.\mathrm{1}\right)\left(\frac{\mathrm{n}+\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{n}} >\mathrm{n}!\:\mathrm{for}\:\forall\mathrm{n}\in\mathrm{N}^{\ast} ,\mathrm{n}>\mathrm{1} \\ $$$$\left.\mathrm{2}\right)\mid\mathrm{sinnx}\mid\leqslant\mathrm{n}\mid\mathrm{sinx}\mid\:\mathrm{for}\:\forall\mathrm{n}\in\mathrm{N}^{\ast} \\ $$ Answered by Dwaipayan Shikari last updated on…
Question Number 184251 by Shrinava last updated on 04/Jan/23 $$\mathrm{If}\:\:\:\mathrm{x}\:\sqrt{\mathrm{y}}\:=\:\mathrm{2904} \\ $$$$\mathrm{Find}:\:\:\:\mathrm{y}=? \\ $$ Answered by SEKRET last updated on 04/Jan/23 $$\:\:\:\boldsymbol{\mathrm{x}}\centerdot\sqrt{\boldsymbol{\mathrm{y}}}\:=\:\mathrm{2}^{\mathrm{3}} \centerdot\mathrm{3}\centerdot\mathrm{11}^{\mathrm{2}} \\ $$$$\:\:\begin{cases}{\boldsymbol{\mathrm{x}}=\mathrm{2}}\\{\sqrt{\boldsymbol{\mathrm{y}}}\:=\mathrm{2}^{\mathrm{2}}…
Question Number 184242 by Shrinava last updated on 04/Jan/23 Answered by SEKRET last updated on 04/Jan/23 $$\:\boldsymbol{\mathrm{metod}}\:\boldsymbol{\mathrm{Kramer}} \\ $$$$\:\boldsymbol{\mathrm{det}}\left(\boldsymbol{\mathrm{A}}\right)\:=\:\:\begin{vmatrix}{\mathrm{4}}&{−\mathrm{5}}&{\:\:\mathrm{2}}\\{\mathrm{3}}&{−\mathrm{2}}&{\:\:\:\mathrm{7}}\\{\mathrm{3}}&{\mathrm{10}}&{−\mathrm{2}}\end{vmatrix}=\:−\mathrm{327} \\ $$$$\:\boldsymbol{\mathrm{det}}\left(\boldsymbol{\mathrm{A}}_{\mathrm{1}} \right)=\:\begin{vmatrix}{\mathrm{9}}&{−\mathrm{5}}&{\mathrm{2}}\\{\mathrm{8}}&{−\mathrm{2}}&{\mathrm{7}}\\{\mathrm{17}}&{\mathrm{10}}&{−\mathrm{2}}\end{vmatrix}=\:−\mathrm{1041} \\ $$$$\:\:\boldsymbol{\mathrm{det}}\left(\boldsymbol{\mathrm{A}}_{\mathrm{2}} \right)=\:\begin{vmatrix}{\mathrm{4}}&{\mathrm{9}}&{\mathrm{2}}\\{\mathrm{3}}&{\mathrm{8}}&{\mathrm{7}}\\{\mathrm{3}}&{\mathrm{17}}&{−\mathrm{2}}\end{vmatrix}=\:−\mathrm{243}…
Question Number 53144 by mr W last updated on 18/Jan/19 $${Find}\:{all}\:{integers}\:{x}\:{and}\:{y}\:{such}\:{that} \\ $$$$\frac{{xy}}{{x}+{y}}\:{is}\:{also}\:{integer}. \\ $$ Commented by mr W last updated on 19/Jan/19 $$\boldsymbol{{x}}=\left(\boldsymbol{{i}}+\mathrm{1}\right)\boldsymbol{{j}} \\…
Question Number 118651 by Sherjon last updated on 18/Oct/20 Commented by Sherjon last updated on 18/Oct/20 $${x}_{\mathrm{1}} +{x}_{\mathrm{2}} +…=?\:\:\:{Why} \\ $$ Commented by bramlexs22 last…
Question Number 53108 by Tawa1 last updated on 17/Jan/19 $$\:\:\sqrt[{\mathrm{3}}]{\mathrm{x}\:+\:\mathrm{2}}\:\:−\:\:\sqrt[{\mathrm{3}}]{\mathrm{x}\:−\:\mathrm{3}}\:\:\:>\:\:\frac{\mathrm{1}}{\mathrm{2}} \\ $$ Answered by kaivan.ahmadi last updated on 17/Jan/19 $$\mathrm{t}^{\mathrm{3}} =\mathrm{x}−\mathrm{3}\Rightarrow\mathrm{t}^{\mathrm{3}} +\mathrm{5}=\mathrm{x}+\mathrm{2}\Rightarrow \\ $$$$\sqrt[{\mathrm{3}}]{\mathrm{t}^{\mathrm{3}} +\mathrm{5}}\rangle\mathrm{t}+\frac{\mathrm{1}}{\mathrm{2}}\Rightarrow\mathrm{power}\:\mathrm{3}…
Question Number 118636 by Algoritm last updated on 18/Oct/20 Commented by TANMAY PANACEA last updated on 18/Oct/20 $${what}\:{is}\:{the}\:{question} \\ $$ Commented by Algoritm last updated…
Question Number 118634 by ajfour last updated on 18/Oct/20 $${Prove}\:{that}\:{the}\:{equation}\:{of}\:{the}\:{circle} \\ $$$${passing}\:{through}\:{the}\:{points}\:{of} \\ $$$${intersection}\:{of}\:{these}\:{two}\:{curves}: \\ $$$$\:\:{y}=\mathrm{1}+\frac{{c}}{{x}}\:;\:\:{y}={x}^{\mathrm{2}} \:\:\:\:\:\left({c}\:<\:\frac{\mathrm{2}}{\mathrm{3}\sqrt{\mathrm{3}}}\:\right)\: \\ $$$${is}\:\:\:\left({x}−\frac{{c}}{\mathrm{2}}\right)^{\mathrm{2}} +\left({y}−\mathrm{1}\right)^{\mathrm{2}} =\mathrm{1}+\frac{{c}^{\mathrm{2}} }{\mathrm{4}}\:\:. \\ $$ Commented…
Question Number 184160 by cortano1 last updated on 03/Jan/23 $$\:\:{Given}\:\begin{cases}{{a}_{\mathrm{0}} =\mathrm{1}}\\{{a}_{{n}+\mathrm{1}} =\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{3}{a}_{{n}} +\sqrt{\mathrm{5}{a}_{{n}} ^{\mathrm{2}} −\mathrm{4}}\:\right)}\end{cases} \\ $$$$\:\forall{n}\geqslant\mathrm{0}\:,\:{n}\in{I}\: \\ $$$$\:\:{find}\:{a}_{{n}} . \\ $$ Commented by mr…
Question Number 53051 by Abror last updated on 16/Jan/19 $$\int\mathrm{sin}\:\left(\mathrm{2}{x}\right)\mathrm{cos}\:{xd}\left({x}\right)= \\ $$ Answered by tanmay.chaudhury50@gmail.com last updated on 16/Jan/19 $$\frac{\mathrm{1}}{\mathrm{2}}\int\mathrm{2}{sin}\mathrm{2}{xcosxdx} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\int\left({sin}\mathrm{3}{x}+{sinx}\right){dx} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left[\frac{−{cos}\mathrm{3}{x}}{\mathrm{3}}+\frac{−{cosx}}{}\right]+{c} \\…