Question Number 209695 by hardmath last updated on 18/Jul/24 $$\mathrm{If}: \\ $$$$\left(\mathrm{a}\:+\:\mathrm{b}\right)\centerdot\sqrt{\mathrm{2}}\:=\:\mathrm{7}\centerdot\left(\mathrm{a}−\mathrm{b}−\mathrm{4}\right) \\ $$$$\mathrm{Find}: \\ $$$$\left(\mathrm{2a}\:+\:\mathrm{b}\right)\:=\:? \\ $$ Answered by mr W last updated on…
Question Number 209633 by Abdullahrussell last updated on 17/Jul/24 Answered by som(math1967) last updated on 17/Jul/24 $$\left({x}+\mathrm{1}\right)\left(\mathrm{3}{x}+\mathrm{2}\right)\left(\mathrm{6}{x}+\mathrm{5}\right)^{\mathrm{2}} =\mathrm{6} \\ $$$$\Rightarrow\left(\mathrm{3}{x}^{\mathrm{2}} +\mathrm{5}{x}+\mathrm{2}\right)\left(\mathrm{36}{x}^{\mathrm{2}} +\mathrm{60}{x}+\mathrm{25}\right)=\mathrm{6} \\ $$$$\Rightarrow\left({a}+\mathrm{2}\right)\left(\mathrm{12}{a}+\mathrm{25}\right)=\mathrm{6} \\…
Question Number 209580 by hardmath last updated on 15/Jul/24 $$\mathrm{If}\:\:\:\mathrm{a}_{\boldsymbol{\mathrm{n}}} >\mathrm{0}\:\:\:\mathrm{and}\:\:\:\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\:\mathrm{a}_{\boldsymbol{\mathrm{n}}} \:=\:\mathrm{0} \\ $$$$\mathrm{Find}:\:\:\:\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{1}}{\mathrm{n}}\:\underset{\boldsymbol{\mathrm{k}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{n}}} {\sum}}\:\mathrm{ln}\:\left(\frac{\mathrm{k}}{\mathrm{n}}\:+\:\mathrm{a}_{\boldsymbol{\mathrm{n}}} \right)\:=\:? \\ $$ Answered by mr W…
Question Number 209531 by hardmath last updated on 13/Jul/24 $$\frac{\mathrm{y}^{''} }{\mathrm{y}}\:\:=\:\:\mathrm{4x}^{\mathrm{2}} \:\:+\:\:\mathrm{2} \\ $$ Commented by hardmath last updated on 13/Jul/24 $$ \\ $$Solve the…
Question Number 209542 by Spillover last updated on 13/Jul/24 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int\frac{{dx}}{{x}^{\mathrm{15}} −{x}^{\mathrm{11}} } \\ $$$$ \\ $$$$ \\ $$ Answered by Frix last updated…
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Question Number 209514 by depressiveshrek last updated on 12/Jul/24 $$\mathrm{If}\:\Sigma\:{a}_{{n}} \:\mathrm{is}\:\mathrm{absolutely}\:\mathrm{convergent},\:\mathrm{prove}\:\mathrm{that} \\ $$$$\Sigma\:\frac{{a}_{{n}} }{{n}}\:\mathrm{is}\:\mathrm{also}\:\mathrm{absolutely}\:\mathrm{convergent}. \\ $$ Answered by MM42 last updated on 13/Jul/24 $$\frac{\mid{a}_{{n}} \mid}{{n}}\:<\mid{a}_{{n}}…
Question Number 209436 by hardmath last updated on 10/Jul/24 $$\begin{cases}{\mathrm{x}\:+\:\mathrm{y}\:+\:\mathrm{z}\:=\:\mathrm{1}}\\{\mathrm{42x}\:+\:\mathrm{44y}\:+\:\mathrm{30z}\:=\:\mathrm{42}}\end{cases} \\ $$$$\left(\mathrm{x},\mathrm{y},\mathrm{z}\right)=\left(\mathrm{1},\mathrm{0},\mathrm{0}\right)\:\mathrm{yes},\:\mathrm{but}\:\mathrm{solution}… \\ $$ Commented by mr W last updated on 10/Jul/24 $${you}\:{mean}\:{integer}\:{solutions}. \\ $$…
Question Number 209456 by peter frank last updated on 10/Jul/24 Answered by Ghisom last updated on 10/Jul/24 $${c}=\mathrm{cos}\:\alpha\:\:\:\:\:{s}=\mathrm{sin}\:\alpha\:\:\:\:\:{t}=\mathrm{tan}\:\alpha\:=\frac{{s}}{{c}} \\ $$$${c}=\sqrt{\mathrm{1}−{s}^{\mathrm{2}} }=\frac{\mathrm{1}}{\:\sqrt{{t}^{\mathrm{2}} +\mathrm{1}}} \\ $$$${s}=\sqrt{\mathrm{1}−{c}^{\mathrm{2}} }=\frac{{t}}{\:\sqrt{{t}^{\mathrm{2}}…
Question Number 209455 by peter frank last updated on 10/Jul/24 Answered by Spillover last updated on 10/Jul/24 Commented by peter frank last updated on 11/Jul/24…