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AllQuestion and Answers: Page 25

Question Number 222141 Answers: 2 Comments: 0

x^2 +((x/(2x−1)))^2 =12

$${x}^{\mathrm{2}} +\left(\frac{{x}}{\mathrm{2}{x}−\mathrm{1}}\right)^{\mathrm{2}} =\mathrm{12} \\ $$

Question Number 222142 Answers: 2 Comments: 0

a=3(√2) ,b=(1/(5^(1/6) (√6))) and x,yεR such that 3x +2y=log _a (18)^(5/4) 2x−y=log _b ((√(1080))) then find the value of 4x+5y

$${a}=\mathrm{3}\sqrt{\mathrm{2}}\:,{b}=\frac{\mathrm{1}}{\mathrm{5}^{\frac{\mathrm{1}}{\mathrm{6}}} \sqrt{\mathrm{6}}}\:{and}\:{x},{y}\epsilon\mathbb{R}\:{such}\:{that} \\ $$$$\mathrm{3}{x}\:+\mathrm{2}{y}=\mathrm{log}\:_{{a}} \left(\mathrm{18}\right)^{\frac{\mathrm{5}}{\mathrm{4}}} \\ $$$$\mathrm{2}{x}−{y}=\mathrm{log}\:_{{b}} \left(\sqrt{\mathrm{1080}}\right) \\ $$$${then}\:{find}\:{the}\:{value}\:{of}\:\: \\ $$$$\mathrm{4}{x}+\mathrm{5}{y} \\ $$

Question Number 222127 Answers: 0 Comments: 0

∫_0 ^( ∞) ((ln(tan(tan^(−1) (e^((1/π) tan^(−1) u) ))) )/(u^2 + 2πu + 2π^2 )) du

$$ \\ $$$$\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\infty} \:\frac{\mathrm{ln}\left(\mathrm{tan}\left(\mathrm{tan}^{−\mathrm{1}} \left({e}^{\frac{\mathrm{1}}{\pi}\:\mathrm{tan}^{−\mathrm{1}} \:{u}} \right)\right)\right)\:}{{u}^{\mathrm{2}} \:+\:\mathrm{2}\pi{u}\:+\:\mathrm{2}\pi^{\mathrm{2}} }\:{du} \\ $$$$ \\ $$

Question Number 222125 Answers: 4 Comments: 2

If: log_3 (5^x + (1/5^x ) + 7) ⇒ min = ?

$$\mathrm{If}:\:\:\:\mathrm{log}_{\mathrm{3}} \left(\mathrm{5}^{\boldsymbol{\mathrm{x}}} \:+\:\frac{\mathrm{1}}{\mathrm{5}^{\boldsymbol{\mathrm{x}}} }\:+\:\mathrm{7}\right)\:\:\:\Rightarrow\:\:\:\mathrm{min}\:=\:? \\ $$

Question Number 222123 Answers: 1 Comments: 1

Question Number 222121 Answers: 2 Comments: 0

∫ acos(((cos(ϱ))/(1+2cos(ϱ)))) dϱ

$$\int\:\mathrm{acos}\left(\frac{\mathrm{cos}\left(\varrho\right)}{\mathrm{1}+\mathrm{2cos}\left(\varrho\right)}\right)\:\mathrm{d}\varrho \\ $$

Question Number 222117 Answers: 0 Comments: 0

problem 1. for a given positive integer m, find all triples(n,x,y) of positive integers,with n relatively prime to m,which satisfy (x^2 +y^2 )^m =(xy)^n hint:utilize AM&GM,diophantine eqn KLIPTO−QUANTA−OOZY

$$\boldsymbol{\mathrm{problem}}\:\mathrm{1}.\:\boldsymbol{\mathrm{for}}\:\boldsymbol{\mathrm{a}}\:\boldsymbol{\mathrm{given}}\:\boldsymbol{\mathrm{positive}}\:\boldsymbol{\mathrm{integer}} \\ $$$$\boldsymbol{\mathrm{m}},\:\boldsymbol{\mathrm{find}}\:\boldsymbol{\mathrm{all}}\:\boldsymbol{\mathrm{triples}}\left(\boldsymbol{\mathrm{n}},\boldsymbol{\mathrm{x}},\boldsymbol{\mathrm{y}}\right)\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{positive}}\:\boldsymbol{\mathrm{integers}},\boldsymbol{\mathrm{with}} \\ $$$$\boldsymbol{\mathrm{n}}\:\boldsymbol{\mathrm{relatively}}\:\boldsymbol{\mathrm{prime}}\:\boldsymbol{\mathrm{to}}\:\boldsymbol{\mathrm{m}},\boldsymbol{\mathrm{which}}\:\boldsymbol{\mathrm{satisfy}} \\ $$$$\left(\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\boldsymbol{\mathrm{y}}^{\mathrm{2}} \right)^{\boldsymbol{\mathrm{m}}} =\left(\boldsymbol{\mathrm{xy}}\right)^{\boldsymbol{\mathrm{n}}} \\ $$$$\boldsymbol{\mathrm{hint}}:\boldsymbol{\mathrm{utilize}}\:\boldsymbol{\mathrm{AM\&GM}},\boldsymbol{\mathrm{diophantine}}\:\boldsymbol{\mathrm{eqn}} \\ $$$$\boldsymbol{\mathrm{KLIPTO}}−\boldsymbol{\mathrm{QUANTA}}−\boldsymbol{\mathrm{OOZY}} \\ $$

Question Number 222113 Answers: 0 Comments: 2

name the following compound

$$\mathrm{name}\:\mathrm{the}\:\mathrm{following}\:\mathrm{compound} \\ $$

Question Number 222105 Answers: 1 Comments: 1

Question Number 222104 Answers: 2 Comments: 0

log _4 x − log _x^2 8 = 1 x =?

$$\:\:\:\:\mathrm{log}\:_{\mathrm{4}} \:\mathrm{x}\:−\:\mathrm{log}\:_{\mathrm{x}^{\mathrm{2}} } \:\mathrm{8}\:=\:\mathrm{1} \\ $$$$\:\:\:\:\mathrm{x}\:=?\: \\ $$

Question Number 222100 Answers: 0 Comments: 0

Question Number 222097 Answers: 0 Comments: 2

Question Number 222095 Answers: 0 Comments: 0

could I consider Y_ν (z)=cot(νπ)J_ν (z)−csc(νπ)J_(−ν) (z) as ∞−∞ form limit when ν∈Z and How can i calculate Y_ν (z)=cot(νπ)J_ν (z)−csc(νπ)J_(−ν) (z)...?? lim_(α→ν) ((cot^2 (απ)J_α ^2 (z)−csc^2 (απ)J_(−α) ^( 2) (z))/(cot(απ)J_α (z)+csc(απ)J_(−α) (z)))..... lim_(α→ν) ((((∂ )/∂α)(cot^2 (απ)J_α ^2 (z)−csc^2 (απ)J_(−α) ^2 (z)))/(((∂ )/∂α)(cot(απ)J_α ^ (z)+csc(απ)J_(−α) (z))))....??.... :(

$$\mathrm{could}\:\:\mathrm{I}\:\mathrm{consider}\:\:{Y}_{\nu} \left({z}\right)=\mathrm{cot}\left(\nu\pi\right){J}_{\nu} \left({z}\right)−\mathrm{csc}\left(\nu\pi\right){J}_{−\nu} \left({z}\right) \\ $$$$\mathrm{as}\:\infty−\infty\:\mathrm{form}\:\mathrm{limit}\:\mathrm{when}\:\nu\in\mathbb{Z} \\ $$$$\mathrm{and}\:\mathrm{How}\:\mathrm{can}\:\mathrm{i}\:\mathrm{calculate} \\ $$$${Y}_{\nu} \left({z}\right)=\mathrm{cot}\left(\nu\pi\right){J}_{\nu} \left({z}\right)−\mathrm{csc}\left(\nu\pi\right){J}_{−\nu} \left({z}\right)...?? \\ $$$$\underset{\alpha\rightarrow\nu} {\mathrm{lim}}\:\frac{\mathrm{cot}^{\mathrm{2}} \left(\alpha\pi\right){J}_{\alpha} ^{\mathrm{2}} \left({z}\right)−\mathrm{csc}^{\mathrm{2}} \left(\alpha\pi\right){J}_{−\alpha} ^{\:\mathrm{2}} \left({z}\right)}{\mathrm{cot}\left(\alpha\pi\right){J}_{\alpha} \left({z}\right)+\mathrm{csc}\left(\alpha\pi\right){J}_{−\alpha} \left({z}\right)}..... \\ $$$$\underset{\alpha\rightarrow\nu} {\mathrm{lim}}\frac{\frac{\partial\:\:}{\partial\alpha}\left(\mathrm{cot}^{\mathrm{2}} \left(\alpha\pi\right){J}_{\alpha} ^{\mathrm{2}} \left({z}\right)−\mathrm{csc}^{\mathrm{2}} \left(\alpha\pi\right){J}_{−\alpha} ^{\mathrm{2}} \left({z}\right)\right)}{\frac{\partial\:\:}{\partial\alpha}\left(\mathrm{cot}\left(\alpha\pi\right){J}_{\alpha} ^{\:} \left({z}\right)+\mathrm{csc}\left(\alpha\pi\right){J}_{−\alpha} \left({z}\right)\right)}....??.... \\ $$$$:\left(\right. \\ $$

Question Number 222076 Answers: 2 Comments: 0

Question Number 222072 Answers: 1 Comments: 2

Question Number 222064 Answers: 0 Comments: 3

Question Number 222066 Answers: 0 Comments: 4

Question Number 222057 Answers: 0 Comments: 2

lim_(n→∞) ((1^1 ×2^2 ×3^3 ......×n^n )/(n^((1/2)n^2 +(1/2)n^2 +(1/(12))) ×e^(−(1/4)n^2 ) ))=??? Help.... i can′t Solve that lim_(n→∞) a_n ...

$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{1}^{\mathrm{1}} ×\mathrm{2}^{\mathrm{2}} ×\mathrm{3}^{\mathrm{3}} ......×{n}^{{n}} }{{n}^{\frac{\mathrm{1}}{\mathrm{2}}{n}^{\mathrm{2}} +\frac{\mathrm{1}}{\mathrm{2}}{n}^{\mathrm{2}} +\frac{\mathrm{1}}{\mathrm{12}}} ×{e}^{−\frac{\mathrm{1}}{\mathrm{4}}{n}^{\mathrm{2}} } }=??? \\ $$$$\mathrm{Help}.... \\ $$$$\mathrm{i}\:\mathrm{can}'\mathrm{t}\:\mathrm{Solve}\:\mathrm{that}\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:{a}_{{n}} ... \\ $$

Question Number 222062 Answers: 2 Comments: 0

Question Number 222044 Answers: 1 Comments: 0

How do you evaluate ∫_(−∞) ^( ∞) ((sin(z+1))/((z+1)(z^2 +1))) dz

$$\mathrm{How}\:\mathrm{do}\:\mathrm{you}\:\mathrm{evaluate} \\ $$$$\int_{−\infty} ^{\:\:\infty} \:\:\frac{\mathrm{sin}\left({z}+\mathrm{1}\right)}{\left({z}+\mathrm{1}\right)\left({z}^{\mathrm{2}} +\mathrm{1}\right)}\:\mathrm{d}{z} \\ $$

Question Number 222031 Answers: 3 Comments: 0

x^x =−1 Number of solutions??

$${x}^{{x}} =−\mathrm{1} \\ $$$${Number}\:{of}\:{solutions}?? \\ $$

Question Number 222026 Answers: 2 Comments: 0

If (1.234)^a =(0.1234)^b =10^c prove that (1/a)−(1/c)=(1/b)

$${If}\:\left(\mathrm{1}.\mathrm{234}\right)^{{a}} =\left(\mathrm{0}.\mathrm{1234}\right)^{{b}} =\mathrm{10}^{{c}} \\ $$$${prove}\:{that}\:\frac{\mathrm{1}}{{a}}−\frac{\mathrm{1}}{{c}}=\frac{\mathrm{1}}{{b}} \\ $$

Question Number 222025 Answers: 1 Comments: 0

(((5cos^2 (π/3)+4sec^2 (π/6)−tan^2 (π/4))/(sin^2 (π/6)+cos^2 (π/6))))=?? [easy mode]

$$\left(\frac{\mathrm{5cos}\:^{\mathrm{2}} \frac{\pi}{\mathrm{3}}+\mathrm{4sec}\:^{\mathrm{2}} \frac{\pi}{\mathrm{6}}−\mathrm{tan}\:^{\mathrm{2}} \frac{\pi}{\mathrm{4}}}{\mathrm{sin}\:^{\mathrm{2}} \frac{\pi}{\mathrm{6}}+\mathrm{cos}\:^{\mathrm{2}} \frac{\pi}{\mathrm{6}}}\right)=?? \\ $$$$\left[{easy}\:{mode}\right] \\ $$

Question Number 222022 Answers: 1 Comments: 2

If ∠P+∠Q =90^0 then prove that (√(((sin P)/(cos Q))−sin Pcos Q))=cos P

$${If}\:\angle{P}+\angle{Q}\:=\mathrm{90}^{\mathrm{0}} \:{then}\:{prove}\:{that} \\ $$$$\sqrt{\frac{\mathrm{sin}\:{P}}{\mathrm{cos}\:{Q}}−\mathrm{sin}\:{P}\mathrm{cos}\:{Q}}=\mathrm{cos}\:{P} \\ $$

Question Number 222019 Answers: 0 Comments: 0

∫_0 ^π tan^(−1) (((ln(sin(x))/x)) dx

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\pi} \:\mathrm{tan}^{−\mathrm{1}} \:\left(\frac{\mathrm{ln}\left(\mathrm{sin}\left({x}\right)\right.}{{x}}\right)\:{dx} \\ $$$$ \\ $$

Question Number 222007 Answers: 0 Comments: 5

x(√(1+x^2 ))+log(x+(√(1+x^2 )))=12.5 find x^2 (answer should not be in decimal)

$$\boldsymbol{\mathrm{x}}\sqrt{\mathrm{1}+\boldsymbol{\mathrm{x}}^{\mathrm{2}} }+\boldsymbol{\mathrm{log}}\left(\boldsymbol{\mathrm{x}}+\sqrt{\mathrm{1}+\boldsymbol{\mathrm{x}}^{\mathrm{2}} }\right)=\mathrm{12}.\mathrm{5} \\ $$$$\mathrm{find}\:\boldsymbol{\mathrm{x}}^{\mathrm{2}} \:\left(\mathrm{answer}\:\mathrm{should}\:\mathrm{not}\:\mathrm{be}\:\mathrm{in}\:\mathrm{decimal}\right) \\ $$

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