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Question Number 161566    Answers: 0   Comments: 0

if x;y;z>0 then prove that: Σ (x/( ((y^3 + 25xyz + z^3 ))^(1/3) )) ≥ 1

$$\mathrm{if}\:\:\mathrm{x};\mathrm{y};\mathrm{z}>\mathrm{0}\:\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\Sigma\:\frac{\mathrm{x}}{\:\sqrt[{\mathrm{3}}]{\mathrm{y}^{\mathrm{3}} \:+\:\mathrm{25xyz}\:+\:\mathrm{z}^{\mathrm{3}} }}\:\geqslant\:\mathrm{1} \\ $$

Question Number 161564    Answers: 1   Comments: 1

Find all values x;y;z>0 such that: { ((x + y + 2z = 6)),(((3/y)∙((2/x) + (1/y)) = 4∙((2/(x + y)) + (1/(2y)))^2 )),((x + 2^y + log_2 z = 4)) :}

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{values}\:\:\mathrm{x};\mathrm{y};\mathrm{z}>\mathrm{0}\:\:\mathrm{such}\:\mathrm{that}: \\ $$$$\begin{cases}{\mathrm{x}\:+\:\mathrm{y}\:+\:\mathrm{2z}\:=\:\mathrm{6}}\\{\frac{\mathrm{3}}{\mathrm{y}}\centerdot\left(\frac{\mathrm{2}}{\mathrm{x}}\:+\:\frac{\mathrm{1}}{\mathrm{y}}\right)\:=\:\mathrm{4}\centerdot\left(\frac{\mathrm{2}}{\mathrm{x}\:+\:\mathrm{y}}\:+\:\frac{\mathrm{1}}{\mathrm{2y}}\right)^{\mathrm{2}} }\\{\mathrm{x}\:+\:\mathrm{2}^{\boldsymbol{\mathrm{y}}} \:+\:\mathrm{log}_{\mathrm{2}} \boldsymbol{\mathrm{z}}\:=\:\mathrm{4}}\end{cases} \\ $$

Question Number 161563    Answers: 1   Comments: 0

Solve the equation: x+(√(x+(√(x+(√(x+ ...)))))) = x∙(√(x∙(√(x∙(√(x∙ ...)))))) where , x>0

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{equation}: \\ $$$$\mathrm{x}+\sqrt{\mathrm{x}+\sqrt{\mathrm{x}+\sqrt{\mathrm{x}+\:...}}}\:=\:\mathrm{x}\centerdot\sqrt{\mathrm{x}\centerdot\sqrt{\mathrm{x}\centerdot\sqrt{\mathrm{x}\centerdot\:...}}} \\ $$$$\mathrm{where}\:,\:\mathrm{x}>\mathrm{0} \\ $$

Question Number 161560    Answers: 1   Comments: 0

lim_(x→+∞) 1+x^2 −2x^2 ln(x)=...?

$${li}\underset{{x}\rightarrow+\infty} {{m}}\mathrm{1}+{x}^{\mathrm{2}} −\mathrm{2}{x}^{\mathrm{2}} {ln}\left({x}\right)=...? \\ $$

Question Number 161559    Answers: 1   Comments: 0

please show that (1/2) + cosx + cos2x + cos3x + ... + cosnx = ((sin[(n+1)(x/2)])/(2sin(x/2)))

$${please}\:{show}\:{that} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\:+\:{cosx}\:+\:{cos}\mathrm{2}{x}\:+\:{cos}\mathrm{3}{x}\:+\:...\:+\:{cosnx}\:=\:\frac{{sin}\left[\left({n}+\mathrm{1}\right)\frac{{x}}{\mathrm{2}}\right]}{\mathrm{2}{sin}\frac{{x}}{\mathrm{2}}} \\ $$

Question Number 161558    Answers: 1   Comments: 0

What′s the value of a for which x^2 +x=a & x^2 −3x+2=a have one root common?

$${What}'{s}\:{the}\:{value}\:{of}\:{a}\:{for}\:{which} \\ $$$${x}^{\mathrm{2}} +{x}={a}\:\&\:{x}^{\mathrm{2}} −\mathrm{3}{x}+\mathrm{2}={a}\:{have}\:{one} \\ $$$${root}\:{common}? \\ $$

Question Number 161553    Answers: 0   Comments: 0

lim_(t→+∞) tΣ_(k=0) ^∞ (((−1)^k )/( (√(k^2 +t^2 ))))=?

$$\underset{\mathrm{t}\rightarrow+\infty} {\mathrm{lim}t}\underset{\mathrm{k}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{\mathrm{k}} }{\:\sqrt{\mathrm{k}^{\mathrm{2}} +\mathrm{t}^{\mathrm{2}} }}=? \\ $$

Question Number 161537    Answers: 2   Comments: 0

∫_0 ^( (π/4)) ((1+tan^4 (x))/(cot^2 (x))) dx =?

$$\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{4}}} \:\frac{\mathrm{1}+\mathrm{tan}\:^{\mathrm{4}} \left({x}\right)}{\mathrm{cot}\:^{\mathrm{2}} \left({x}\right)}\:{dx}\:=? \\ $$

Question Number 161533    Answers: 1   Comments: 1

Question Number 161529    Answers: 0   Comments: 0

lim_(n→∞) n((∫_1 ^(+∞) ((x^n −x^2 +1)/(x^2 (x^n +1)))dx)^n −(1/2))=?

$$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}n}\left(\left(\int_{\mathrm{1}} ^{+\infty} \frac{\mathrm{x}^{\mathrm{n}} −\mathrm{x}^{\mathrm{2}} +\mathrm{1}}{\mathrm{x}^{\mathrm{2}} \left(\mathrm{x}^{\mathrm{n}} +\mathrm{1}\right)}\mathrm{dx}\right)^{\mathrm{n}} −\frac{\mathrm{1}}{\mathrm{2}}\right)=? \\ $$

Question Number 161528    Answers: 1   Comments: 0

PROVE that the numbers of types 4k+2 & 4k+3 are NOT perfect □s.

$$\mathrm{PROVE}\:\mathrm{that}\:\mathrm{the}\:\mathrm{numbers}\:\mathrm{of}\:\mathrm{types} \\ $$$$\mathrm{4k}+\mathrm{2}\:\&\:\mathrm{4k}+\mathrm{3}\:\mathrm{are}\:\mathrm{NOT}\:\mathrm{perfect}\:\Box\mathrm{s}. \\ $$

Question Number 161527    Answers: 0   Comments: 0

Question Number 161521    Answers: 0   Comments: 3

x^8 +ax^4 +1=0 a=? is x_1 +x_2 +x_3 +x_4 =?

$$\boldsymbol{\mathrm{x}}^{\mathrm{8}} +\boldsymbol{\mathrm{ax}}^{\mathrm{4}} +\mathrm{1}=\mathrm{0} \\ $$$$\boldsymbol{{a}}=?\: \\ $$$$\boldsymbol{{is}}\:\:\boldsymbol{{x}}_{\mathrm{1}} +\boldsymbol{{x}}_{\mathrm{2}} +\boldsymbol{{x}}_{\mathrm{3}} +\boldsymbol{{x}}_{\mathrm{4}} =? \\ $$

Question Number 161516    Answers: 0   Comments: 2

f′(a) is derivative of function f(a) . lim_(h→0) ((f(a−2h^2 )−f(a+h^3 ))/h^2 ) = ?

$${f}'\left({a}\right)\:\:{is}\:\:{derivative}\:\:{of}\:\:{function}\:\:{f}\left({a}\right)\:. \\ $$$$\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\:\frac{{f}\left({a}−\mathrm{2}{h}^{\mathrm{2}} \right)−{f}\left({a}+{h}^{\mathrm{3}} \right)}{{h}^{\mathrm{2}} }\:\:=\:\:? \\ $$

Question Number 161513    Answers: 0   Comments: 0

find the value of −1−1/3^2 −1/5^2 −1/7^2 −...

$$\boldsymbol{{find}}\:\boldsymbol{{the}}\:\boldsymbol{{value}}\:\boldsymbol{{of}} \\ $$$$−\mathrm{1}−\mathrm{1}/\mathrm{3}^{\mathrm{2}} −\mathrm{1}/\mathrm{5}^{\mathrm{2}} −\mathrm{1}/\mathrm{7}^{\mathrm{2}} −... \\ $$

Question Number 161507    Answers: 2   Comments: 0

Question Number 161505    Answers: 1   Comments: 3

((a + ((a+8)/3) (√((a−1)/3))))^(1/3) + ((a − ((a+8)/3) (√((a−1)/3))))^(1/3) = ?

$$\sqrt[{\mathrm{3}}]{{a}\:+\:\frac{{a}+\mathrm{8}}{\mathrm{3}}\:\sqrt{\frac{{a}−\mathrm{1}}{\mathrm{3}}}}\:+\:\sqrt[{\mathrm{3}}]{{a}\:−\:\frac{{a}+\mathrm{8}}{\mathrm{3}}\:\sqrt{\frac{{a}−\mathrm{1}}{\mathrm{3}}}}\:\:=\:\:? \\ $$

Question Number 161504    Answers: 1   Comments: 0

Montrer a^ partir du crite^ re de Cauchy que U_n =Σ_(k=1) ^n (1/k^2 ) est une de Cauchy. −−−−−−−−−−−−−−−− Show by using Cauchy′s sequence definition that U_n =Σ_(k=1) ^n (1/k^2 ) is a sequence of Cauchy.

$${Montrer}\:\grave {{a}}\:{partir}\:{du}\:{crit}\grave {{e}re}\:{de}\: \\ $$$${Cauchy}\:{que}\:{U}_{{n}} =\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\mathrm{1}}{{k}^{\mathrm{2}} }\:{est}\:{une} \\ $$$${de}\:{Cauchy}. \\ $$$$−−−−−−−−−−−−−−−− \\ $$$${Show}\:{by}\:{using}\:{Cauchy}'{s}\:{sequence} \\ $$$${definition}\:{that}\:{U}_{{n}} =\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\mathrm{1}}{{k}^{\mathrm{2}} }\:{is}\:{a}\: \\ $$$${sequence}\:{of}\:{Cauchy}. \\ $$

Question Number 161500    Answers: 1   Comments: 0

x^6 - 6x^5 + ax^4 + bx^3 + cx^2 + dx + 1 = 0 all the roots of the equation are positive find a+b+c+d=?

$$\mathrm{x}^{\mathrm{6}} \:-\:\mathrm{6x}^{\mathrm{5}} \:+\:\mathrm{ax}^{\mathrm{4}} \:+\:\mathrm{bx}^{\mathrm{3}} \:+\:\mathrm{cx}^{\mathrm{2}} \:+\:\mathrm{dx}\:+\:\mathrm{1}\:=\:\mathrm{0} \\ $$$$\mathrm{all}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{are}\:\mathrm{positive} \\ $$$$\mathrm{find}\:\:\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{d}=? \\ $$

Question Number 161485    Answers: 0   Comments: 0

Find range of function y=((cos 4x+4sin 4x+1)/(cos 4x+2))

$$\:\mathrm{Find}\:\mathrm{range}\:\mathrm{of}\:\mathrm{function}\:\mathrm{y}=\frac{\mathrm{cos}\:\mathrm{4x}+\mathrm{4sin}\:\mathrm{4x}+\mathrm{1}}{\mathrm{cos}\:\mathrm{4x}+\mathrm{2}} \\ $$

Question Number 161484    Answers: 2   Comments: 0

{ (((1/a)+(1/b)=9)),((((1/( (a)^(1/3) ))+(1/( (b)^(1/3) )))(1+(1/( (a)^(1/3) )))(1+(1/( (b)^(1/3) )))=18)) :} 8a+4b=?

$$\:\:\begin{cases}{\frac{\mathrm{1}}{\mathrm{a}}+\frac{\mathrm{1}}{\mathrm{b}}=\mathrm{9}}\\{\left(\frac{\mathrm{1}}{\:\sqrt[{\mathrm{3}}]{\mathrm{a}}}+\frac{\mathrm{1}}{\:\sqrt[{\mathrm{3}}]{\mathrm{b}}}\right)\left(\mathrm{1}+\frac{\mathrm{1}}{\:\sqrt[{\mathrm{3}}]{\mathrm{a}}}\right)\left(\mathrm{1}+\frac{\mathrm{1}}{\:\sqrt[{\mathrm{3}}]{\mathrm{b}}}\right)=\mathrm{18}}\end{cases} \\ $$$$\:\:\:\mathrm{8a}+\mathrm{4b}=? \\ $$

Question Number 161482    Answers: 3   Comments: 1

Question Number 161476    Answers: 0   Comments: 2

Between (3/6) and −(4/5) How do i list two rational numbers please?

$$\:{Between}\:\frac{\mathrm{3}}{\mathrm{6}}\:\:{and}\:−\frac{\mathrm{4}}{\mathrm{5}} \\ $$$$\:{How}\:{do}\:{i}\:{list}\:{two}\:{rational}\: \\ $$$$\:{numbers}\:{please}? \\ $$

Question Number 161538    Answers: 2   Comments: 0

lim_( x → −2 ) (((2+ 3x + 3x^( 2) + x^( 3) )/( sin ( ((πx)/2) ))) )=? −−−−

$$ \\ $$$${lim}_{\:{x}\:\rightarrow\:−\mathrm{2}\:\:} \left(\frac{\mathrm{2}+\:\mathrm{3}{x}\:+\:\mathrm{3}{x}^{\:\mathrm{2}} \:+\:{x}^{\:\mathrm{3}} }{\:{sin}\:\left(\:\frac{\pi{x}}{\mathrm{2}}\:\right)}\:\right)=? \\ $$$$\:\:\:\:−−−− \\ $$

Question Number 161491    Answers: 0   Comments: 0

Question Number 161464    Answers: 0   Comments: 1

Given that in ΔABC, (sin A+sin B):(sin B+sin C):(sin C+sin A)= 6: 4: 5 Find the angle A.

$$\mathrm{Given}\:\mathrm{that}\:\mathrm{in}\:\Delta\mathrm{ABC}, \\ $$$$\left(\mathrm{sin}\:\mathrm{A}+\mathrm{sin}\:\mathrm{B}\right):\left(\mathrm{sin}\:\mathrm{B}+\mathrm{sin}\:\mathrm{C}\right):\left(\mathrm{sin}\:\mathrm{C}+\mathrm{sin}\:\mathrm{A}\right)=\:\mathrm{6}:\:\mathrm{4}:\:\mathrm{5} \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{angle}\:\mathrm{A}. \\ $$

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