Question and Answers Forum

AllQuestion and Answers: Page 500

Question Number 160080 Answers: 1 Comments: 0

lim_(x→0) ((x−∫_0 ^x e^t^2 dt)/(x(1−cos x)))=?

$$\underset{\mathrm{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{x}−\int_{\mathrm{0}} ^{\mathrm{x}} \mathrm{e}^{\mathrm{t}^{\mathrm{2}} } \mathrm{dt}}{\mathrm{x}\left(\mathrm{1}−\mathrm{cos}\:\mathrm{x}\right)}=? \\ $$

Question Number 160077 Answers: 1 Comments: 4

lim_(x→1) (((654)/(1−x^(654) ))−((678)/(1−x^(678) )))=?

$$\underset{\mathrm{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\left(\frac{\mathrm{654}}{\mathrm{1}−\mathrm{x}^{\mathrm{654}} }−\frac{\mathrm{678}}{\mathrm{1}−\mathrm{x}^{\mathrm{678}} }\right)=? \\ $$

Question Number 160065 Answers: 1 Comments: 8

The largest value of non-negative integer a for which lim_(x→1) {((−ax+sin(x−1)+a)/(x+sin(x−1)−1))}^((1−x)/( 1−(√x))) =(1/4) is ........?

$$\mathrm{The}\:\mathrm{largest}\:\mathrm{value}\:\mathrm{of}\:\mathrm{non}-\mathrm{negative}\:\mathrm{integer}\:{a} \\ $$$$\mathrm{for}\:\mathrm{which}\:\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\left\{\frac{−{ax}+\mathrm{sin}\left({x}−\mathrm{1}\right)+{a}}{{x}+\mathrm{sin}\left({x}−\mathrm{1}\right)−\mathrm{1}}\right\}^{\frac{\mathrm{1}−{x}}{\:\mathrm{1}−\sqrt{{x}}}} =\frac{\mathrm{1}}{\mathrm{4}} \\ $$$$\mathrm{is}\:........? \\ $$

Question Number 160064 Answers: 0 Comments: 2

Find the least positive integer n for which 2^n + 5^n - n is a multiple of 1000

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{least}\:\mathrm{positive}\:\mathrm{integer}\:\:\boldsymbol{\mathrm{n}}\:\:\mathrm{for} \\ $$$$\mathrm{which}\:\:\mathrm{2}^{\boldsymbol{\mathrm{n}}} \:+\:\mathrm{5}^{\boldsymbol{\mathrm{n}}} \:-\:\boldsymbol{\mathrm{n}}\:\:\mathrm{is}\:\mathrm{a}\:\mathrm{multiple}\:\mathrm{of}\:\mathrm{1000} \\ $$

Question Number 160063 Answers: 0 Comments: 0

Find: Ω =lim_(n→∞) (1/(n!)) ∫_( 0) ^( 1) ((1 - x)^n + cosnx)e^x dx

$$\mathrm{Find}: \\ $$$$\Omega\:=\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{1}}{\mathrm{n}!}\:\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\:\left(\left(\mathrm{1}\:-\:\mathrm{x}\right)^{\boldsymbol{\mathrm{n}}} \:+\:\mathrm{cos}\boldsymbol{\mathrm{nx}}\right)\mathrm{e}^{\boldsymbol{\mathrm{x}}} \:\mathrm{dx} \\ $$

Question Number 160062 Answers: 0 Comments: 0

Find: Ω =Π_(n=1) ^∞ ((n^(1/(n+1)) /2)) = ?

$$\mathrm{Find}: \\ $$$$\Omega\:=\underset{\boldsymbol{\mathrm{n}}=\mathrm{1}} {\overset{\infty} {\prod}}\left(\frac{\mathrm{n}^{\frac{\mathrm{1}}{\boldsymbol{\mathrm{n}}+\mathrm{1}}} }{\mathrm{2}}\right)\:=\:? \\ $$

Question Number 160061 Answers: 1 Comments: 0

Find out some pairs (a,b) such that for some n≥1 a^n +b^n ,a^(2n) +b^(2n) ,a^(4n) +b^(4n) ,a^(8n) +b^(8n) ∈P

$$ \\ $$$${Find}\:{out}\:{some}\:{pairs}\:\left({a},{b}\right)\:{such}\:{that} \\ $$$${for}\:{some}\:{n}\geqslant\mathrm{1} \\ $$$${a}^{{n}} +{b}^{{n}} ,{a}^{\mathrm{2}{n}} +{b}^{\mathrm{2}{n}} ,{a}^{\mathrm{4}{n}} +{b}^{\mathrm{4}{n}} ,{a}^{\mathrm{8}{n}} +{b}^{\mathrm{8}{n}} \in\mathbb{P} \\ $$$$ \\ $$

Question Number 160056 Answers: 0 Comments: 0

Question Number 160058 Answers: 1 Comments: 2

Find n so that ((a^(n+1) +b^(n+1) )/(a^n +b^n )) may be the arithmetic mean between a and b.

$${Find}\:{n}\:{so}\:{that}\:\frac{{a}^{{n}+\mathrm{1}} +{b}^{{n}+\mathrm{1}} }{{a}^{{n}} +{b}^{{n}} }\:{may}\:{be} \\ $$$${the}\:{arithmetic}\:{mean}\:{between}\:{a} \\ $$$${and}\:{b}. \\ $$

Question Number 160052 Answers: 1 Comments: 0

find Φ(k)=Σ_(n=1) ^∞ (n^k /(n!)) with k≥1.

$${find}\:\Phi\left({k}\right)=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{n}^{{k}} }{{n}!}\:{with}\:{k}\geqslant\mathrm{1}. \\ $$

Question Number 160050 Answers: 1 Comments: 0

Question Number 160048 Answers: 0 Comments: 0

Prove that: ∫_( 0) ^( 1) ((x sin^(-1) x)/(1 + sin^(-1) x)) dx < (1/4)

$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\:\frac{\mathrm{x}\:\mathrm{sin}^{-\mathrm{1}} \:\mathrm{x}}{\mathrm{1}\:+\:\mathrm{sin}^{-\mathrm{1}} \:\mathrm{x}}\:\mathrm{dx}\:<\:\frac{\mathrm{1}}{\mathrm{4}} \\ $$

Question Number 160045 Answers: 0 Comments: 0

Question Number 160036 Answers: 0 Comments: 0

Question Number 160035 Answers: 0 Comments: 0

Question Number 160025 Answers: 0 Comments: 1

Question Number 160023 Answers: 0 Comments: 0

Question Number 160014 Answers: 1 Comments: 2

Question Number 160013 Answers: 0 Comments: 1

∫(1/(4sin x+3cos x))dx evaluate

$$\int\frac{\mathrm{1}}{\mathrm{4}{sin}\:{x}+\mathrm{3}{cos}\:{x}}{dx} \\ $$$${evaluate} \\ $$

Question Number 160009 Answers: 1 Comments: 0

Find: lim_(n→∞) (((n!))^(1/n) ∙∫_((1/1^2 ) + (1/2^2 ) + ... + (1/n^2 )) ^( (𝛑^2 /6)) e^x^2 dx)

$$\mathrm{Find}: \\ $$$$\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\left(\sqrt[{\boldsymbol{\mathrm{n}}}]{\mathrm{n}!}\:\centerdot\underset{\frac{\mathrm{1}}{\mathrm{1}^{\mathrm{2}} }\:+\:\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{2}} }\:+\:...\:+\:\frac{\mathrm{1}}{\boldsymbol{\mathrm{n}}^{\mathrm{2}} }} {\overset{\:\frac{\boldsymbol{\pi}^{\mathrm{2}} }{\mathrm{6}}} {\int}}\:\mathrm{e}^{\boldsymbol{\mathrm{x}}^{\mathrm{2}} } \:\mathrm{dx}\right) \\ $$

Question Number 160008 Answers: 2 Comments: 2

x_1 =3 ; n(x_1 +x_2 +...+x_n )=x_n ; n∈N ; n≥1 Find: Ω =Σ_(n=1) ^∞ (-1)^(n+1) x_n

$$\mathrm{x}_{\mathrm{1}} =\mathrm{3}\:;\:\mathrm{n}\left(\mathrm{x}_{\mathrm{1}} +\mathrm{x}_{\mathrm{2}} +...+\mathrm{x}_{\boldsymbol{\mathrm{n}}} \right)=\mathrm{x}_{\boldsymbol{\mathrm{n}}} \:;\:\mathrm{n}\in\mathbb{N}\:;\:\mathrm{n}\geqslant\mathrm{1} \\ $$$$\mathrm{Find}: \\ $$$$\Omega\:=\underset{\boldsymbol{\mathrm{n}}=\mathrm{1}} {\overset{\infty} {\sum}}\left(-\mathrm{1}\right)^{\boldsymbol{\mathrm{n}}+\mathrm{1}} \:\mathrm{x}_{\boldsymbol{\mathrm{n}}} \\ $$

Question Number 160007 Answers: 0 Comments: 0

Find: lim_(n→∞) (n(((1 + (1/n))^n - e - 1)^n - e^(- (e/2)) ))

$$\mathrm{Find}: \\ $$$$\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\left(\mathrm{n}\left(\left(\left(\mathrm{1}\:+\:\frac{\mathrm{1}}{\mathrm{n}}\right)^{\boldsymbol{\mathrm{n}}} -\:\mathrm{e}\:-\:\mathrm{1}\right)^{\boldsymbol{\mathrm{n}}} -\:\mathrm{e}^{-\:\frac{\mathrm{e}}{\mathrm{2}}} \right)\right) \\ $$$$ \\ $$

Question Number 160006 Answers: 0 Comments: 2

Evaluate: lim_(n→∞) ∫_n ^(n+1) e^(1/x) dx = ?

$$\mathrm{Evaluate}: \\ $$$$\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\:\underset{\boldsymbol{\mathrm{n}}} {\overset{\boldsymbol{\mathrm{n}}+\mathrm{1}} {\int}}\:\mathrm{e}^{\frac{\mathrm{1}}{\boldsymbol{\mathrm{x}}}} \:\mathrm{dx}\:=\:? \\ $$$$ \\ $$

Question Number 159999 Answers: 1 Comments: 0

Question Number 159994 Answers: 1 Comments: 0

montrer que le quotient d′un nombe rationnel et dun nombre irrationnel est irrationnel

$$ \\ $$$$\mathrm{montrer}\:\mathrm{que}\:\mathrm{le}\:\mathrm{quotient}\:\mathrm{d}'\mathrm{un} \\ $$$$\mathrm{nombe}\:\mathrm{rationnel}\:\mathrm{et}\:\mathrm{dun}\:\mathrm{nombre}\: \\ $$$$\mathrm{irr}{a}\mathrm{tionnel}\:\mathrm{est}\:\mathrm{irrationnel} \\ $$

Question Number 159973 Answers: 0 Comments: 6

Can anyone please resolve the Q 159787 in details..

$${Can}\:{anyone}\:{please}\:{resolve}\:{the} \\ $$$${Q}\:\mathrm{159787}\:{in}\:{details}.. \\ $$

Pg 495 Pg 496 Pg 497 Pg 498 Pg 499 Pg 500 Pg 501 Pg 502 Pg 503 Pg 504

Contact: info@tinkutara.com