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AllQuestion and Answers: Page 1539

Question Number 55933 Answers: 2 Comments: 1

Question Number 55930 Answers: 2 Comments: 0

∫_0 ^( π) ((xtan x)/(sec x+tan x))dx = (is it (π^2 /2)−π)?

$$\int_{\mathrm{0}} ^{\:\:\pi} \frac{{x}\mathrm{tan}\:{x}}{\mathrm{sec}\:{x}+\mathrm{tan}\:{x}}{dx}\:=\:\left({is}\:{it}\:\frac{\pi^{\mathrm{2}} }{\mathrm{2}}−\pi\right)? \\ $$

Question Number 55926 Answers: 1 Comments: 0

Ancient Roman Number Magic: take 5 matches or picks and form a roman 3 ∣∣∣_(−) ^(−) now subtract 2 so that a little more than 3 is left

$$\mathrm{Ancient}\:\mathrm{Roman}\:\mathrm{Number}\:\mathrm{Magic}: \\ $$$$\mathrm{take}\:\mathrm{5}\:\mathrm{matches}\:\mathrm{or}\:\mathrm{picks}\:\mathrm{and}\:\mathrm{form}\:\mathrm{a}\:\mathrm{roman}\:\mathrm{3} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{−} {\overline {\mid\mid\mid}} \\ $$$$\mathrm{now}\:\mathrm{subtract}\:\mathrm{2}\:\mathrm{so}\:\mathrm{that}\:\mathrm{a}\:\mathrm{little}\:\mathrm{more}\:\mathrm{than}\:\mathrm{3}\:\mathrm{is}\:\mathrm{left} \\ $$

Question Number 55920 Answers: 0 Comments: 1

Calculate. Σ_(a=1) ^∞ Σ_(b=1) ^∞ Σ_(c=1) ^∞ ((ab(3a+c))/(4^(a+b+c) (a+b)(b+c)(c+a))).

$$\boldsymbol{\mathrm{Calculate}}. \\ $$$$\underset{\boldsymbol{{a}}=\mathrm{1}} {\overset{\infty} {\sum}}\underset{\boldsymbol{{b}}=\mathrm{1}} {\overset{\infty} {\sum}}\underset{\boldsymbol{{c}}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\boldsymbol{{ab}}\left(\mathrm{3}\boldsymbol{{a}}+\boldsymbol{{c}}\right)}{\mathrm{4}^{\boldsymbol{{a}}+\boldsymbol{{b}}+\boldsymbol{{c}}} \left(\boldsymbol{{a}}+\boldsymbol{{b}}\right)\left(\boldsymbol{{b}}+\boldsymbol{{c}}\right)\left(\boldsymbol{{c}}+\boldsymbol{{a}}\right)}. \\ $$

Question Number 55918 Answers: 1 Comments: 0

Two similar spheres of the same material have masses of 12kg and250kg respectively. find the radius of the smaller sphere if the radius of the bigger shere is 12.5cm

Question Number 55914 Answers: 1 Comments: 0

Question Number 55913 Answers: 1 Comments: 0

Question Number 55909 Answers: 0 Comments: 0

If E={f ∣f : R→R continoues function f(x) ∈Q, ∀x ∈R} then E=...

$$\mathrm{If}\:{E}=\left\{{f}\:\mid{f}\::\:\mathbb{R}\rightarrow\mathbb{R}\:\mathrm{continoues}\:\mathrm{function}\right. \\ $$$$\left.{f}\left({x}\right)\:\in\mathrm{Q},\:\forall{x}\:\in\mathbb{R}\right\}\:\mathrm{then}\:{E}=... \\ $$

Question Number 55908 Answers: 1 Comments: 0

known a < (π/2) . If M<1 with ∣cos x−cos y∣≤M ∣x−y∣ for every x, y ∈ [0,a], then M=..

$$\mathrm{known}\:{a}\:<\:\frac{\pi}{\mathrm{2}}\:. \\ $$$$\mathrm{If}\:\:\mathrm{M}<\mathrm{1}\:\mathrm{with}\:\mid\mathrm{cos}\:{x}−\mathrm{cos}\:{y}\mid\leqslant\mathrm{M}\:\mid{x}−{y}\mid \\ $$$$\mathrm{for}\:\mathrm{every}\:{x},\:{y}\:\in\:\left[\mathrm{0},{a}\right],\:\mathrm{then}\:\mathrm{M}=.. \\ $$

Question Number 55907 Answers: 1 Comments: 1

for every n ∈ N , f_n (x)=nx(1−x^2 )^n , for every x, 0≤x≤1 and a_n =∫_0 ^1 f_n (x) dx. If S_n =sin (πa_n ), for every n∈ N, then lim_(n→∞) s_n =...

$$\mathrm{for}\:\mathrm{every}\:{n}\:\in\:\mathbb{N}\:,\:{f}_{{n}} \left({x}\right)={nx}\left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{{n}} , \\ $$$$\mathrm{for}\:\mathrm{every}\:{x},\:\mathrm{0}\leqslant{x}\leqslant\mathrm{1} \\ $$$$\mathrm{and}\:{a}_{{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} {f}_{{n}} \left({x}\right)\:{dx}. \\ $$$$\mathrm{If}\:\mathrm{S}_{\mathrm{n}} =\mathrm{sin}\:\left(\pi{a}_{{n}} \right),\:\mathrm{for}\:\mathrm{every} \\ $$$$\mathrm{n}\in\:\mathbb{N},\:\mathrm{then}\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\mathrm{s}_{\mathrm{n}} =... \\ $$

Question Number 55906 Answers: 1 Comments: 0

lim_(n→∞) Σ_(k=1) ^(n) ((8n^2 )/(n^4 +1))=..

$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\overset{{n}} {\underset{{k}=\mathrm{1}} {\sum}}\frac{\mathrm{8}{n}^{\mathrm{2}} }{{n}^{\mathrm{4}} +\mathrm{1}}=.. \\ $$

Question Number 55905 Answers: 1 Comments: 0

lim_(n→∞) (x_(2n) +x_(2n+1) )=315 lim_(n→∞) (x_(2n) +x_(2n−1) )=2016 lim_(n→∞) (x_(2n) /x_(2n+1) )=...

$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\left({x}_{\mathrm{2}{n}} +{x}_{\mathrm{2}{n}+\mathrm{1}} \right)=\mathrm{315} \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\left({x}_{\mathrm{2}{n}} +{x}_{\mathrm{2}{n}−\mathrm{1}} \right)=\mathrm{2016} \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{{x}_{\mathrm{2}{n}} }{{x}_{\mathrm{2}{n}+\mathrm{1}} }=... \\ $$

Question Number 55904 Answers: 2 Comments: 0

Let a_i >0, ∀_i =1, 2, 3, …2016 If (a_1 a_2 …a_(2016) )^(1/(2016)) =2 then (1+a_1 )(1+a_2 )…(1+a_(2016) )≥...

$$\mathrm{Let}\:{a}_{{i}} >\mathrm{0},\:\forall_{{i}} =\mathrm{1},\:\mathrm{2},\:\mathrm{3},\:\ldots\mathrm{2016} \\ $$$$\mathrm{If}\:\left({a}_{\mathrm{1}} {a}_{\mathrm{2}} \ldots{a}_{\mathrm{2016}} \right)^{\frac{\mathrm{1}}{\mathrm{2016}}} =\mathrm{2} \\ $$$$\mathrm{then} \\ $$$$\left(\mathrm{1}+{a}_{\mathrm{1}} \right)\left(\mathrm{1}+{a}_{\mathrm{2}} \right)\ldots\left(\mathrm{1}+{a}_{\mathrm{2016}} \right)\geqslant... \\ $$

Question Number 55902 Answers: 1 Comments: 0

Question Number 55893 Answers: 1 Comments: 0

Three numbers are in G.P such that their sum is p and the sum of their square is q. Find the middle term of the G.P

$$\mathrm{Three}\:\mathrm{numbers}\:\mathrm{are}\:\mathrm{in}\:\mathrm{G}.\mathrm{P}\:\mathrm{such}\:\mathrm{that}\:\mathrm{their}\:\mathrm{sum}\:\mathrm{is}\:\:\boldsymbol{\mathrm{p}}\:\:\mathrm{and}\:\mathrm{the} \\ $$$$\mathrm{sum}\:\mathrm{of}\:\mathrm{their}\:\mathrm{square}\:\mathrm{is}\:\:\boldsymbol{\mathrm{q}}.\:\:\mathrm{Find}\:\mathrm{the}\:\mathrm{middle}\:\mathrm{term}\:\mathrm{of}\:\mathrm{the}\:\mathrm{G}.\mathrm{P} \\ $$

Question Number 55889 Answers: 1 Comments: 0

a=b^2 +bc+c^2 b=a^2 +ac+c^2 c=a^2 +ab+b^2 solve for : a, b, c.

Question Number 55877 Answers: 0 Comments: 0

Question Number 55873 Answers: 2 Comments: 1

Integrate..∫(√(1+(√(1+(√x))))) dx

$${Integrate}..\int\sqrt{\mathrm{1}+\sqrt{\mathrm{1}+\sqrt{{x}}}}\:{dx} \\ $$

Question Number 55869 Answers: 2 Comments: 2

if a_(n+2) =(a_(n+1) ^3 /a_n ^2 ) and a_1 =2, a_2 =4 find a_n =?

$${if}\:\boldsymbol{{a}}_{\boldsymbol{{n}}+\mathrm{2}} =\frac{\boldsymbol{{a}}_{\boldsymbol{{n}}+\mathrm{1}} ^{\mathrm{3}} }{\boldsymbol{{a}}_{\boldsymbol{{n}}} ^{\mathrm{2}} }\:{and}\:{a}_{\mathrm{1}} =\mathrm{2},\:{a}_{\mathrm{2}} =\mathrm{4} \\ $$$${find}\:\boldsymbol{{a}}_{\boldsymbol{{n}}} =? \\ $$

Question Number 55888 Answers: 1 Comments: 0

a=1+b+b^2 b=1+c+c^2 c=ab+a^2 +b^2 solve for : a, b, c.

Question Number 55887 Answers: 2 Comments: 0

a^2 +1=b^2 b^2 +c^2 =b^4 ab=c solve for : a, b, c.

$$\:\:\:{a}^{\mathrm{2}} +\mathrm{1}={b}^{\mathrm{2}} \\ $$$$\:\:\:{b}^{\mathrm{2}} +{c}^{\mathrm{2}} ={b}^{\mathrm{4}} \\ $$$$\:\:\:{ab}={c} \\ $$$$\:\:\:\:\boldsymbol{\mathrm{solve}}\:\boldsymbol{\mathrm{for}}\::\:\:\:\boldsymbol{\mathrm{a}},\:\boldsymbol{\mathrm{b}},\:\boldsymbol{\mathrm{c}}. \\ $$

Question Number 55860 Answers: 2 Comments: 0

Question Number 55859 Answers: 1 Comments: 0

Question Number 55858 Answers: 0 Comments: 0

Question Number 55857 Answers: 0 Comments: 0

Let A and B are matrices in R^(2017×2017) such that A^(−1) = (A + B)^(−1) − B^(−1) and det(A^(−1) ) = 2017 Find det(B)

$$\mathrm{Let}\:{A}\:\mathrm{and}\:{B}\:\mathrm{are}\:\mathrm{matrices}\:\mathrm{in}\:\mathbb{R}^{\mathrm{2017}×\mathrm{2017}} \:\mathrm{such}\:\mathrm{that}\: \\ $$$${A}^{−\mathrm{1}} \:=\:\left({A}\:+\:{B}\right)^{−\mathrm{1}} \:−\:{B}^{−\mathrm{1}} \\ $$$$\mathrm{and}\: \\ $$$$\mathrm{det}\left({A}^{−\mathrm{1}} \right)\:=\:\mathrm{2017} \\ $$$$\mathrm{Find}\:\:\mathrm{det}\left({B}\right) \\ $$

Question Number 55856 Answers: 1 Comments: 1

For what values of p the integral ∫_0 ^1 x^p ln x dx converge?

$$\mathrm{For}\:\mathrm{what}\:\mathrm{values}\:\mathrm{of}\:{p}\:\mathrm{the}\:\mathrm{integral} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \:{x}^{{p}} \:\mathrm{ln}\:{x}\:{dx} \\ $$$$\mathrm{converge}? \\ $$

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