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AllQuestion and Answers: Page 1519

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?

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

Question Number 55855 Answers: 0 Comments: 1

How to integrate ∫_0 ^1 ((sec^2 x)/(x(√x))) dx ?

$$\mathrm{How}\:\mathrm{to}\:\mathrm{integrate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{\mathrm{sec}^{\mathrm{2}} \:{x}}{{x}\sqrt{{x}}}\:{dx}\:\:? \\ $$

Question Number 55853 Answers: 0 Comments: 0

Question Number 55846 Answers: 1 Comments: 0

If x, y, z are in AP. Then the value of the determinant determinant (((a+2),(a+3),(a+2x)),((a+3),(a+4),(a+2y)),((a+4),(a+5),(a+2z))) is

$$\mathrm{If}\:{x},\:{y},\:{z}\:\mathrm{are}\:\mathrm{in}\:\mathrm{AP}.\:\mathrm{Then}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{determinant} \\ $$$$\begin{vmatrix}{{a}+\mathrm{2}}&{{a}+\mathrm{3}}&{{a}+\mathrm{2}{x}}\\{{a}+\mathrm{3}}&{{a}+\mathrm{4}}&{{a}+\mathrm{2}{y}}\\{{a}+\mathrm{4}}&{{a}+\mathrm{5}}&{{a}+\mathrm{2}{z}}\end{vmatrix}\:\mathrm{is} \\ $$

Question Number 55845 Answers: 0 Comments: 0

If A is 3×4 matrix and B is a matrix such that A′B and BA′ are both defined. Then B is of the type

$$\mathrm{If}\:{A}\:\mathrm{is}\:\mathrm{3}×\mathrm{4}\:\mathrm{matrix}\:\mathrm{and}\:{B}\:\mathrm{is}\:\mathrm{a}\:\mathrm{matrix}\:\mathrm{such} \\ $$$$\mathrm{that}\:{A}'{B}\:\mathrm{and}\:{BA}'\:\mathrm{are}\:\mathrm{both}\:\mathrm{defined}.\:\mathrm{Then} \\ $$$${B}\:\:\:\mathrm{is}\:\mathrm{of}\:\mathrm{the}\:\mathrm{type} \\ $$

Question Number 55844 Answers: 1 Comments: 0

If A is an involutory matrix, then (I+A)(I−A)=0.

$$\mathrm{If}\:{A}\:\mathrm{is}\:\mathrm{an}\:\mathrm{involutory}\:\mathrm{matrix},\:\mathrm{then}\: \\ $$$$\left({I}+{A}\right)\left({I}−{A}\right)=\mathrm{0}. \\ $$

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