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AllQuestion and Answers: Page 1687

Question Number 34438 Answers: 0 Comments: 0

Question Number 34985 Answers: 1 Comments: 0

Question Number 34433 Answers: 1 Comments: 1

find lim_(n→+∞) (1/n)Σ_(k=1) ^(n−1) (√((n+k)/(n−k)))

$${find}\:{lim}_{{n}\rightarrow+\infty} \frac{\mathrm{1}}{{n}}\sum_{{k}=\mathrm{1}} ^{{n}−\mathrm{1}} \:\sqrt{\frac{{n}+{k}}{{n}−{k}}} \\ $$

Question Number 34429 Answers: 2 Comments: 0

Prove that 3^m +3^n +1 is not a perfect square. where m and n are positive integers.

$${Prove}\:{that} \\ $$$$\mathrm{3}^{{m}} +\mathrm{3}^{{n}} +\mathrm{1}\:{is}\:{not}\:{a}\:{perfect}\:{square}. \\ $$$${where}\:{m}\:{and}\:{n}\:{are}\:{positive}\:{integers}. \\ $$

Question Number 34422 Answers: 2 Comments: 0

Show that ((1+x)/(1+(√(1+x)) )) +((1−x)/(1−(√(1−x )))) =1 when x=((√(3 ))/2)

$${Show}\:{that} \\ $$$$\frac{\mathrm{1}+{x}}{\mathrm{1}+\sqrt{\mathrm{1}+{x}}\:}\:+\frac{\mathrm{1}−{x}}{\mathrm{1}−\sqrt{\mathrm{1}−{x}\:}}\:=\mathrm{1}\:{when}\:{x}=\frac{\sqrt{\mathrm{3}\:}}{\mathrm{2}} \\ $$

Question Number 34420 Answers: 1 Comments: 1

let A = arctanx −arctany give another form of A if xy≠−1 .

$${let}\:{A}\:\:=\:{arctanx}\:−{arctany} \\ $$$${give}\:{another}\:{form}\:{of}\:{A}\:{if}\:\:{xy}\neq−\mathrm{1}\:. \\ $$

Question Number 34421 Answers: 0 Comments: 1

let A = ∫_(−∞) ^(+∞) (dx/(x^2 −j)) with j=e^(i((2π)/3)) extract ReA and Im(A) and calculste its values.

$${let}\:{A}\:\:=\:\int_{−\infty} ^{+\infty} \:\:\:\:\frac{{dx}}{{x}^{\mathrm{2}} \:−{j}}\:\:\:\:{with}\:{j}={e}^{{i}\frac{\mathrm{2}\pi}{\mathrm{3}}} \\ $$$${extract}\:\:{ReA}\:{and}\:{Im}\left({A}\right)\:{and}\:{calculste}\:{its}\:{values}. \\ $$

Question Number 34411 Answers: 0 Comments: 5

Question Number 34410 Answers: 0 Comments: 0

Prove that (((n^2 )!)/((n!)^(n+1) )) is always an integer for n∈N.

$${Prove}\:{that}\:\frac{\left({n}^{\mathrm{2}} \right)!}{\left({n}!\right)^{{n}+\mathrm{1}} }\:{is}\:{always}\:{an}\:{integer} \\ $$$${for}\:{n}\in{N}. \\ $$

Question Number 34408 Answers: 0 Comments: 1

what is the value of (√(5(√(5(√(5(√(5(√(5(√5))))))))))) ignoring 64.

$${what}\:{is}\:{the}\:{value}\:{of}\: \\ $$$$\:\sqrt{\mathrm{5}\sqrt{\mathrm{5}\sqrt{\mathrm{5}\sqrt{\mathrm{5}\sqrt{\mathrm{5}\sqrt{\mathrm{5}}}}}}} \\ $$$${ignoring}\:\mathrm{64}. \\ $$

Question Number 34385 Answers: 0 Comments: 0

p,q∈P m,n∈{0,1,2,...} How many pairs are there,whose LCM is p^m q^(n ) ,when: (i)(a,b) & (b,a) are considered same. (ii)(a,b) & (b,a) are considered different. (Generalization of Q# 34358)

$$\mathrm{p},\mathrm{q}\in\mathbb{P} \\ $$$$\mathrm{m},\mathrm{n}\in\left\{\mathrm{0},\mathrm{1},\mathrm{2},...\right\} \\ $$$$\mathrm{How}\:\mathrm{many}\:\mathrm{pairs}\:\mathrm{are}\:\mathrm{there},\mathrm{whose} \\ $$$$\mathrm{LCM}\:\mathrm{is}\:\mathrm{p}^{\mathrm{m}} \mathrm{q}^{\mathrm{n}\:} ,\mathrm{when}: \\ $$$$\left(\mathrm{i}\right)\left(\mathrm{a},\mathrm{b}\right)\:\&\:\left(\mathrm{b},\mathrm{a}\right)\:\mathrm{are}\:\mathrm{considered}\:\mathrm{same}. \\ $$$$\left(\mathrm{ii}\right)\left(\mathrm{a},\mathrm{b}\right)\:\&\:\left(\mathrm{b},\mathrm{a}\right)\:\mathrm{are}\:\mathrm{considered}\:\mathrm{different}. \\ $$$$\:\:\:\:\left(\mathrm{Generalization}\:\mathrm{of}\:\mathrm{Q}#\:\mathrm{34358}\right) \\ $$

Question Number 34382 Answers: 1 Comments: 0

how do i prove ∫_o ^t cos 2(ωt+α)dt=0

$$\mathrm{how}\:\mathrm{do}\:\mathrm{i}\:\mathrm{prove} \\ $$$$\int_{\mathrm{o}} ^{\mathrm{t}} \mathrm{cos}\:\mathrm{2}\left(\omega\mathrm{t}+\alpha\right)\mathrm{dt}=\mathrm{0} \\ $$

Question Number 34376 Answers: 0 Comments: 0

How to do A4 size page layout in equation editor ?

$$\boldsymbol{{H}}{ow}\:{to}\:{do}\:{A}\mathrm{4}\:{size}\:{page}\:{layout}\:{in}\:{equation}\:{editor}\:? \\ $$

Question Number 34375 Answers: 1 Comments: 0

The value of lim_(x→(π/2)) (([(x/2)])/(log (sin x))) = ? [.]= greatest integer function.

$$\boldsymbol{{T}}{he}\:{value}\:{of}\: \\ $$$${lim}_{{x}\rightarrow\frac{\pi}{\mathrm{2}}} \:\:\:\frac{\left[\frac{{x}}{\mathrm{2}}\right]}{\mathrm{log}\:\left(\mathrm{sin}\:{x}\right)}\:=\:? \\ $$$$\left[.\right]=\:{greatest}\:{integer}\:{function}. \\ $$

Question Number 34374 Answers: 2 Comments: 0

lim_(x→1) {1−x+[x−1]+[1−x]} = ? [.]= greatest integer function.

$${lim}_{{x}\rightarrow\mathrm{1}} \left\{\mathrm{1}−{x}+\left[{x}−\mathrm{1}\right]+\left[\mathrm{1}−{x}\right]\right\}\:=\:? \\ $$$$\left[.\right]=\:{greatest}\:{integer}\:{function}. \\ $$

Question Number 34369 Answers: 1 Comments: 0

Determine number of possible pairs,whose GCD is 144 in case: (i) when (a,b) and (b,a) is considerd same. (ii) when (a,b) and (b,a) is considerd different.

$$\mathrm{Determine}\:\mathrm{number}\:\mathrm{of}\:\mathrm{possible}\:\mathrm{pairs},\mathrm{whose} \\ $$$$\mathrm{GCD}\:\mathrm{is}\:\mathrm{144}\:\mathrm{in}\:\mathrm{case}: \\ $$$$\left(\mathrm{i}\right)\:\mathrm{when}\:\left(\mathrm{a},\mathrm{b}\right)\:\mathrm{and}\:\left(\mathrm{b},\mathrm{a}\right)\:\mathrm{is}\:\mathrm{considerd} \\ $$$$\:\:\:\:\:\:\:\mathrm{same}. \\ $$$$\left(\mathrm{ii}\right)\:\mathrm{when}\:\left(\mathrm{a},\mathrm{b}\right)\:\mathrm{and}\:\left(\mathrm{b},\mathrm{a}\right)\:\mathrm{is}\:\mathrm{considerd} \\ $$$$\:\:\:\:\:\:\:\:\:\mathrm{different}. \\ $$

Question Number 34367 Answers: 1 Comments: 0

Evaluate lim_(n→∞) ((n/(n^2 +1^2 ))+(n/(n^2 +2^2 ))+.....+(n/(n^2 +n^2 ))).

$$\boldsymbol{\mathrm{Evaluate}}\: \\ $$$$\mathrm{lim}_{\mathrm{n}\rightarrow\infty} \left(\frac{{n}}{{n}^{\mathrm{2}} +\mathrm{1}^{\mathrm{2}} }+\frac{{n}}{{n}^{\mathrm{2}} +\mathrm{2}^{\mathrm{2}} }+.....+\frac{{n}}{{n}^{\mathrm{2}} +{n}^{\mathrm{2}} }\right). \\ $$

Question Number 34365 Answers: 1 Comments: 0

if a>b>0 prove that b<((ae^x +be^(−x) )/(e^x +e^(−x) ))<a

$$\boldsymbol{\mathrm{if}}\:\boldsymbol{\mathrm{a}}>\boldsymbol{\mathrm{b}}>\mathrm{0}\:\boldsymbol{\mathrm{prove}}\:\boldsymbol{\mathrm{that}} \\ $$$$\boldsymbol{\mathrm{b}}<\frac{\boldsymbol{\mathrm{ae}}^{\boldsymbol{{x}}} +\boldsymbol{\mathrm{be}}^{−\boldsymbol{{x}}} }{\boldsymbol{\mathrm{e}}^{\boldsymbol{{x}}} +\boldsymbol{\mathrm{e}}^{−\boldsymbol{{x}}} }<\boldsymbol{\mathrm{a}} \\ $$

Question Number 34361 Answers: 1 Comments: 0

A right prism has a regular hexagonal base with sides of length 15cm,and a height of 20cm. find its volume and total surface area.

Question Number 34358 Answers: 1 Comments: 1

Determine number of possible pairs whose LCM is 144 in case, (i)when (a,b) & (b,a) are considered same. (ii)when(a,b) & (b,a) are considered different.

$$\mathrm{Determine}\:\mathrm{number}\:\mathrm{of}\:\mathrm{possible}\:\mathrm{pairs}\:\mathrm{whose} \\ $$$$\mathrm{LCM}\:\mathrm{is}\:\mathrm{144}\:\mathrm{in}\:\mathrm{case}, \\ $$$$\left(\mathrm{i}\right)\mathrm{when}\:\left(\mathrm{a},\mathrm{b}\right)\:\&\:\left(\mathrm{b},\mathrm{a}\right)\:\mathrm{are}\:\mathrm{considered}\:\mathrm{same}. \\ $$$$\left(\mathrm{ii}\right)\mathrm{when}\left(\mathrm{a},\mathrm{b}\right)\:\&\:\left(\mathrm{b},\mathrm{a}\right)\:\mathrm{are}\:\mathrm{considered}\:\mathrm{different}. \\ $$

Question Number 34357 Answers: 1 Comments: 0

If a+b+c=0 prove that i)((a/(b+c))+ (b/(c+a)) +(c/(a+b)))(((b+c)/a) +((c+a)/b) +((a+b)/c))=9 ii)((a/(b−c)) +(b/(c−a)) +(c/(a−b)))(((b−c)/a) +((c−a)/b) +((a−b)/c))=9

$${If}\:{a}+{b}+{c}=\mathrm{0}\:\:{prove}\:{that} \\ $$$$\left.{i}\right)\left(\frac{{a}}{{b}+{c}}+\:\frac{{b}}{{c}+{a}}\:+\frac{{c}}{{a}+{b}}\right)\left(\frac{{b}+{c}}{{a}}\:+\frac{{c}+{a}}{{b}}\:+\frac{{a}+{b}}{{c}}\right)=\mathrm{9} \\ $$$$\left.{ii}\right)\left(\frac{{a}}{{b}−{c}}\:+\frac{{b}}{{c}−{a}}\:+\frac{{c}}{{a}−{b}}\right)\left(\frac{{b}−{c}}{{a}}\:+\frac{{c}−{a}}{{b}}\:+\frac{{a}−{b}}{{c}}\right)=\mathrm{9} \\ $$

Question Number 34355 Answers: 2 Comments: 1

Question Number 34425 Answers: 2 Comments: 0

a, b, c ∈ R, a≠0 and the quadratic equation ax^2 +bx+c=0 has no real roots, then

$${a},\:{b},\:{c}\:\in\:{R},\:{a}\neq\mathrm{0}\:\mathrm{and}\:\mathrm{the}\:\mathrm{quadratic} \\ $$$$\mathrm{equation}\:{ax}^{\mathrm{2}} +{bx}+{c}=\mathrm{0}\:\mathrm{has}\:\mathrm{no}\:\mathrm{real} \\ $$$$\mathrm{roots},\:\mathrm{then} \\ $$

Question Number 34328 Answers: 2 Comments: 1

solve for x and y (i) 2^(x−1) .3^(y+1) =25 (ii)2^(n−1) .3^(m+1) =113

$$\boldsymbol{\mathrm{solve}}\:\boldsymbol{\mathrm{for}}\:\boldsymbol{{x}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{{y}} \\ $$$$\left(\boldsymbol{\mathrm{i}}\right)\:\mathrm{2}^{\boldsymbol{{x}}−\mathrm{1}} .\mathrm{3}^{\boldsymbol{{y}}+\mathrm{1}} =\mathrm{25} \\ $$$$\left(\boldsymbol{\mathrm{ii}}\right)\mathrm{2}^{\boldsymbol{{n}}−\mathrm{1}} .\mathrm{3}^{\boldsymbol{{m}}+\mathrm{1}} =\mathrm{113} \\ $$

Question Number 34320 Answers: 0 Comments: 2

calculate ∫_(−∞) ^(+∞) (dx/(x^2 +1 −i))

$${calculate}\:\:\:\int_{−\infty} ^{+\infty} \:\:\:\:\:\frac{{dx}}{{x}^{\mathrm{2}} \:+\mathrm{1}\:−{i}} \\ $$

Question Number 34316 Answers: 0 Comments: 0

find a eajivalent of u_n = ∫_0 ^∞ e^(−(t/n)) arcctant dt .

$${find}\:{a}\:{eajivalent}\:{of} \\ $$$${u}_{{n}} =\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:{e}^{−\frac{{t}}{{n}}} \:\:\:{arcctant}\:{dt}\:. \\ $$

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