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AllQuestion and Answers: Page 1631

Question Number 38675 Answers: 1 Comments: 3

Question Number 38679 Answers: 3 Comments: 0

Question Number 38651 Answers: 1 Comments: 0

If ∫_0 ^1 e^(−x^2 ) dx = a , then find the value of ∫_0 ^1 x^2 e^(−x^2 ) dx in terms of ′a′ ?

$$\mathrm{If}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\mathrm{e}^{−{x}^{\mathrm{2}} } {dx}\:=\:{a}\:,\:\mathrm{then}\:\mathrm{find}\:\mathrm{the}\:\mathrm{value} \\ $$$$\mathrm{of}\:\int_{\mathrm{0}} ^{\mathrm{1}} {x}^{\mathrm{2}} {e}^{−{x}^{\mathrm{2}} } {dx}\:{in}\:{terms}\:{of}\:'{a}'\:? \\ $$

Question Number 38643 Answers: 1 Comments: 4

calculate lim_(n→+∞) ((1+2+3+...+n)/(1+2^4 +3^4 +...+n^4 ))

$${calculate}\:{lim}_{{n}\rightarrow+\infty} \:\:\:\frac{\mathrm{1}+\mathrm{2}+\mathrm{3}+...+{n}}{\mathrm{1}+\mathrm{2}^{\mathrm{4}} \:+\mathrm{3}^{\mathrm{4}} \:+...+{n}^{\mathrm{4}} } \\ $$

Question Number 38642 Answers: 0 Comments: 3

calculate lim_(n→+∞) ((1 +2^2 +3^2 +....+n^2 )/(1+2^4 +3^4 +....+n^4 ))

$${calculate}\:{lim}_{{n}\rightarrow+\infty} \:\:\frac{\mathrm{1}\:+\mathrm{2}^{\mathrm{2}} \:+\mathrm{3}^{\mathrm{2}} \:+....+{n}^{\mathrm{2}} }{\mathrm{1}+\mathrm{2}^{\mathrm{4}} \:+\mathrm{3}^{\mathrm{4}} \:+....+{n}^{\mathrm{4}} } \\ $$

Question Number 38641 Answers: 0 Comments: 1

calculate lim_(n→+∞) ((1+2^3 +3^3 +....+n^3 )/(1+2^4 +3^4 +...+n^4 )) .

$${calculate}\:{lim}_{{n}\rightarrow+\infty} \:\:\:\frac{\mathrm{1}+\mathrm{2}^{\mathrm{3}} \:+\mathrm{3}^{\mathrm{3}} \:+....+{n}^{\mathrm{3}} }{\mathrm{1}+\mathrm{2}^{\mathrm{4}} \:+\mathrm{3}^{\mathrm{4}} \:+...+{n}^{\mathrm{4}} }\:. \\ $$

Question Number 38640 Answers: 0 Comments: 1

calculate Σ_(k=1) ^n k^4 interms of n.

$${calculate}\:\sum_{{k}=\mathrm{1}} ^{{n}} {k}^{\mathrm{4}} \:\:{interms}\:{of}\:{n}. \\ $$

Question Number 38636 Answers: 1 Comments: 0

Three variables u,v and w are related such that u varies directly as v and inversely as the square of w. If v increases by 15% and w decreased by 10%, find the percentage change in u.

$$\mathrm{Three}\:\mathrm{variables}\:\mathrm{u},\mathrm{v}\:\mathrm{and}\:\mathrm{w}\:\mathrm{are}\:\mathrm{related}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{u}\:\mathrm{varies}\:\mathrm{directly}\:\mathrm{as}\:\mathrm{v}\:\mathrm{and}\:\mathrm{inversely}\:\mathrm{as}\:\mathrm{the}\:\mathrm{square}\:\mathrm{of}\: \\ $$$$\mathrm{w}.\:\mathrm{If}\:\mathrm{v}\:\mathrm{increases}\:\mathrm{by}\:\mathrm{15\%}\:\mathrm{and}\:\mathrm{w}\:\mathrm{decreased}\:\mathrm{by}\:\mathrm{10\%}, \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{percentage}\:\mathrm{change}\:\mathrm{in}\:\mathrm{u}. \\ $$

Question Number 38618 Answers: 1 Comments: 2

Question Number 38613 Answers: 1 Comments: 0

The radius of the largest circle which passes through (1,2) and (3,4) and lies completely in the first quadrant is A) 3 B) 2 C) (√6) D) 2(√5) I got the answer as 2 but the answer given is 2(√5).

$$\mathrm{The}\:\mathrm{radius}\:\mathrm{of}\:\mathrm{the}\:\mathrm{largest}\:\mathrm{circle}\:\mathrm{which} \\ $$$$\mathrm{passes}\:\mathrm{through}\:\left(\mathrm{1},\mathrm{2}\right)\:\mathrm{and}\:\left(\mathrm{3},\mathrm{4}\right)\:\mathrm{and}\:\mathrm{lies} \\ $$$$\mathrm{completely}\:\mathrm{in}\:\mathrm{the}\:\mathrm{first}\:\mathrm{quadrant}\:\mathrm{is} \\ $$$$\left.\mathrm{A}\right)\:\mathrm{3} \\ $$$$\left.\mathrm{B}\right)\:\mathrm{2} \\ $$$$\left.\mathrm{C}\right)\:\sqrt{\mathrm{6}} \\ $$$$\left.\mathrm{D}\right)\:\mathrm{2}\sqrt{\mathrm{5}} \\ $$$$\mathrm{I}\:\mathrm{got}\:\mathrm{the}\:\mathrm{answer}\:\mathrm{as}\:\mathrm{2}\:\mathrm{but}\:\mathrm{the}\:\mathrm{answer}\: \\ $$$$\mathrm{given}\:\mathrm{is}\:\mathrm{2}\sqrt{\mathrm{5}}. \\ $$

Question Number 39491 Answers: 0 Comments: 0

Question Number 38600 Answers: 3 Comments: 1

Question Number 38593 Answers: 1 Comments: 3

Question Number 38587 Answers: 2 Comments: 2

solve for x: e^x + e^x^2 + e^x^3 = 3 + x + x^2 + x^3

$$\mathrm{solve}\:\mathrm{for}\:\mathrm{x}:\:\:\:\:\:\mathrm{e}^{\mathrm{x}} \:+\:\mathrm{e}^{\mathrm{x}^{\mathrm{2}} } \:+\:\mathrm{e}^{\mathrm{x}^{\mathrm{3}} } \:=\:\:\mathrm{3}\:+\:\mathrm{x}\:+\:\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{x}^{\mathrm{3}} \\ $$

Question Number 38570 Answers: 0 Comments: 2

Question Number 38568 Answers: 0 Comments: 18

Find the area common to min{[x], [y] } =2 and max{[x], [y] } =4 . [x] denotes the greatest integer less than or equal to x.

$${Find}\:{the}\:{area}\:{common}\:{to} \\ $$$${min}\left\{\left[{x}\right],\:\left[{y}\right]\:\right\}\:=\mathrm{2}\:\:\:{and} \\ $$$${max}\left\{\left[{x}\right],\:\left[{y}\right]\:\right\}\:=\mathrm{4}\:. \\ $$$$\left[{x}\right]\:{denotes}\:{the}\:{greatest}\:{integer} \\ $$$${less}\:{than}\:{or}\:{equal}\:{to}\:{x}. \\ $$

Question Number 38562 Answers: 1 Comments: 0

((√2) +i)(1−(√(2i)) )

$$\left(\sqrt{\mathrm{2}}\:+{i}\right)\left(\mathrm{1}−\sqrt{\mathrm{2}{i}}\:\right) \\ $$

Question Number 38559 Answers: 1 Comments: 0

in a geometric series, the first term =a, common ratio=r. If S_n denotes the sum of the n terms and U_n =Σ_(n=1) ^n S_(n,) then rS_n +(1−r)U_(n ) equals to (a) 0 (b) n (c) na (d)nar

$${in}\:{a}\:{geometric}\:{series},\:{the}\:{first}\:{term} \\ $$$$={a},\:{common}\:{ratio}={r}.\:{If}\:{S}_{{n}} \:{denotes} \\ $$$${the}\:{sum}\:{of}\:{the}\:{n}\:{terms}\:{and}\:{U}_{{n}} =\underset{{n}=\mathrm{1}} {\overset{{n}} {\sum}}{S}_{{n},} \\ $$$${then}\:{rS}_{{n}} +\left(\mathrm{1}−{r}\right){U}_{{n}\:\:} {equals}\:{to} \\ $$$$\left({a}\right)\:\:\mathrm{0}\:\:\:\:\:\:\left({b}\right)\:\:{n}\:\:\:\:\:\left({c}\right)\:\:\:\:{na}\:\:\:\:\left({d}\right){nar} \\ $$

Question Number 38557 Answers: 1 Comments: 0

prove that ((2 cos 2^n θ + 1)/(2 cos θ + 1)) = (2 cos θ − 1)(2 cos 2θ − 1)(2 cos 2^2 θ− 1) ...(2 cos 2^(n − 1) θ − 1)

Question Number 38536 Answers: 2 Comments: 0

∫_0 ^π (dx/(√(3−cos x)))=

$$\int_{\mathrm{0}} ^{\pi} \frac{{dx}}{\sqrt{\mathrm{3}−\mathrm{cos}\:{x}}}= \\ $$

Question Number 38535 Answers: 0 Comments: 0

Given the function f(x) where f(x)= { ((∫x^2 + 1 ,for {x:x D(f) 2)),((∫x^3 − 1,for y = f′(x))) :} a) Evaluate f(2) if f(a)= 2 + a^(n−1) find the value of a hence the domain of f(x).

$${Given}\:{the}\:{function} \\ $$$${f}\left({x}\right)\:{where}\: \\ $$$$ \\ $$$${f}\left({x}\right)=\:\begin{cases}{\int{x}^{\mathrm{2}} \:+\:\mathrm{1}\:,{for}\:\left\{{x}:{x}\:{D}\left({f}\right)\:\mathrm{2}\right.}\\{\int{x}^{\mathrm{3}} \:−\:\mathrm{1},{for}\:{y}\:=\:{f}'\left({x}\right)}\end{cases} \\ $$$$\left.{a}\right)\:{Evaluate}\:{f}\left(\mathrm{2}\right) \\ $$$${if}\:{f}\left({a}\right)=\:\mathrm{2}\:+\:{a}^{{n}−\mathrm{1}} \\ $$$${find}\:{the}\:{value}\:{of}\:{a} \\ $$$${hence}\:{the}\:{domain}\:{of}\:{f}\left({x}\right). \\ $$

Question Number 38534 Answers: 0 Comments: 0

∫∫_R (2x + 3y)^2 dA=??

$$\int\underset{{R}} {\int}\left(\mathrm{2}{x}\:+\:\mathrm{3}{y}\right)^{\mathrm{2}} \:{dA}=?? \\ $$

Question Number 38533 Answers: 2 Comments: 2

Question Number 38521 Answers: 0 Comments: 1

letf(x) = ((2x+1)/((x−2)(x^2 +x+1))) 1) calculate f^((n)) (x) 2) find f^((n)) (0) 3) developp f at integr serie. (

$${letf}\left({x}\right)\:=\:\frac{\mathrm{2}{x}+\mathrm{1}}{\left({x}−\mathrm{2}\right)\left({x}^{\mathrm{2}} \:+{x}+\mathrm{1}\right)} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}^{\left({n}\right)} \left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{3}\right)\:{developp}\:{f}\:{at}\:{integr}\:{serie}. \\ $$$$\left(\right. \\ $$

Question Number 38520 Answers: 0 Comments: 1

find the value of Σ_(n=2) ^∞ ((3n^2 +1)/((n−1)^3 (n+1)^3 ))

$${find}\:{the}\:{value}\:{of}\:\sum_{{n}=\mathrm{2}} ^{\infty} \:\:\frac{\mathrm{3}{n}^{\mathrm{2}} \:+\mathrm{1}}{\left({n}−\mathrm{1}\right)^{\mathrm{3}} \left({n}+\mathrm{1}\right)^{\mathrm{3}} } \\ $$

Question Number 38518 Answers: 0 Comments: 1

calculate Σ_(n=1) ^∞ (1/(n^2 (2n−1)^2 ))

$${calculate}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\mathrm{1}}{{n}^{\mathrm{2}} \left(\mathrm{2}{n}−\mathrm{1}\right)^{\mathrm{2}} } \\ $$

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