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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.

$$\boldsymbol{\mathrm{A}}\:\boldsymbol{\mathrm{right}}\:\boldsymbol{\mathrm{prism}}\:\boldsymbol{\mathrm{has}}\:\boldsymbol{\mathrm{a}}\:\boldsymbol{\mathrm{regular}} \\ $$$$\boldsymbol{\mathrm{hexagonal}}\:\boldsymbol{\mathrm{base}}\:\boldsymbol{\mathrm{with}}\:\boldsymbol{\mathrm{sides}}\:\boldsymbol{\mathrm{of}} \\ $$$$\boldsymbol{\mathrm{length}}\:\mathrm{15}\boldsymbol{\mathrm{cm}},\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{a}}\:\boldsymbol{\mathrm{height}}\: \\ $$$$\boldsymbol{\mathrm{of}}\:\mathrm{20}\boldsymbol{\mathrm{cm}}.\:\boldsymbol{\mathrm{find}}\:\boldsymbol{\mathrm{its}}\:\boldsymbol{\mathrm{volume}}\:\boldsymbol{\mathrm{and}} \\ $$$$\boldsymbol{\mathrm{total}}\:\boldsymbol{\mathrm{surface}}\:\boldsymbol{\mathrm{area}}. \\ $$

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 .

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

Question Number 34315    Answers: 0   Comments: 2

1) find F(x)= ∫_0 ^(+∞) ((e^(−at) −e^(−bt) )/t)sin(xt)dt with a>0 ,b>0 .

$$\left.\mathrm{1}\right)\:{find}\:\:{F}\left({x}\right)=\:\int_{\mathrm{0}} ^{+\infty} \:\:\frac{{e}^{−{at}} \:−{e}^{−{bt}} }{{t}}{sin}\left({xt}\right){dt} \\ $$$${with}\:{a}>\mathrm{0}\:,{b}>\mathrm{0}\:. \\ $$

Question Number 34314    Answers: 0   Comments: 1

let f(x)= ∫_0 ^(+∞) ((1−cos(xt))/t^2 ) e^(−t) dt calculate f(x) .

$${let}\:{f}\left({x}\right)=\:\int_{\mathrm{0}} ^{+\infty} \:\:\frac{\mathrm{1}−{cos}\left({xt}\right)}{{t}^{\mathrm{2}} }\:{e}^{−{t}} {dt}\: \\ $$$${calculate}\:{f}\left({x}\right)\:. \\ $$

Question Number 34313    Answers: 0   Comments: 0

let u_0 =x ≠o and u_(n+1) =ln(((e^u_n −1)/u_n )) 1) study the convervence of (u_n ) 2)find Σ_(n=0) ^∞ (Π_(k=0) ^n u_k ) .

$${let}\:{u}_{\mathrm{0}} ={x}\:\neq{o}\:\:{and}\:{u}_{{n}+\mathrm{1}} ={ln}\left(\frac{{e}^{{u}_{{n}} } \:−\mathrm{1}}{{u}_{{n}} }\right) \\ $$$$\left.\mathrm{1}\right)\:{study}\:{the}\:{convervence}\:{of}\:\left({u}_{{n}} \right) \\ $$$$\left.\mathrm{2}\right){find}\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\left(\prod_{{k}=\mathrm{0}} ^{{n}} \:{u}_{{k}} \right)\:. \\ $$

Question Number 34312    Answers: 0   Comments: 1

calculate I = ∫∫_D x^3 dxdy on the domain D ={(x,y)∈R^2 /1≤x≤2 , x^2 −y^2 −1≥0}

$${calculate}\:{I}\:\:=\:\int\int_{{D}} {x}^{\mathrm{3}} {dxdy}\:\:\:{on}\:{the}\:{domain} \\ $$$${D}\:=\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} /\mathrm{1}\leqslant{x}\leqslant\mathrm{2}\:,\:{x}^{\mathrm{2}} −{y}^{\mathrm{2}} −\mathrm{1}\geqslant\mathrm{0}\right\} \\ $$

Question Number 34311    Answers: 1   Comments: 0

let give the d.e. (1+x^2 )y^(′′) +3xy^′ +y =0find a solution y(x) deveppable at integr serie with∣x∣<1 .

$${let}\:{give}\:{the}\:{d}.{e}.\:\left(\mathrm{1}+{x}^{\mathrm{2}} \right){y}^{''} \:+\mathrm{3}{xy}^{'} \:+{y}\:=\mathrm{0}{find} \\ $$$${a}\:{solution}\:{y}\left({x}\right)\:{deveppable}\:{at}\:{integr}\:{serie}\: \\ $$$${with}\mid{x}\mid<\mathrm{1}\:. \\ $$

Question Number 34310    Answers: 0   Comments: 0

let f(x)= ∫_(−∞) ^x (dt/(1+t^2 +t^4 )) 1) prove that f id derivsble and calculate f^′ (x) 2)devellpp f at integr serie at o.

$${let}\:{f}\left({x}\right)=\:\int_{−\infty} ^{{x}} \:\:\frac{{dt}}{\mathrm{1}+{t}^{\mathrm{2}} \:+{t}^{\mathrm{4}} } \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{f}\:{id}\:{derivsble}\:{and}\:{calculate}\:{f}^{'} \left({x}\right) \\ $$$$\left.\mathrm{2}\right){devellpp}\:{f}\:{at}\:{integr}\:{serie}\:{at}\:{o}. \\ $$

Question Number 34309    Answers: 0   Comments: 3

let S(x)= Σ_(n=1) ^∞ (−1)^(n−1) (x^(2n+1) /(4n^2 −1)) 1) find the radius of convergence 2) calculate the sum S(x).

$${let}\:{S}\left({x}\right)=\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} \:\:\:\frac{{x}^{\mathrm{2}{n}+\mathrm{1}} }{\mathrm{4}{n}^{\mathrm{2}} \:−\mathrm{1}} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{the}\:{radius}\:{of}\:{convergence} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{the}\:{sum}\:\:{S}\left({x}\right). \\ $$

Question Number 34308    Answers: 0   Comments: 0

let I = ∫_0 ^(+∞) (((1+x)^(−(1/4)) −(1+x)^(−(3/4)) )/x)dx prove that I isconvergent and find its value .

$${let}\:\:{I}\:=\:\int_{\mathrm{0}} ^{+\infty} \:\:\frac{\left(\mathrm{1}+{x}\right)^{−\frac{\mathrm{1}}{\mathrm{4}}} \:\:−\left(\mathrm{1}+{x}\right)^{−\frac{\mathrm{3}}{\mathrm{4}}} }{{x}}{dx} \\ $$$${prove}\:{that}\:{I}\:{isconvergent}\:{and}\:{find}\:{its}\:{value}\:. \\ $$

Question Number 34307    Answers: 0   Comments: 1

let give A_n = (((1 (α/n))),((−(α/n) 1)) ) calculate lim_(n→+∞) A_n ^n .

$${let}\:{give}\:{A}_{{n}} =\:\:\begin{pmatrix}{\mathrm{1}\:\:\:\:\:\:\:\:\:\frac{\alpha}{{n}}}\\{−\frac{\alpha}{{n}}\:\:\:\:\:\:\mathrm{1}}\end{pmatrix} \\ $$$${calculate}\:{lim}_{{n}\rightarrow+\infty} \:{A}_{{n}} ^{{n}} \:\:\:\:. \\ $$

Question Number 34305    Answers: 0   Comments: 1

let] A = (((1 1 0)),((1 1 1)) ) (0 1 1 1)find the caractetistic polynom of A 2) calculate A^n

$$\left.{let}\right]\:{A}\:=\begin{pmatrix}{\mathrm{1}\:\:\:\:\mathrm{1}\:\:\:\:\:\:\:\mathrm{0}}\\{\mathrm{1}\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\:\mathrm{1}}\end{pmatrix} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{0}\:\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\:\mathrm{1}\:\right. \\ $$$$\left.\mathrm{1}\right){find}\:{the}\:{caractetistic}\:{polynom}\:{of}\:{A} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{A}^{{n}} \\ $$

Question Number 34301    Answers: 1   Comments: 0

If log_(12) 18=a ,find log_(24) 16 in term of a

$${If}\:{log}_{\mathrm{12}} \mathrm{18}={a}\:,{find}\:{log}_{\mathrm{24}} \mathrm{16}\:{in}\:{term} \\ $$$${of}\:\:{a} \\ $$

Question Number 34298    Answers: 0   Comments: 2

let A_ = ∫_0 ^∞ e^(−x) cos[x]dx and B = ∫_0 ^∞ e^(−[x]) cosxdx calculate A−B .

$${let}\:{A}_{\:} =\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−{x}} {cos}\left[{x}\right]{dx}\:\:{and}\:{B}\:=\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−\left[{x}\right]} \:{cosxdx} \\ $$$${calculate}\:{A}−{B}\:\:. \\ $$

Question Number 34297    Answers: 1   Comments: 1

find ∫_(−∞) ^(+∞) e^(−z t^2 ) dt with z=r e^(iθ) ∈ C .

$${find}\:\:\int_{−\infty} ^{+\infty} \:\:{e}^{−{z}\:{t}^{\mathrm{2}} } {dt}\:\:\:{with}\:{z}={r}\:{e}^{{i}\theta} \:\:\in\:{C}\:. \\ $$

Question Number 34296    Answers: 0   Comments: 3

find ∫_(−∞) ^(+∞) e^(−jx^2 ) with j =e^(i((2π)/3))

$${find}\:\:\int_{−\infty} ^{+\infty} \:\:{e}^{−{jx}^{\mathrm{2}} } \:\:\:\:{with}\:\:{j}\:={e}^{{i}\frac{\mathrm{2}\pi}{\mathrm{3}}} \\ $$

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