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Question Number 33120    Answers: 1   Comments: 0

let give α>0 find the value of ∫_0 ^1 (dx/(√((1−x)(1+αx)))) .

$${let}\:{give}\:\alpha>\mathrm{0}\:{find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\frac{{dx}}{\sqrt{\left(\mathrm{1}−{x}\right)\left(\mathrm{1}+\alpha{x}\right)}}\:. \\ $$

Question Number 33119    Answers: 0   Comments: 1

find ∫_0 ^∞ (t^n /(e^t −1)) dt by using ξ(x) for n integr ξ(x)=Σ_(n=1) ^∞ (1/n^x ) with x>1 .

$${find}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{t}^{{n}} }{{e}^{{t}} \:−\mathrm{1}}\:{dt}\:{by}\:{using}\:\xi\left({x}\right)\:{for}\:{n}\:{integr} \\ $$$$\xi\left({x}\right)=\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{{n}^{{x}} }\:\:\:{with}\:{x}>\mathrm{1}\:. \\ $$

Question Number 33116    Answers: 1   Comments: 0

solve at [0,π] cosα +cos(2α) +cos(3α)=0

$${solve}\:{at}\:\left[\mathrm{0},\pi\right]\:\:{cos}\alpha\:+{cos}\left(\mathrm{2}\alpha\right)\:+{cos}\left(\mathrm{3}\alpha\right)=\mathrm{0} \\ $$

Question Number 33134    Answers: 0   Comments: 0

describe geometrically the transformation with matrix 1) (((1 0)),((0 −1)) ) 2) (((1 0)),((2 1)) )

$${describe}\:{geometrically}\:{the}\:{transformation} \\ $$$${with}\:{matrix} \\ $$$$\left.\mathrm{1}\left.\right)\:\begin{pmatrix}{\mathrm{1}\:\:\:\:\:\:\:\:\mathrm{0}}\\{\mathrm{0}\:\:\:\:\:\:\:−\mathrm{1}}\end{pmatrix}\:\:\:\mathrm{2}\right)\:\:\begin{pmatrix}{\mathrm{1}\:\:\:\:\:\:\:\:\:\mathrm{0}}\\{\mathrm{2}\:\:\:\:\:\:\:\:\:\mathrm{1}}\end{pmatrix} \\ $$

Question Number 33103    Answers: 1   Comments: 1

Question Number 33100    Answers: 1   Comments: 0

Question Number 33135    Answers: 0   Comments: 0

the triangle with vertices A(−1,−3) ,B(2,1),and C(−2,2), is transformed by matrix (((a b)),((c d)) ) into the triangle with vertices A(−2,−3)^ , B(4,1),C(−4^ ,2) find the values of a,b,c and d

$${the}\:{triangle}\:{with}\:{vertices}\:{A}\left(−\mathrm{1},−\mathrm{3}\right) \\ $$$$,{B}\left(\mathrm{2},\mathrm{1}\right),{and}\:{C}\left(−\mathrm{2},\mathrm{2}\right),\:{is}\:{transformed} \\ $$$${by}\:{matrix}\:\begin{pmatrix}{{a}\:\:\:\:\:\:{b}}\\{{c}\:\:\:\:\:\:\:{d}}\end{pmatrix}\:{into}\:{the}\:{triangle}\:{with}\:{vertices} \\ $$$${A}\left(−\mathrm{2},−\mathrm{3}\bar {\right)},\:{B}\left(\mathrm{4},\mathrm{1}\right),{C}\left(−\bar {\mathrm{4}},\mathrm{2}\right)\:{find}\:{the}\:{values}\:{of}\: \\ $$$${a},{b},{c}\:{and}\:{d} \\ $$$$ \\ $$

Question Number 33096    Answers: 1   Comments: 0

z and w ∈ C proof ∣∣z∣−∣w∣∣ ≤ ∣z−w∣ and ∣z∣−∣w∣≤ ∣z+w∣

$${z}\:{and}\:{w}\:\in\:\mathbb{C} \\ $$$${proof}\:\mid\mid{z}\mid−\mid{w}\mid\mid\:\leqslant\:\mid{z}−{w}\mid\:{and}\:\mid{z}\mid−\mid{w}\mid\leqslant\:\mid{z}+{w}\mid \\ $$

Question Number 33095    Answers: 1   Comments: 0

z=x+yi ∈ C z^− =x−y ∈ C proof ∣z∣^2 =∣z^2 ∣=zz^− , so z≠0 →(1/z)=(z^− /(∣z∣^2 ))

$${z}={x}+{yi}\:\in\:\mathbb{C} \\ $$$$\overset{−} {{z}}={x}−{y}\:\in\:\mathbb{C} \\ $$$${proof}\:\mid{z}\mid^{\mathrm{2}} =\mid{z}^{\mathrm{2}} \mid={z}\overset{−} {{z}},\:{so}\:{z}\neq\mathrm{0}\:\rightarrow\frac{\mathrm{1}}{{z}}=\frac{\overset{−} {{z}}}{\mid{z}\mid^{\mathrm{2}} } \\ $$

Question Number 33094    Answers: 0   Comments: 1

let f(x)= (1/(1+x+x^2 )) dvelopp f at integr serie.

$${let}\:{f}\left({x}\right)=\:\frac{\mathrm{1}}{\mathrm{1}+{x}+{x}^{\mathrm{2}} }\:\:{dvelopp}\:{f}\:{at}\:{integr}\:{serie}. \\ $$

Question Number 33090    Answers: 1   Comments: 0

Question Number 33089    Answers: 1   Comments: 5

The LCM and GCF of three numbers is 360 and 6 respectively. if the two numbers are 18 and 60. find the third number.

$$\:\boldsymbol{\mathrm{T}}\mathrm{he}\:\boldsymbol{\mathrm{LCM}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{GCF}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{three}}\:\boldsymbol{\mathrm{numbers}}\:\boldsymbol{\mathrm{is}} \\ $$$$\:\mathrm{360}\:\boldsymbol{\mathrm{and}}\:\mathrm{6}\:\boldsymbol{\mathrm{respectively}}.\:\boldsymbol{\mathrm{if}}\:\boldsymbol{\mathrm{the}} \\ $$$$\:\boldsymbol{\mathrm{two}}\:\boldsymbol{\mathrm{numbers}}\:\boldsymbol{\mathrm{are}}\:\mathrm{18}\:\boldsymbol{\mathrm{and}}\:\mathrm{60}. \\ $$$$\:\boldsymbol{\mathrm{find}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{third}}\:\boldsymbol{\mathrm{number}}. \\ $$

Question Number 33088    Answers: 0   Comments: 3

Find the value of Σ_(n=1) ^∞ (n^2 /2^(n−1) )

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{{n}^{\mathrm{2}} }{\mathrm{2}^{{n}−\mathrm{1}} } \\ $$

Question Number 33074    Answers: 1   Comments: 1

find interms of n the sum Σ_(k=0) ^n k^2 C_n ^k

$${find}\:{interms}\:{of}\:{n}\:\:{the}\:{sum}\:\sum_{{k}=\mathrm{0}} ^{{n}} \:{k}^{\mathrm{2}} \:\:{C}_{{n}} ^{{k}} \\ $$

Question Number 33073    Answers: 2   Comments: 1

lim_(x→∞) x^2 e^(−x)

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:{x}^{\mathrm{2}} {e}^{−{x}} \\ $$

Question Number 33072    Answers: 1   Comments: 1

find Σ_(k=0) ^n k C_n ^k .

$${find}\:\:\sum_{{k}=\mathrm{0}} ^{{n}} \:{k}\:{C}_{{n}} ^{{k}} \:. \\ $$

Question Number 33069    Answers: 0   Comments: 0

by using residus theorem prove that ∫_0 ^∞ (t^(a−1) /(1+t)) dt = (π/(sin(πa))) with 0<a<1 .

$${by}\:\:{using}\:{residus}\:{theorem}\:{prove}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{t}^{{a}−\mathrm{1}} }{\mathrm{1}+{t}}\:{dt}\:=\:\frac{\pi}{{sin}\left(\pi{a}\right)}\:{with}\:\:\mathrm{0}<{a}<\mathrm{1}\:. \\ $$

Question Number 33064    Answers: 1   Comments: 2

Question Number 33153    Answers: 0   Comments: 1

it is given that Σ_(r=1 ) ^(20) [f(r)−10]=200 and Σ_(r=1) ^(20) [f(r)−10]^2 =2800 find the value of Σ_(r=1) ^(20) [f(r)]^2

$${it}\:{is}\:{given}\:{that} \\ $$$$\:\:\underset{{r}=\mathrm{1}\:} {\overset{\mathrm{20}} {\sum}}\left[{f}\left({r}\right)−\mathrm{10}\right]=\mathrm{200} \\ $$$${and} \\ $$$$\:\underset{{r}=\mathrm{1}} {\overset{\mathrm{20}} {\sum}}\left[{f}\left({r}\right)−\mathrm{10}\right]^{\mathrm{2}} =\mathrm{2800} \\ $$$${find}\:{the}\:{value}\:{of} \\ $$$$\underset{{r}=\mathrm{1}} {\overset{\mathrm{20}} {\sum}}\left[{f}\left({r}\right)\right]^{\mathrm{2}} \\ $$$$ \\ $$

Question Number 33052    Answers: 0   Comments: 1

Question Number 33051    Answers: 0   Comments: 3

Question Number 33048    Answers: 0   Comments: 7

Let f:N→R be a function sarisfying following conditions: f(1)=1. f(1)+2f(2)+....+nf(n)=n(n+1)f(n). Then find the value of 49f(49) ?

$${Let}\:{f}:{N}\rightarrow{R}\:{be}\:{a}\:{function}\:{sarisfying} \\ $$$${following}\:{conditions}: \\ $$$${f}\left(\mathrm{1}\right)=\mathrm{1}. \\ $$$${f}\left(\mathrm{1}\right)+\mathrm{2}{f}\left(\mathrm{2}\right)+....+{nf}\left({n}\right)={n}\left({n}+\mathrm{1}\right){f}\left({n}\right). \\ $$$${Then}\:{find}\:{the}\:{value}\:{of}\:\mathrm{49}{f}\left(\mathrm{49}\right)\:? \\ $$

Question Number 33043    Answers: 1   Comments: 0

equal squares as large as possible are drawn on a rectangular ceiling board measuring 54cm by 78cm,find (a)The size of the squares (b)The total number of squares

$$\:\boldsymbol{\mathrm{equal}}\:\boldsymbol{\mathrm{squares}}\:\boldsymbol{\mathrm{as}}\:\boldsymbol{\mathrm{large}}\:\boldsymbol{\mathrm{as}}\:\boldsymbol{\mathrm{possible}} \\ $$$$\:\boldsymbol{\mathrm{are}}\:\boldsymbol{\mathrm{drawn}}\:\boldsymbol{\mathrm{on}}\:\boldsymbol{\mathrm{a}}\:\boldsymbol{\mathrm{rectangular}}\:\boldsymbol{\mathrm{ceiling}}\:\boldsymbol{\mathrm{board}} \\ $$$$\:\boldsymbol{\mathrm{measuring}}\:\mathrm{54}\boldsymbol{\mathrm{cm}}\:\boldsymbol{\mathrm{by}}\:\mathrm{78}\boldsymbol{\mathrm{cm}},\boldsymbol{\mathrm{find}} \\ $$$$\:\left(\boldsymbol{\mathrm{a}}\right)\boldsymbol{\mathrm{The}}\:\boldsymbol{\mathrm{size}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{squares}} \\ $$$$\:\left(\boldsymbol{\mathrm{b}}\right)\boldsymbol{\mathrm{The}}\:\boldsymbol{\mathrm{total}}\:\boldsymbol{\mathrm{number}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{squares}} \\ $$

Question Number 33036    Answers: 1   Comments: 0

If range of f(x)= (x+1)(x+2)(x+3)(x+4)+5 x∈ [−6,6] is [a,b] ,a,b∈N, find a+b ?

$${If}\:{range}\:{of}\: \\ $$$${f}\left({x}\right)=\:\left({x}+\mathrm{1}\right)\left({x}+\mathrm{2}\right)\left({x}+\mathrm{3}\right)\left({x}+\mathrm{4}\right)+\mathrm{5} \\ $$$${x}\in\:\left[−\mathrm{6},\mathrm{6}\right]\:{is}\:\left[{a},{b}\right]\:,{a},{b}\in{N},\:{find}\:{a}+{b}\:? \\ $$

Question Number 33032    Answers: 1   Comments: 5

f:N→R f(1)=2005. and f(1)+f(2)+......+f(n)= n^2 f(n),n>1. Then f(2004)=?

$${f}:{N}\rightarrow{R} \\ $$$${f}\left(\mathrm{1}\right)=\mathrm{2005}. \\ $$$${and}\: \\ $$$${f}\left(\mathrm{1}\right)+{f}\left(\mathrm{2}\right)+......+{f}\left({n}\right)=\:{n}^{\mathrm{2}} \:{f}\left({n}\right),{n}>\mathrm{1}. \\ $$$${Then}\:{f}\left(\mathrm{2004}\right)=? \\ $$

Question Number 33028    Answers: 0   Comments: 0

find the value of∫_0 ^∞ (e^(−[t]) /(t+1))dt .

$${find}\:{the}\:{value}\:{of}\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{e}^{−\left[{t}\right]} }{{t}+\mathrm{1}}{dt}\:\:. \\ $$

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