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Question Number 26751 Answers: 0 Comments: 1

by using fourier serie find the value of Σ_(n=0) ^∝ (1/((2n+1)^2 ))

$${by}\:{using}\:{fourier}\:{serie}\:{find}\:{the}\:{value}\:{of}\:\sum_{{n}=\mathrm{0}} ^{\propto} \:\frac{\mathrm{1}}{\left(\mathrm{2}{n}+\mathrm{1}\right)^{\mathrm{2}} } \\ $$

Question Number 26749 Answers: 0 Comments: 1

let give S_n = Σ_(1≤i<j≤n) (1/(i^2 j^2 )) find lim_(n−>∝) S_n .

$${let}\:{give}\:\:{S}_{{n}} \:=\:\:\sum_{\mathrm{1}\leqslant{i}<{j}\leqslant{n}} \:\:\:\frac{\mathrm{1}}{{i}^{\mathrm{2}} {j}^{\mathrm{2}} }\:\:\:{find}\:{lim}_{{n}−>\propto} \:\:{S}_{{n}} \:\:. \\ $$

Question Number 26738 Answers: 1 Comments: 1

Question Number 26733 Answers: 1 Comments: 1

Question Number 26732 Answers: 0 Comments: 1

Question Number 27002 Answers: 2 Comments: 4

Question Number 26723 Answers: 0 Comments: 2

Take a point on a given circle as a center and draw an arc which divide the given circle into two equal(in area) regions.Use only Eucledean tools.

$$\mathcal{T}{ake}\:{a}\:{point}\:\boldsymbol{{on}}\:{a}\:{given}\:{circle} \\ $$$${as}\:{a}\:{center}\:{and}\:{draw}\:{an}\:{arc} \\ $$$${which}\:{divide}\:{the}\:{given}\:{circle} \\ $$$${into}\:{two}\:{equal}\left(\mathrm{in}\:\mathrm{area}\right)\:{regions}.{Use}\:{only} \\ $$$${Eucledean}\:{tools}. \\ $$

Question Number 26722 Answers: 0 Comments: 0

lim_(x→−1^+ ) (1+x+x^2 +x^3 +...up to ∞)=? lim_(x→−1^− ) (1+x+x^2 +x^3 +...up to ∞)=?

$$\underset{{x}\rightarrow−\mathrm{1}^{+} } {\mathrm{lim}}\left(\mathrm{1}+{x}+{x}^{\mathrm{2}} +{x}^{\mathrm{3}} +...\mathrm{up}\:\mathrm{to}\:\infty\right)=? \\ $$$$\underset{{x}\rightarrow−\mathrm{1}^{−} } {\mathrm{lim}}\left(\mathrm{1}+{x}+{x}^{\mathrm{2}} +{x}^{\mathrm{3}} +...\mathrm{up}\:\mathrm{to}\:\infty\right)=? \\ $$

Question Number 26721 Answers: 0 Comments: 1

−2y(y−12)(y−1)or(y−12)(−2y^2 −2y) both are same.

$$−\mathrm{2}{y}\left({y}−\mathrm{12}\right)\left({y}−\mathrm{1}\right){or}\left({y}−\mathrm{12}\right)\left(−\mathrm{2}{y}^{\mathrm{2}} −\mathrm{2}{y}\right)\:\:{both}\:{are}\:{same}. \\ $$

Question Number 26713 Answers: 0 Comments: 0

2sin (3x)<1

$$\mathrm{2sin}\:\left(\mathrm{3}{x}\right)<\mathrm{1} \\ $$

Question Number 26712 Answers: 1 Comments: 0

If ∣x∣ < 1, then the coefficient of x^n in the expansion of (1+x+x^2 +x^3 +x^4 +...)^2 is

$$\mathrm{If}\:\:\mid{x}\mid\:<\:\mathrm{1},\:\mathrm{then}\:\mathrm{the}\:\mathrm{coefficient}\:\mathrm{of}\:{x}^{{n}} \:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{expansion}\:\mathrm{of}\:\left(\mathrm{1}+{x}+{x}^{\mathrm{2}} +{x}^{\mathrm{3}} +{x}^{\mathrm{4}} +...\right)^{\mathrm{2}} \:\mathrm{is} \\ $$

Question Number 26711 Answers: 0 Comments: 1

ABC is a right tringle.prove it?

$$\mathrm{ABC}\:\mathrm{is}\:\mathrm{a}\:\mathrm{right}\:\mathrm{tringle}.\mathrm{prove}\:\mathrm{it}? \\ $$

Question Number 26703 Answers: 0 Comments: 0

The total number of dissimilar terms in the expansion of (x_1 +x_2 +...+x_n )^3 is

$$\mathrm{The}\:\mathrm{total}\:\mathrm{number}\:\mathrm{of}\:\mathrm{dissimilar}\:\mathrm{terms}\:\mathrm{in} \\ $$$$\mathrm{the}\:\mathrm{expansion}\:\mathrm{of}\:\left({x}_{\mathrm{1}} +{x}_{\mathrm{2}} +...+{x}_{{n}} \right)^{\mathrm{3}} \:\mathrm{is} \\ $$

Question Number 26694 Answers: 1 Comments: 0

divide x^6 −y^6 by the product of x^2 +x^ y+y^(2 ) and x−y.

$${divide}\:{x}^{\mathrm{6}} −{y}^{\mathrm{6}} \:{by}\:{the}\:{product}\:{of}\:{x}^{\mathrm{2}} +{x}^{} {y}+{y}^{\mathrm{2}\:} \:{and}\:{x}−{y}. \\ $$

Question Number 26693 Answers: 0 Comments: 3

STATEMENT-1: The angle between one of the lines represented by ax^2 + 2hxy + by^2 = 0 and one of the lines represented by (a + 2008)x^2 + 2hxy + (b + 2008)y^2 = 0 is equal to angle between other two lines of the system. and STATEMENT-2: The pair of lines given by a^2 x^2 + 2008(a + b)xy + b^2 y^2 = 0 is equally inclined to the pair given by ax^2 + 2008xy + by^2 = 0.

$${STATEMENT}-\mathrm{1}:\:{The}\:{angle}\:{between} \\ $$$${one}\:{of}\:{the}\:{lines}\:{represented}\:{by}\:{ax}^{\mathrm{2}} \:+ \\ $$$$\mathrm{2}{hxy}\:+\:{by}^{\mathrm{2}} \:=\:\mathrm{0}\:{and}\:{one}\:{of}\:{the}\:{lines} \\ $$$${represented}\:{by}\:\left({a}\:+\:\mathrm{2008}\right){x}^{\mathrm{2}} \:+\:\mathrm{2}{hxy} \\ $$$$+\:\left({b}\:+\:\mathrm{2008}\right){y}^{\mathrm{2}} \:=\:\mathrm{0}\:{is}\:{equal}\:{to}\:{angle} \\ $$$${between}\:{other}\:{two}\:{lines}\:{of}\:{the} \\ $$$${system}. \\ $$$$\boldsymbol{{and}} \\ $$$${STATEMENT}-\mathrm{2}:\:{The}\:{pair}\:{of}\:{lines} \\ $$$${given}\:{by}\:{a}^{\mathrm{2}} {x}^{\mathrm{2}} \:+\:\mathrm{2008}\left({a}\:+\:{b}\right){xy}\:+\:{b}^{\mathrm{2}} {y}^{\mathrm{2}} \\ $$$$=\:\mathrm{0}\:{is}\:{equally}\:{inclined}\:{to}\:{the}\:{pair} \\ $$$${given}\:{by}\:{ax}^{\mathrm{2}} \:+\:\mathrm{2008}{xy}\:+\:{by}^{\mathrm{2}} \:=\:\mathrm{0}. \\ $$

Question Number 26686 Answers: 1 Comments: 1

Question Number 26681 Answers: 0 Comments: 1

f:R×R→R such that f(x+iy)=(√(x^2 +y^2 .)) Then f is a) many−one and into function b) one−one and onto function c) many−one and onto function d) one−one and into function

$${f}:{R}×{R}\rightarrow{R}\:{such}\:{that}\:{f}\left({x}+{iy}\right)=\sqrt{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} .} \\ $$$${Then}\:{f}\:{is} \\ $$$$\left.{a}\right)\:{many}−{one}\:{and}\:{into}\:{function} \\ $$$$\left.{b}\right)\:{one}−{one}\:{and}\:{onto}\:{function} \\ $$$$\left.{c}\right)\:{many}−{one}\:{and}\:{onto}\:{function} \\ $$$$\left.{d}\right)\:{one}−{one}\:{and}\:{into}\:{function} \\ $$

Question Number 26680 Answers: 1 Comments: 0

(b−c)x^2 +(c−a)x+(a−b)=0 if the eqation roots are eqal you proved that 2b=a+c.

$$\left({b}−{c}\right){x}^{\mathrm{2}} +\left({c}−{a}\right){x}+\left({a}−{b}\right)=\mathrm{0}\:{if}\:{the}\: \\ $$$${eqation}\:{roots}\:\:{are}\:{eqal}\:{you}\:{proved}\:{that} \\ $$$$\mathrm{2}{b}={a}+{c}. \\ $$

Question Number 26652 Answers: 0 Comments: 2