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Question Number 164599 Answers: 0 Comments: 1

180<θ<270 and 2sinθ−cos θ=0 faind volue of sin θ×cos θ=?

$$\mathrm{180}<\theta<\mathrm{270}\:\:\:\:{and} \\ $$$$\mathrm{2}{sin}\theta−\mathrm{cos}\:\theta=\mathrm{0} \\ $$$${faind}\:\:\:{volue}\:{of} \\ $$$$\mathrm{sin}\:\theta×\mathrm{cos}\:\theta=? \\ $$

Question Number 164598 Answers: 0 Comments: 0

Prove that; ∫_(−∞) ^∞ y tan x + y^3 tan x dx = undefined

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{Prove}}\:\boldsymbol{\mathrm{that}}; \\ $$$$\:\:\:\:\:\:\int_{−\infty} ^{\infty} \:\boldsymbol{{y}}\:\boldsymbol{{tan}}\:\boldsymbol{{x}}\:+\:\boldsymbol{{y}}^{\mathrm{3}} \:\:\boldsymbol{{tan}}\:\:\boldsymbol{{x}}\:\boldsymbol{{dx}}\:=\:\boldsymbol{{undefined}} \\ $$

Question Number 164591 Answers: 1 Comments: 0

pour quelle valeur α la serie converge Σ_(n=2) (ln(n)+αln(n−(1/n))

$${pour}\:{quelle}\:{valeur}\:\alpha\:{la}\:{serie}\:{converge} \\ $$$$\underset{{n}=\mathrm{2}} {\sum}\left({ln}\left({n}\right)+\alpha{ln}\left({n}−\frac{\mathrm{1}}{{n}}\right)\right. \\ $$

Question Number 164590 Answers: 1 Comments: 0

∫ ((In(x^2 .e^(cos2) ))/x) dx

$$\int\:\frac{\mathrm{In}\left(\mathrm{x}^{\mathrm{2}} .\boldsymbol{{e}}^{\boldsymbol{{cos}}\mathrm{2}} \right)}{\boldsymbol{\mathrm{x}}}\:\boldsymbol{\mathrm{dx}} \\ $$

Question Number 164589 Answers: 0 Comments: 0

∫ A.^5 (√(x^3 )) dx

$$\int\:\mathrm{A}.\:^{\mathrm{5}} \sqrt{\boldsymbol{\mathrm{x}}^{\mathrm{3}} \:\:}\:\boldsymbol{\mathrm{dx}} \\ $$

Question Number 164588 Answers: 0 Comments: 0

soit K un corps; pour toute permutation σ de S_n , on note P(σ) sa matrice dans la base canonique de K^n . montrer que deux permutations σ_1 et σ_2 sont conjugues dans S_n si et seulement si P(σ_1 ) et P(σ_2 ) sont semblables.

$${soit}\:{K}\:{un}\:{corps};\:{pour}\:{toute}\:{permutation} \\ $$$$\sigma\:{de}\:{S}_{{n}} ,\:{on}\:{note}\:{P}\left(\sigma\right)\:{sa}\:{matrice}\:{dans}\:{la}\:{base} \\ $$$${canonique}\:{de}\:{K}^{{n}} . \\ $$$${montrer}\:{que}\:{deux}\:{permutations}\:\sigma_{\mathrm{1}} \:{et}\:\sigma_{\mathrm{2}} \:{sont} \\ $$$${conjugues}\:{dans}\:{S}_{{n}} \:{si}\:{et}\:{seulement}\:{si}\: \\ $$$${P}\left(\sigma_{\mathrm{1}} \right)\:{et}\:{P}\left(\sigma_{\mathrm{2}} \right)\:{sont}\:{semblables}. \\ $$

Question Number 164585 Answers: 2 Comments: 0

I = ∫_0 ^( 1) (( Li_( 2) ( x ))/(1 + x)) dx = ? −−−−−−

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\mathcal{I}\:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:\mathrm{Li}_{\:\mathrm{2}} \:\left(\:{x}\:\right)}{\mathrm{1}\:+\:{x}}\:{dx}\:=\:? \\ $$$$\:\:\:\:−−−−−−\: \\ $$

Question Number 164576 Answers: 1 Comments: 0

Question Number 164568 Answers: 2 Comments: 0

(1/( (√(2−x)))) > (1/(x−1))

$$\:\:\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}−{x}}}\:>\:\frac{\mathrm{1}}{{x}−\mathrm{1}} \\ $$

Question Number 164564 Answers: 0 Comments: 0

when F does not contain y explicity . Find the extremals of ∫_x_1 ^x_2 ((y′^2 )/x^2 )dx

$$ \\ $$$${when}\:{F}\:{does}\:{not}\:{contain}\:{y}\:{explicity}\:.\:{Find}\:{the}\:{extremals}\:{of} \\ $$$$\underset{{x}_{\mathrm{1}} } {\overset{{x}_{\mathrm{2}} } {\int}}\frac{{y}'^{\mathrm{2}} }{{x}^{\mathrm{2}} }{dx} \\ $$

Question Number 164560 Answers: 2 Comments: 1

6 boys and 6 girls go to an exhibition and the cost of ticket is Rs 10.Each girl has a 10 rupees note while each boy has a 20 rupees note. They stand in a queue at the counter and the cashier does not have any money at the begining , then the number of ways of arranging the boys and girls so that no one waits for a change is A) 132 B) 264 C) 132(720)^2 D)264(720)^2

$$\mathrm{6}\:\mathrm{boys}\:\mathrm{and}\:\mathrm{6}\:\mathrm{girls}\:\mathrm{go}\:\mathrm{to}\:\mathrm{an}\:\mathrm{exhibition}\:\mathrm{and}\:\mathrm{the}\:\mathrm{cost} \\ $$$$\mathrm{of}\:\mathrm{ticket}\:\mathrm{is}\:\mathrm{Rs}\:\mathrm{10}.\mathrm{Each}\:\mathrm{girl}\:\mathrm{has}\:\mathrm{a}\:\mathrm{10}\:\mathrm{rupees}\:\mathrm{note} \\ $$$$\mathrm{while}\:\mathrm{each}\:\mathrm{boy}\:\mathrm{has}\:\mathrm{a}\:\mathrm{20}\:\mathrm{rupees}\:\mathrm{note}.\:\mathrm{They}\:\mathrm{stand}\: \\ $$$$\mathrm{in}\:\mathrm{a}\:\mathrm{queue}\:\mathrm{at}\:\mathrm{the}\:\mathrm{counter}\:\mathrm{and}\:\mathrm{the}\:\mathrm{cashier}\:\mathrm{does} \\ $$$$\mathrm{not}\:\mathrm{have}\:\mathrm{any}\:\mathrm{money}\:\mathrm{at}\:\mathrm{the}\:\mathrm{begining}\:,\:\mathrm{then}\:\mathrm{the}\: \\ $$$$\mathrm{number}\:\mathrm{of}\:\mathrm{ways}\:\mathrm{of}\:\mathrm{arranging}\:\mathrm{the}\:\mathrm{boys}\:\mathrm{and}\:\mathrm{girls} \\ $$$$\mathrm{so}\:\mathrm{that}\:\mathrm{no}\:\mathrm{one}\:\mathrm{waits}\:\mathrm{for}\:\mathrm{a}\:\mathrm{change}\:\mathrm{is} \\ $$$$\left.\mathrm{A}\left.\right)\left.\:\left.\mathrm{132}\:\:\:\:\:\mathrm{B}\right)\:\:\:\:\mathrm{264}\:\mathrm{C}\right)\:\:\mathrm{132}\left(\mathrm{720}\right)^{\mathrm{2}} \:\:\:\:\mathrm{D}\right)\mathrm{264}\left(\mathrm{720}\right)^{\mathrm{2}} \\ $$

Question Number 164555 Answers: 2 Comments: 0

prove that ( _( k−1) ^(2k) ) =^? Σ_(r=0) ^(k−1) ( _( r) ^( k) ) (^( ) _( r+1) ^( k) ) −−−−

$$ \\ $$$$\:\:\:{prove}\:\:{that} \\ $$$$\: \\ $$$$\:\:\:\:\:\:\left(\underset{\:{k}−\mathrm{1}} {\overset{\mathrm{2}{k}} {\:}}\:\right)\:\overset{?} {=}\:\underset{{r}=\mathrm{0}} {\overset{{k}−\mathrm{1}} {\sum}}\:\left(\:\underset{\:{r}} {\overset{\:{k}} {\:}}\:\:\:\right)\:\overset{\:\:} {\left(}\underset{\:{r}+\mathrm{1}} {\overset{\:\:{k}} {\:}}\:\right) \\ $$$$\:\:−−−− \\ $$

Question Number 164553 Answers: 3 Comments: 1

Question Number 164549 Answers: 2 Comments: 0

Question Number 164547 Answers: 1 Comments: 0

prove Ω= ∫_0 ^( ∞) (( (√x))/(( 1+x +x^( 2) )^( 3) )) dx =^? ((π(√3))/(36)) −−m.n−−

$$ \\ $$$$\:\:\:\:\:\:\:\:{prove} \\ $$$$\: \\ $$$$\:\:\Omega=\:\int_{\mathrm{0}} ^{\:\infty} \frac{\:\sqrt{{x}}}{\left(\:\mathrm{1}+{x}\:+{x}^{\:\mathrm{2}} \right)^{\:\mathrm{3}} \:}\:{dx}\:\overset{?} {=}\:\frac{\pi\sqrt{\mathrm{3}}}{\mathrm{36}}\: \\ $$$$\:\:\:\:\:\:−−{m}.{n}−−\: \\ $$$$ \\ $$

Question Number 164546 Answers: 1 Comments: 4

Question Number 164544 Answers: 2 Comments: 0

prove that Σ_(k=1) ^n ((n),(k) )^2 =(((2n)!)/((n!)^2 ))−1

$${prove}\:{that}\:\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\begin{pmatrix}{{n}}\\{{k}}\end{pmatrix}^{\mathrm{2}} =\frac{\left(\mathrm{2}{n}\right)!}{\left({n}!\right)^{\mathrm{2}} }−\mathrm{1} \\ $$

Question Number 164543 Answers: 1 Comments: 0

((4(√(1−x)))/x) +((√(2x−1))/(1−x)) = 3(√3)

$$\:\:\frac{\mathrm{4}\sqrt{\mathrm{1}−{x}}}{{x}}\:+\frac{\sqrt{\mathrm{2}{x}−\mathrm{1}}}{\mathrm{1}−{x}}\:=\:\mathrm{3}\sqrt{\mathrm{3}} \\ $$

Question Number 164538 Answers: 0 Comments: 0

Question Number 164533 Answers: 2 Comments: 0

Prove, that 1) Σ_(k=1) ^α (1/(k(k+1))) = (α/(α+1)) 2) lim_(x→∞) Σ_(k=1) ^∞ (1/(k(k+x))) = 0

$$\mathrm{Prove},\:\mathrm{that} \\ $$$$\left.\mathrm{1}\right)\:\underset{{k}=\mathrm{1}} {\overset{\alpha} {\sum}}\frac{\mathrm{1}}{{k}\left({k}+\mathrm{1}\right)}\:=\:\frac{\alpha}{\alpha+\mathrm{1}} \\ $$$$\left.\mathrm{2}\right)\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\underset{{k}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{k}\left({k}+{x}\right)}\:=\:\mathrm{0} \\ $$

Question Number 164530 Answers: 1 Comments: 1

Question Number 164524 Answers: 1 Comments: 0

Question Number 164520 Answers: 0 Comments: 0

A circle is drawn with center (0,1) and radius 1. A line OAB is drawn, making an angle θ with the x-axis to cut the circle at A and the tangent to the circle at (0,2) at B. Lines are now drawn through A and B parallel to the x- and y-axes respectively to intersect at P. Prove that (i) OA=2 sin θ and (ii)the coordinates of P are (2 cot θ, 2 sin^2 θ) Hence, find the Cartesian equation of the locus of P.

$$\mathrm{A}\:\mathrm{circle}\:\mathrm{is}\:\mathrm{drawn}\:\mathrm{with}\:\mathrm{center}\:\left(\mathrm{0},\mathrm{1}\right)\:\mathrm{and}\:\mathrm{radius}\:\mathrm{1}. \\ $$$$\mathrm{A}\:\mathrm{line}\:\mathrm{OAB}\:\mathrm{is}\:\mathrm{drawn},\:\mathrm{making}\:\mathrm{an}\:\mathrm{angle}\:\theta\:\mathrm{with}\:\mathrm{the} \\ $$$${x}-\mathrm{axis}\:\mathrm{to}\:\mathrm{cut}\:\mathrm{the}\:\mathrm{circle}\:\mathrm{at}\:{A}\:\mathrm{and}\:\mathrm{the}\:\mathrm{tangent}\:\mathrm{to}\:\mathrm{the} \\ $$$$\mathrm{circle}\:\mathrm{at}\:\left(\mathrm{0},\mathrm{2}\right)\:\mathrm{at}\:{B}.\:\mathrm{Lines}\:\mathrm{are}\:\mathrm{now}\:\mathrm{drawn}\:\mathrm{through} \\ $$$${A}\:\mathrm{and}\:{B}\:\:\mathrm{parallel}\:\mathrm{to}\:\mathrm{the}\:{x}-\:\mathrm{and}\:{y}-\mathrm{axes}\:\mathrm{respectively} \\ $$$$\mathrm{to}\:\mathrm{intersect}\:\mathrm{at}\:{P}.\:\:\:\mathrm{Prove}\:\mathrm{that} \\ $$$$\left(\mathrm{i}\right)\:\mathrm{OA}=\mathrm{2}\:\mathrm{sin}\:\theta\:\:\:\:\:\mathrm{and}\: \\ $$$$\left(\mathrm{ii}\right)\mathrm{the}\:\mathrm{coordinates}\:\mathrm{of}\:{P}\:\mathrm{are}\:\left(\mathrm{2}\:\mathrm{cot}\:\theta,\:\mathrm{2}\:\mathrm{sin}^{\mathrm{2}} \theta\right) \\ $$$$\mathrm{Hence},\:\mathrm{find}\:\mathrm{the}\:\mathrm{Cartesian}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{the}\:\mathrm{locus}\:\mathrm{of}\:{P}. \\ $$

Question Number 164542 Answers: 0 Comments: 0

((x^2 −x)/(3−2x)) + (1/2) < (5/(−3x+x^2 −2))

$$\:\:\:\:\:\:\:\frac{{x}^{\mathrm{2}} −{x}}{\mathrm{3}−\mathrm{2}{x}}\:+\:\frac{\mathrm{1}}{\mathrm{2}}\:<\:\frac{\mathrm{5}}{−\mathrm{3}{x}+{x}^{\mathrm{2}} −\mathrm{2}} \\ $$

Question Number 164515 Answers: 0 Comments: 3

Question Number 164511 Answers: 1 Comments: 0

sin∙(π sin x) - cos∙(π sin x) = 1 find x=?

$$\mathrm{sin}\centerdot\left(\pi\:\mathrm{sin}\:\mathrm{x}\right)\:-\:\mathrm{cos}\centerdot\left(\pi\:\mathrm{sin}\:\mathrm{x}\right)\:=\:\mathrm{1} \\ $$$$\mathrm{find}\:\:\boldsymbol{\mathrm{x}}=? \\ $$

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