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Question Number 173820 Answers: 0 Comments: 0

Prove ((cos(π/(19))cos((7π)/(19))cos((8π)/(19))))^(1/3) +((cos((2π)/(19))cos((3π)/(19))cos((5π)/(19))))^(1/3) − −((cos((4π)/(19))cos((6π)/(19))cos((9π)/(19))))^(1/3) =(1/2)((3((19))^(1/3) −1))^(1/3)

$$ \\ $$ Prove $$ \\ $$$$\sqrt[{\mathrm{3}}]{\mathrm{cos}\frac{\pi}{\mathrm{19}}\mathrm{cos}\frac{\mathrm{7}\pi}{\mathrm{19}}\mathrm{cos}\frac{\mathrm{8}\pi}{\mathrm{19}}}+\sqrt[{\mathrm{3}}]{\mathrm{cos}\frac{\mathrm{2}\pi}{\mathrm{19}}\mathrm{cos}\frac{\mathrm{3}\pi}{\mathrm{19}}\mathrm{cos}\frac{\mathrm{5}\pi}{\mathrm{19}}}− \\ $$$$−\sqrt[{\mathrm{3}}]{\mathrm{cos}\frac{\mathrm{4}\pi}{\mathrm{19}}\mathrm{cos}\frac{\mathrm{6}\pi}{\mathrm{19}}\mathrm{cos}\frac{\mathrm{9}\pi}{\mathrm{19}}}=\frac{\mathrm{1}}{\mathrm{2}}\sqrt[{\mathrm{3}}]{\mathrm{3}\sqrt[{\mathrm{3}}]{\mathrm{19}}−\mathrm{1}} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Question Number 173815 Answers: 0 Comments: 2

prove (((n+1)/3))^n <n!

$${prove}\: \\ $$$$\left(\frac{{n}+\mathrm{1}}{\mathrm{3}}\right)^{{n}} <{n}! \\ $$

Question Number 173807 Answers: 1 Comments: 0

Question Number 173798 Answers: 3 Comments: 0

( ϕ^( 5) + ϕ^( 4) + 1)^( 2) = mϕ + n m,n=? ........

$$ \\ $$$$\:\:\:\:\:\:\:\left(\:\varphi^{\:\mathrm{5}} +\:\varphi^{\:\mathrm{4}} +\:\mathrm{1}\right)^{\:\mathrm{2}} =\:{m}\varphi\:+\:{n} \\ $$$$\:\:\:\:\:\:\:\:\:{m},{n}=? \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:........ \\ $$$$\:\:\:\:\:\:\:\:\:\: \\ $$

Question Number 173795 Answers: 1 Comments: 1

Soient a et b deux re^ els. Pour tout n ∈ N on pose u_n =(√n)+a(√(n+1))+b(√(n+2)). 1. Ve^ rifier que la suite (u_n ) tend vers 0 si et seulement si a+b=−1. 2. De^ terminer a et b pour que la se^ rie Σu_n soit convergente.

$$\:\:\:\:\mathrm{Soient}\:{a}\:\mathrm{et}\:{b}\:\mathrm{deux}\:\mathrm{r}\acute {\mathrm{e}els}.\:\mathrm{Pour}\:\mathrm{tout}\:{n}\:\in\:\mathbb{N}\:\mathrm{on}\:\mathrm{pose} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{u}_{{n}} =\sqrt{{n}}+{a}\sqrt{{n}+\mathrm{1}}+{b}\sqrt{{n}+\mathrm{2}}. \\ $$$$\:\:\mathrm{1}.\:\:\mathrm{V}\acute {\mathrm{e}rifier}\:\mathrm{que}\:\mathrm{la}\:\mathrm{suite}\:\left({u}_{{n}} \right)\:\mathrm{tend}\:\mathrm{vers}\:\mathrm{0}\:\mathrm{si}\:\mathrm{et}\:\mathrm{seulement}\:\mathrm{si}\:{a}+{b}=−\mathrm{1}. \\ $$$$\:\:\mathrm{2}.\:\:\mathrm{D}\acute {\mathrm{e}terminer}\:{a}\:\mathrm{et}\:{b}\:\mathrm{pour}\:\mathrm{que}\:\mathrm{la}\:\mathrm{s}\acute {\mathrm{e}rie}\:\Sigma{u}_{{n}} \:\mathrm{soit}\:\mathrm{convergente}.\:\: \\ $$

Question Number 173787 Answers: 2 Comments: 11

4sin^( 2) (x)− 4sin(x)=cos^( 2) (x)−2cos(x) , 0<x<(π/2) . Find sin(x)=?

$$\:\mathrm{4}{sin}^{\:\mathrm{2}} \left({x}\right)−\:\mathrm{4}{sin}\left({x}\right)={cos}^{\:\mathrm{2}} \left({x}\right)−\mathrm{2}{cos}\left({x}\right) \\ $$$$\:\:\:,\:\mathrm{0}<{x}<\frac{\pi}{\mathrm{2}}\:.\:\:\:{Find}\:\:\:\:\:\:\:{sin}\left({x}\right)=? \\ $$

Question Number 173776 Answers: 2 Comments: 0

Question Number 173767 Answers: 0 Comments: 2

prouve ,nεN ∫_a ^a (t−a)(t−b)=(((b−a)^(n+2) )/((n+1)(n+2)))

$${prouve}\:,{n}\epsilon{N} \\ $$$$\int_{{a}} ^{{a}} \left({t}−{a}\right)\left({t}−{b}\right)=\frac{\left({b}−{a}\right)^{{n}+\mathrm{2}} }{\left({n}+\mathrm{1}\right)\left({n}+\mathrm{2}\right)} \\ $$

Question Number 173765 Answers: 4 Comments: 0

Question Number 176804 Answers: 0 Comments: 1

Question Number 173736 Answers: 1 Comments: 0

Show that x^2 + 2xy + 3y^2 = 1, is a solution to the differential equation (x + 3y)^2 (d^2 y/dx^2 ) + 2(x^2 + 2xy + 2y^3 ) = 0

$$\mathrm{Show}\:\mathrm{that}\:\:\:\:\:\mathrm{x}^{\mathrm{2}} \:\:\:+\:\:\:\mathrm{2xy}\:\:\:+\:\:\:\mathrm{3y}^{\mathrm{2}} \:\:\:=\:\:\:\mathrm{1},\:\:\:\:\mathrm{is}\:\mathrm{a}\:\mathrm{solution}\:\mathrm{to}\:\mathrm{the} \\ $$$$\mathrm{differential}\:\mathrm{equation}\:\:\:\:\:\left(\mathrm{x}\:\:\:+\:\:\:\mathrm{3y}\right)^{\mathrm{2}} \:\frac{\mathrm{d}^{\mathrm{2}} \mathrm{y}}{\mathrm{dx}^{\mathrm{2}} }\:\:\:\:\:+\:\:\:\mathrm{2}\left(\mathrm{x}^{\mathrm{2}} \:\:\:+\:\:\:\mathrm{2xy}\:\:\:+\:\:\:\mathrm{2y}^{\mathrm{3}} \right)\:\:\:=\:\:\:\mathrm{0} \\ $$

Question Number 173734 Answers: 1 Comments: 0

lim_( n→∞) (n +2) ∫_0 ^( 1) x^( n) ln( 1+x )dx=?

$$ \\ $$$$\:\mathrm{lim}_{\:{n}\rightarrow\infty} \:\left({n}\:+\mathrm{2}\right)\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \:{x}^{\:{n}} \:\mathrm{ln}\left(\:\mathrm{1}+{x}\:\right){dx}=? \\ $$$$ \\ $$

Question Number 173733 Answers: 0 Comments: 3

Show that x^3 + y^3 = 1 is a solution to the differential equation 20x^3 + 3y^2 (d^2 y/dx^2 ) + 6y((dy/dx))^2 = 0

$$\mathrm{Show}\:\mathrm{that}\:\:\:\:\mathrm{x}^{\mathrm{3}} \:\:+\:\:\mathrm{y}^{\mathrm{3}} \:\:=\:\:\mathrm{1}\:\:\:\:\mathrm{is}\:\mathrm{a}\:\mathrm{solution}\:\mathrm{to}\:\mathrm{the}\:\mathrm{differential} \\ $$$$\mathrm{equation}\:\:\:\:\mathrm{20x}^{\mathrm{3}} \:\:\:+\:\:\:\mathrm{3y}^{\mathrm{2}} \:\frac{\mathrm{d}^{\mathrm{2}} \mathrm{y}}{\mathrm{dx}^{\mathrm{2}} }\:\:\:\:+\:\:\:\mathrm{6y}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)^{\mathrm{2}} \:\:\:=\:\:\:\:\mathrm{0} \\ $$

Question Number 173732 Answers: 4 Comments: 1

solve for x∈R (√(x^2 −4))+2(√(x^2 −1))=x^2

$${solve}\:{for}\:{x}\in{R} \\ $$$$\sqrt{{x}^{\mathrm{2}} −\mathrm{4}}+\mathrm{2}\sqrt{{x}^{\mathrm{2}} −\mathrm{1}}={x}^{\mathrm{2}} \\ $$

Question Number 173729 Answers: 2 Comments: 0

Evaluate lim_(x→0) ((x−(1/( (√x)−x))+1)/(x^2 +((√x)/( (√x)−1))−1))

$${Evaluate} \\ $$$${lim}_{{x}\rightarrow\mathrm{0}} \frac{{x}−\frac{\mathrm{1}}{\:\sqrt{{x}}−{x}}+\mathrm{1}}{{x}^{\mathrm{2}} +\frac{\sqrt{{x}}}{\:\sqrt{{x}}−\mathrm{1}}−\mathrm{1}} \\ $$

Question Number 173768 Answers: 0 Comments: 0

Total factors of 20! = ?

$${Total}\:{factors}\:\:{of}\:\:\mathrm{20}!\:=\:? \\ $$

Question Number 173721 Answers: 1 Comments: 0

Question Number 173709 Answers: 0 Comments: 0

In triangle △ABC prove: cos(∡(A/2))>(1/4)∙((2a+b+c)/( (√(R(R+2r_a )))))

$$\mathrm{In}\:\mathrm{triangle}\:\bigtriangleup\mathrm{ABC}\:\mathrm{prove}: \\ $$$$\mathrm{cos}\left(\measuredangle\frac{\mathrm{A}}{\mathrm{2}}\right)>\frac{\mathrm{1}}{\mathrm{4}}\centerdot\frac{\mathrm{2a}+\mathrm{b}+\mathrm{c}}{\:\sqrt{\mathrm{R}\left(\mathrm{R}+\mathrm{2r}_{\mathrm{a}} \right)}} \\ $$

Question Number 173707 Answers: 3 Comments: 0

Let x,y∈R such that 2x^2 +3y^2 =5 Find the minimum and maximum of expression: P=x^3 −y^3 +x−2y

$$\mathrm{Let}\:\mathrm{x},\mathrm{y}\in\mathbb{R}\:\mathrm{such}\:\mathrm{that}\:\mathrm{2x}^{\mathrm{2}} +\mathrm{3y}^{\mathrm{2}} =\mathrm{5} \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{minimum}\:\mathrm{and} \\ $$$$\mathrm{maximum}\:\mathrm{of}\:\mathrm{expression}: \\ $$$$\mathrm{P}=\mathrm{x}^{\mathrm{3}} −\mathrm{y}^{\mathrm{3}} +\mathrm{x}−\mathrm{2y} \\ $$

Question Number 173706 Answers: 2 Comments: 1

Question Number 173697 Answers: 1 Comments: 0

Question Number 173696 Answers: 0 Comments: 0

Question Number 173695 Answers: 1 Comments: 0

Question Number 176812 Answers: 1 Comments: 1

tan(a−b)=x and tan(a+b)=y then tan2α=?

$${tan}\left({a}−{b}\right)={x}\:\:\:\:{and}\:\:\:{tan}\left({a}+{b}\right)={y} \\ $$$${then}\:\:\:\:{tan}\mathrm{2}\alpha=? \\ $$

Question Number 176811 Answers: 0 Comments: 1

Question Number 176806 Answers: 0 Comments: 0

Σ_(k=0) ^(2n) cos^(2n) (θ+((kπ)/(2n)))

$$\underset{{k}=\mathrm{0}} {\overset{\mathrm{2}{n}} {\sum}}{cos}^{\mathrm{2}{n}} \left(\theta+\frac{{k}\pi}{\mathrm{2}{n}}\right) \\ $$

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