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Question Number 154662 Answers: 0 Comments: 1

Question Number 154661 Answers: 1 Comments: 0

Question Number 154652 Answers: 1 Comments: 0

lim_(x→0) ((1−tan (x+(π/4))tan (2x+(π/4))tan ((π/4)−3x))/x^3 )=?

$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}−\mathrm{tan}\:\left({x}+\frac{\pi}{\mathrm{4}}\right)\mathrm{tan}\:\left(\mathrm{2}{x}+\frac{\pi}{\mathrm{4}}\right)\mathrm{tan}\:\left(\frac{\pi}{\mathrm{4}}−\mathrm{3}{x}\right)}{{x}^{\mathrm{3}} }=? \\ $$

Question Number 154651 Answers: 1 Comments: 0

let be A = ((1,1),(0,1) ) ; B = ((1,0),(1,1) ) find 𝛀 = e^A ∙ (e^B )^(−1) (e^A - exponential matrix)

$$\mathrm{let}\:\mathrm{be}\:\:\boldsymbol{\mathrm{A}}\:=\:\begin{pmatrix}{\mathrm{1}}&{\mathrm{1}}\\{\mathrm{0}}&{\mathrm{1}}\end{pmatrix}\:\:;\:\:\boldsymbol{\mathrm{B}}\:=\:\begin{pmatrix}{\mathrm{1}}&{\mathrm{0}}\\{\mathrm{1}}&{\mathrm{1}}\end{pmatrix} \\ $$$$\mathrm{find}\:\:\boldsymbol{\Omega}\:=\:\mathrm{e}^{\boldsymbol{\mathrm{A}}} \:\centerdot\:\left(\mathrm{e}^{\boldsymbol{\mathrm{B}}} \right)^{−\mathrm{1}} \\ $$$$\left(\mathrm{e}^{\boldsymbol{\mathrm{A}}} \:-\:\mathrm{exponential}\:\mathrm{matrix}\right) \\ $$

Question Number 154648 Answers: 0 Comments: 0

Question Number 154647 Answers: 1 Comments: 0

f : Q → Q f(x + f(y)) = y + f(x) ∀ x;y ∈ Q

$$\mathrm{f}\::\:\mathrm{Q}\:\rightarrow\:\mathrm{Q} \\ $$$$\mathrm{f}\left(\mathrm{x}\:+\:\mathrm{f}\left(\mathrm{y}\right)\right)\:=\:\mathrm{y}\:+\:\mathrm{f}\left(\mathrm{x}\right) \\ $$$$\forall\:\mathrm{x};\mathrm{y}\:\in\:\mathrm{Q} \\ $$

Question Number 154641 Answers: 2 Comments: 0

∫((1−(√x))/( (√(1−x))))dx

$$\int\frac{\mathrm{1}−\sqrt{{x}}}{\:\sqrt{\mathrm{1}−{x}}}{dx} \\ $$

Question Number 154640 Answers: 2 Comments: 0

Σ_1 ^(89) sin^2 (x)=?

$$\underset{\mathrm{1}} {\overset{\mathrm{89}} {\sum}}{sin}^{\mathrm{2}} \left({x}\right)=? \\ $$

Question Number 154622 Answers: 0 Comments: 0

Question Number 154613 Answers: 0 Comments: 3

There are 3 or more profiles of this person and I mentioned it, he knowes who he and that person paints what I share in red, do his best, nothing will chang!

$$\mathrm{There}\:\mathrm{are}\:\mathrm{3}\:\mathrm{or}\:\mathrm{more}\:\mathrm{profiles}\:\mathrm{of}\:\mathrm{this} \\ $$$$\mathrm{person}\:\mathrm{and}\:\mathrm{I}\:\mathrm{mentioned}\:\mathrm{it},\:\mathrm{he}\:\mathrm{knowes} \\ $$$$\mathrm{who}\:\mathrm{he}\:\mathrm{and}\:\mathrm{that}\:\mathrm{person}\:\mathrm{paints}\:\mathrm{what} \\ $$$$\mathrm{I}\:\mathrm{share}\:\mathrm{in}\:\mathrm{red},\:\mathrm{do}\:\mathrm{his}\:\mathrm{best},\:\mathrm{nothing}\:\mathrm{will} \\ $$$$\mathrm{chang}! \\ $$

Question Number 154621 Answers: 2 Comments: 1

∫ ((√(x^2 +x+1))/(x^2 +x)) dx

$$\:\int\:\frac{\sqrt{{x}^{\mathrm{2}} +{x}+\mathrm{1}}}{{x}^{\mathrm{2}} +{x}}\:{dx}\: \\ $$

Question Number 154599 Answers: 1 Comments: 1

Question Number 154596 Answers: 0 Comments: 0

let x;y;z>0 prove that: (z/( (√(x^2 +y^2 )))) + (√2) ≥ 2 (√(z/(x+y))) + ((√(2xy))/(x+y))

$$\mathrm{let}\:\:\mathrm{x};\mathrm{y};\mathrm{z}>\mathrm{0}\:\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\frac{\mathrm{z}}{\:\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} }}\:+\:\sqrt{\mathrm{2}}\:\geqslant\:\mathrm{2}\:\sqrt{\frac{\mathrm{z}}{\mathrm{x}+\mathrm{y}}}\:+\:\frac{\sqrt{\mathrm{2xy}}}{\mathrm{x}+\mathrm{y}} \\ $$

Question Number 154593 Answers: 1 Comments: 1

Question Number 154594 Answers: 0 Comments: 2

Question Number 154561 Answers: 1 Comments: 0

Question Number 154557 Answers: 0 Comments: 0

∫_0 ^( ∞) (1/((Σ_(n=0) ^(⌈x^2 ⌉) n)^2 )) dx

$$\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\infty} \:\frac{\mathrm{1}}{\left(\underset{{n}=\mathrm{0}} {\overset{\lceil{x}^{\mathrm{2}} \rceil} {\sum}}\:{n}\right)^{\mathrm{2}} }\:{dx} \\ $$$$\: \\ $$

Question Number 154554 Answers: 2 Comments: 1

Question Number 154553 Answers: 0 Comments: 0

Π_(n=1) ^∞ (( Γ(n+ (1/n^2 )) )/( Γ(n+ (1/n)) ))

$$\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\prod}}\:\frac{\:\Gamma\left({n}+\:\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\right)\:}{\:\Gamma\left({n}+\:\frac{\mathrm{1}}{{n}}\right)\:} \\ $$$$\: \\ $$

Question Number 154552 Answers: 2 Comments: 28

Question Number 154549 Answers: 4 Comments: 1

A particle is projected with velocity 2(√(gh)) so that it just clears two walls of equal heigh(h) in the t_1 and t_(2 ) respectively.The two walls are at a distance of 2h from each other.If time passing between the two walls is 2(√(h/g)) show that (i) angle projected 60^° (ii)t_1 +t_2 =2(√((3h)/g))

$$\mathrm{A}\:\mathrm{particle}\:\mathrm{is}\:\mathrm{projected}\:\:\mathrm{with}\:\mathrm{velocity} \\ $$$$\mathrm{2}\sqrt{\mathrm{gh}}\:\:\:\mathrm{so}\:\mathrm{that}\:\:\mathrm{it}\:\mathrm{just}\:\mathrm{clears}\:\mathrm{two} \\ $$$$\mathrm{walls}\:\mathrm{of}\:\mathrm{equal}\:\mathrm{heigh}\left(\mathrm{h}\right)\:\:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{t}_{\mathrm{1}} \:\mathrm{and}\:\mathrm{t}_{\mathrm{2}\:} \:\mathrm{respectively}.\mathrm{The}\:\mathrm{two} \\ $$$$\mathrm{walls}\:\mathrm{are}\:\mathrm{at}\:\mathrm{a}\:\mathrm{distance}\:\mathrm{of}\:\:\mathrm{2h}\:\:\mathrm{from} \\ $$$$\mathrm{each}\:\mathrm{other}.\mathrm{If}\:\mathrm{time}\:\mathrm{passing}\: \\ $$$$\mathrm{between}\:\mathrm{the}\:\mathrm{two}\:\mathrm{walls}\:\mathrm{is}\:\mathrm{2}\sqrt{\frac{\mathrm{h}}{\mathrm{g}}} \\ $$$$\mathrm{show}\:\mathrm{that}\:\left(\mathrm{i}\right)\:\mathrm{angle}\:\mathrm{projected}\:\mathrm{60}^{°} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{ii}\right)\mathrm{t}_{\mathrm{1}} +\mathrm{t}_{\mathrm{2}} =\mathrm{2}\sqrt{\frac{\mathrm{3h}}{\mathrm{g}}} \\ $$$$ \\ $$

Question Number 154547 Answers: 1 Comments: 0

A particle is projected inside the tunnel which is 4m high.if the initial speed is V_o .show that the maximum range inside the tunnel is given by R=4(√2) (√((V_o ^2 /g)−8))

$$\mathrm{A}\:\mathrm{particle}\:\mathrm{is}\:\mathrm{projected}\:\mathrm{inside}\:\mathrm{the}\:\mathrm{tunnel} \\ $$$$\mathrm{which}\:\mathrm{is}\:\mathrm{4m}\:\mathrm{high}.\mathrm{if}\:\:\mathrm{the}\:\mathrm{initial}\:\mathrm{speed} \\ $$$$\mathrm{is}\:\mathrm{V}_{\mathrm{o}} .\mathrm{show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{maximum} \\ $$$$\mathrm{range}\:\mathrm{inside}\:\mathrm{the}\:\mathrm{tunnel}\:\mathrm{is}\:\mathrm{given} \\ $$$$\mathrm{by}\:\:\:\:\:\:\mathrm{R}=\mathrm{4}\sqrt{\mathrm{2}}\:\sqrt{\frac{\mathrm{V}_{\mathrm{o}} ^{\mathrm{2}} }{\mathrm{g}}−\mathrm{8}} \\ $$

Question Number 154573 Answers: 2 Comments: 1

Question Number 154544 Answers: 0 Comments: 0

determinant (((prove that)),((Σ_(n=1) ^∞ ((H_n H_n ^((2)) )/n^3 )+Σ_(n=1) ^∞ ((H_n H_n ^((3)) )/n^2 )=((21)/8)𝛇(6)+𝛇^2 (3)))) by Math.Amin 11.fb.96

$$\begin{array}{|c|c|}{\boldsymbol{\mathrm{prove}}\:\boldsymbol{\mathrm{that}}}\\{\underset{\boldsymbol{\mathrm{n}}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\boldsymbol{\mathrm{H}}_{\boldsymbol{\mathrm{n}}} \boldsymbol{\mathrm{H}}_{\boldsymbol{\mathrm{n}}} ^{\left(\mathrm{2}\right)} }{\boldsymbol{\mathrm{n}}^{\mathrm{3}} }+\underset{\boldsymbol{\mathrm{n}}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\boldsymbol{\mathrm{H}}_{\boldsymbol{\mathrm{n}}} \boldsymbol{\mathrm{H}}_{\boldsymbol{\mathrm{n}}} ^{\left(\mathrm{3}\right)} }{\boldsymbol{\mathrm{n}}^{\mathrm{2}} }=\frac{\mathrm{21}}{\mathrm{8}}\boldsymbol{\zeta}\left(\mathrm{6}\right)+\boldsymbol{\zeta}^{\mathrm{2}} \left(\mathrm{3}\right)}\\\hline\end{array} \\ $$$${by}\:{Math}.{Amin}\:\:\mathrm{11}.{fb}.\mathrm{96} \\ $$

Question Number 154542 Answers: 1 Comments: 0

Question Number 154530 Answers: 0 Comments: 0

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