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Question Number 114967    Answers: 0   Comments: 1

Question Number 114965    Answers: 1   Comments: 0

find all irational numbers x such that x 2+20x+20 and x^(3 ) −2020x+1 both is a rasional number.

$${find}\:{all}\:{irational}\:{numbers}\:{x}\:{such}\:{that}\:{x} \\ $$$$\mathrm{2}+\mathrm{20}{x}+\mathrm{20}\:{and}\:{x}^{\mathrm{3}\:} −\mathrm{2020}{x}+\mathrm{1}\:\:\:{both}\:{is}\:{a} \\ $$$${rasional}\:{number}. \\ $$

Question Number 114996    Answers: 2   Comments: 1

...nice mathematics... prove that::: i:: Σ_(n=1) ^∞ (1/(sinh^2 (πn))) =(1/6) −(1/(2π)) ✓ ii:: Σ_(n=1) ^∞ (n/(e^(2πn) −1))=(1/(24)) −(1/(8π)) ✓✓ iii::Σ_(n=1) ^∞ (1/( nsinh(πn))) =(π/(12))−((ln(2))/4) ✓✓✓ .... M..n..july..1970 ....

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...{nice}\:\:\:{mathematics}...\: \\ $$$$\:\:\:\:{prove}\:{that}::: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{i}::\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{sinh}^{\mathrm{2}} \left(\pi{n}\right)}\:=\frac{\mathrm{1}}{\mathrm{6}}\:−\frac{\mathrm{1}}{\mathrm{2}\pi}\:\:\:\:\checkmark \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{ii}::\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{n}}{{e}^{\mathrm{2}\pi{n}} −\mathrm{1}}=\frac{\mathrm{1}}{\mathrm{24}}\:−\frac{\mathrm{1}}{\mathrm{8}\pi}\:\:\checkmark\checkmark \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{iii}::\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\:{nsinh}\left(\pi{n}\right)}\:=\frac{\pi}{\mathrm{12}}−\frac{{ln}\left(\mathrm{2}\right)}{\mathrm{4}}\:\checkmark\checkmark\checkmark \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:....\:\:\:\:\mathscr{M}..{n}..{july}..\mathrm{1970}\:.... \\ $$$$ \\ $$

Question Number 114959    Answers: 2   Comments: 0

long time question proposed by math abdo ∫_0 ^∞ ((lnx)/(1+x^2 +x^4 ))dx

$${long}\:{time}\:{question}\:{proposed}\:{by} \\ $$$${math}\:{abdo} \\ $$$$\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{ln}{x}}{\mathrm{1}+{x}^{\mathrm{2}} +{x}^{\mathrm{4}} }{dx} \\ $$$$ \\ $$

Question Number 114956    Answers: 0   Comments: 5

Question Number 114950    Answers: 0   Comments: 0

note the tringle ABC is not isosceles with the elevtions of AA1,BB1, and CC1.suppose BA amd CA respectively point at BB1 and CC1 so that A1BA is pependiculer to BB1 and A1CA perpendiculer to CC1.the read and BC lines intersect at the TA point. define in the same way TB and TC poits. prove that TA,TB,and TC are collinear.

$${note}\:{the}\:{tringle}\:{ABC}\:\:{is}\:{not}\:{isosceles}\:{with}\: \\ $$$${the}\:{elevtions}\:{of}\:{AA}\mathrm{1},{BB}\mathrm{1},\:{and}\:{CC}\mathrm{1}.{suppose} \\ $$$${BA}\:\:{amd}\:{CA}\:{respectively}\:{point}\:{at}\:{BB}\mathrm{1}\:{and} \\ $$$${CC}\mathrm{1}\:{so}\:{that}\:{A}\mathrm{1}{BA}\:{is}\:{pependiculer}\:{to}\:{BB}\mathrm{1} \\ $$$${and}\:{A}\mathrm{1}{CA}\:{perpendiculer}\:{to}\:{CC}\mathrm{1}.{the}\:{read}\: \\ $$$${and}\:{BC}\:{lines}\:{intersect}\:{at}\:{the}\:{TA}\:{point}. \\ $$$${define}\:{in}\:{the}\:{same}\:{way}\:\:{TB}\:{and}\:{TC}\:{poits}. \\ $$$${prove}\:{that}\:{TA},{TB},{and}\:{TC}\:{are}\:{collinear}. \\ $$

Question Number 114945    Answers: 1   Comments: 0

Question Number 114943    Answers: 1   Comments: 0

If ^m C_1 =^n C_2 , express m in terms of n.

$$\mathrm{If}\:\:^{{m}} {C}_{\mathrm{1}} =\:^{{n}} {C}_{\mathrm{2}} \:, \\ $$$$\mathrm{express}\:{m}\:\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:{n}. \\ $$

Question Number 114971    Answers: 0   Comments: 2

Question Number 114933    Answers: 2   Comments: 0

find the value of ∫_0 ^∞ ((cos(2x))/(x^4 +x^2 +1))dx

$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{\mathrm{cos}\left(\mathrm{2x}\right)}{\mathrm{x}^{\mathrm{4}} \:+\mathrm{x}^{\mathrm{2}} \:+\mathrm{1}}\mathrm{dx} \\ $$

Question Number 114922    Answers: 2   Comments: 0

find minimum value of function y=(√((x+6)^2 +25)) +(√((x−6)^2 +121))

$${find}\:{minimum}\:{value}\:{of}\:{function} \\ $$$${y}=\sqrt{\left({x}+\mathrm{6}\right)^{\mathrm{2}} +\mathrm{25}}\:+\sqrt{\left({x}−\mathrm{6}\right)^{\mathrm{2}} +\mathrm{121}} \\ $$

Question Number 114919    Answers: 0   Comments: 0

Question Number 114912    Answers: 0   Comments: 0

Question Number 114917    Answers: 0   Comments: 3

Question Number 114906    Answers: 3   Comments: 1

Question Number 114880    Answers: 1   Comments: 1

∫ln (sin (x))dx=?

$$\int\mathrm{ln}\:\left(\mathrm{sin}\:\left({x}\right)\right){dx}=? \\ $$

Question Number 114879    Answers: 1   Comments: 0

If the product of the matrices (((1 1)),((0 1)) ) (((1 2)),((0 1)) ) (((1 3)),((0 1)) )... (((1 k)),((0 1)) )= (((1 378)),((0 1)) ) then k =

$${If}\:{the}\:{product}\:{of}\:{the}\:{matrices}\: \\ $$$$\begin{pmatrix}{\mathrm{1}\:\:\:\mathrm{1}}\\{\mathrm{0}\:\:\:\mathrm{1}}\end{pmatrix}\begin{pmatrix}{\mathrm{1}\:\:\:\:\mathrm{2}}\\{\mathrm{0}\:\:\:\:\mathrm{1}}\end{pmatrix}\begin{pmatrix}{\mathrm{1}\:\:\:\:\:\mathrm{3}}\\{\mathrm{0}\:\:\:\:\:\mathrm{1}}\end{pmatrix}...\begin{pmatrix}{\mathrm{1}\:\:\:\:\:{k}}\\{\mathrm{0}\:\:\:\:\:\mathrm{1}}\end{pmatrix}=\begin{pmatrix}{\mathrm{1}\:\:\:\:\mathrm{378}}\\{\mathrm{0}\:\:\:\:\:\:\:\:\mathrm{1}}\end{pmatrix} \\ $$$${then}\:{k}\:=\: \\ $$

Question Number 114878    Answers: 2   Comments: 0

Given f(x) = ∫_0 ^x (dt/( (√(1+t^3 )))) and g(x) be the inverse function of f(x), then g ′′(x)=λg^2 (x). then the value of λ =

$${Given}\:{f}\left({x}\right)\:=\:\underset{\mathrm{0}} {\overset{{x}} {\int}}\:\frac{{dt}}{\:\sqrt{\mathrm{1}+{t}^{\mathrm{3}} }}\:{and}\:{g}\left({x}\right)\:{be}\:{the} \\ $$$${inverse}\:{function}\:{of}\:{f}\left({x}\right),\:{then}\:{g}\:''\left({x}\right)=\lambda{g}^{\mathrm{2}} \left({x}\right). \\ $$$${then}\:{the}\:{value}\:{of}\:\lambda\:= \\ $$

Question Number 114876    Answers: 2   Comments: 1

Question Number 114875    Answers: 2   Comments: 0

A man sent 7 letters to his 7 friend . the letters are kept in addressed envelopes at random. the probability that 3 friends receive correct letters and 4 letters go to wrong destination is _ (old question unanswered)

$${A}\:{man}\:{sent}\:\mathrm{7}\:{letters}\:{to}\:{his}\:\mathrm{7}\:{friend}\:. \\ $$$${the}\:{letters}\:{are}\:{kept}\:{in}\:{addressed}\:{envelopes} \\ $$$${at}\:{random}.\:{the}\:{probability}\:{that}\:\mathrm{3}\:{friends} \\ $$$${receive}\:{correct}\:{letters}\:{and}\:\mathrm{4}\:{letters}\:{go} \\ $$$${to}\:{wrong}\:{destination}\:{is}\:\_ \\ $$$$ \\ $$$$\left({old}\:{question}\:{unanswered}\right) \\ $$

Question Number 114863    Answers: 1   Comments: 0

find four consecutive multiples of 5 such that twice the sum of the two greatest integer exceed five times the least by 5

$${find}\:{four}\:{consecutive}\:{multiples}\:{of}\:\mathrm{5} \\ $$$${such}\:{that}\:{twice}\:{the}\:{sum}\:{of}\:{the}\:{two} \\ $$$${greatest}\:{integer}\:{exceed}\:{five}\:{times}\:{the} \\ $$$${least}\:{by}\:\mathrm{5} \\ $$

Question Number 114860    Answers: 0   Comments: 3

Question Number 114858    Answers: 0   Comments: 0

prove Σ_(k=1) ^∞ (((H_k )^2 )/(k2^k ))=((7ζ(3))/8)

$${prove} \\ $$$$\underset{{k}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left({H}_{{k}} \right)^{\mathrm{2}} }{{k}\mathrm{2}^{{k}} }=\frac{\mathrm{7}\zeta\left(\mathrm{3}\right)}{\mathrm{8}} \\ $$

Question Number 114857    Answers: 0   Comments: 7

we know that e^(πi) = −1 ⇒ ln (e^(πi) ) = ln(−1) πi = ln (−1). How good is this prove?

$$\mathrm{we}\:\mathrm{know}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\:{e}^{\pi{i}} \:=\:−\mathrm{1}\: \\ $$$$\Rightarrow\:\mathrm{ln}\:\left({e}^{\pi{i}} \right)\:=\:\mathrm{ln}\left(−\mathrm{1}\right) \\ $$$$\:\:\pi{i}\:=\:\mathrm{ln}\:\left(−\mathrm{1}\right).\: \\ $$$$\mathrm{How}\:\mathrm{good}\:\mathrm{is}\:\mathrm{this}\:\mathrm{prove}? \\ $$

Question Number 114850    Answers: 2   Comments: 1

a man hits a golf ball at the top of a cliff which is 40.0 m high. given that the ball falls into a water down cliff and he hears the sound 3 s after he hit the ball. what is the initial speed of the ball. take the speed of sound in air as 343 m/s. neglect air resistance.

$$\mathrm{a}\:\mathrm{man}\:\mathrm{hits}\:\mathrm{a}\:\mathrm{golf}\:\mathrm{ball}\:\mathrm{at}\:\mathrm{the}\:\mathrm{top}\:\mathrm{of}\:\mathrm{a}\:\mathrm{cliff} \\ $$$$\mathrm{which}\:\mathrm{is}\:\mathrm{40}.\mathrm{0}\:\mathrm{m}\:\mathrm{high}.\:\mathrm{given}\:\mathrm{that}\:\mathrm{the} \\ $$$$\mathrm{ball}\:\mathrm{falls}\:\mathrm{into}\:\mathrm{a}\:\mathrm{water}\:\mathrm{down}\:\mathrm{cliff}\:\mathrm{and}\:\mathrm{he} \\ $$$$\mathrm{hears}\:\mathrm{the}\:\mathrm{sound}\:\mathrm{3}\:\mathrm{s}\:\mathrm{after}\:\mathrm{he}\:\mathrm{hit}\:\mathrm{the}\:\mathrm{ball}. \\ $$$$\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{initial}\:\mathrm{speed}\:\mathrm{of}\:\mathrm{the}\:\mathrm{ball}.\:\mathrm{take} \\ $$$$\mathrm{the}\:\mathrm{speed}\:\mathrm{of}\:\mathrm{sound}\:\mathrm{in}\:\mathrm{air}\:\mathrm{as}\:\mathrm{343}\:\mathrm{m}/\mathrm{s}.\:\mathrm{neglect} \\ $$$$\mathrm{air}\:\mathrm{resistance}. \\ $$

Question Number 114840    Answers: 1   Comments: 0

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