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Question Number 45649 Answers: 1 Comments: 1

Let consider A(3,5), B(6,4) and C(3,−2), d : x−5y+7=0 Consider a dot D such as ABCD is a trapezium and (AD) and (BC) are parallel lines. Q1) Give the equation of the line which contains the point D. Q2) Considering that the trapezium should be convex, are all the points D of the lines correct for the Trapezium ? Which ones are ? Give a proof. I have some difficulties to anzwer the question n°2, could someone please, help me. Thank you. H.T.

$$\mathrm{Let}\:\mathrm{consider}\:\mathrm{A}\left(\mathrm{3},\mathrm{5}\right),\:\mathrm{B}\left(\mathrm{6},\mathrm{4}\right)\:\mathrm{and}\:\mathrm{C}\left(\mathrm{3},−\mathrm{2}\right), \\ $$$$\mathrm{d}\::\:{x}−\mathrm{5}{y}+\mathrm{7}=\mathrm{0} \\ $$$$ \\ $$$$\mathrm{Consider}\:\mathrm{a}\:\mathrm{dot}\:\mathrm{D}\:\mathrm{such}\:\mathrm{as}\:\mathrm{ABCD}\:\mathrm{is}\:\mathrm{a}\:\mathrm{trapezium} \\ $$$$\mathrm{and}\:\left(\mathrm{AD}\right)\:\mathrm{and}\:\left(\mathrm{BC}\right)\:\mathrm{are}\:\mathrm{parallel}\:\mathrm{lines}. \\ $$$$ \\ $$$$\left.\mathrm{Q1}\right)\:\:\:\mathrm{Give}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{the}\:\mathrm{line}\:\mathrm{which} \\ $$$$\mathrm{contains}\:\mathrm{the}\:\mathrm{point}\:\mathrm{D}. \\ $$$$ \\ $$$$\left.\mathrm{Q2}\right)\:\:\:\mathrm{Considering}\:\mathrm{that}\:\mathrm{the}\:\mathrm{trapezium} \\ $$$$\mathrm{should}\:\mathrm{be}\:\mathrm{convex},\:\mathrm{are}\:\mathrm{all}\:\mathrm{the}\:\mathrm{points}\:\mathrm{D}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{lines}\:\mathrm{correct}\:\mathrm{for}\:\mathrm{the}\:\mathrm{Trapezium}\:? \\ $$$$\mathrm{Which}\:\mathrm{ones}\:\mathrm{are}\:? \\ $$$$\mathrm{Give}\:\mathrm{a}\:\mathrm{proof}. \\ $$$$ \\ $$$$ \\ $$$$\mathrm{I}\:\mathrm{have}\:\mathrm{some}\:\mathrm{difficulties}\:\mathrm{to}\:\mathrm{anzwer}\:\mathrm{the} \\ $$$$\mathrm{question}\:\mathrm{n}°\mathrm{2},\:\mathrm{could}\:\mathrm{someone}\:\mathrm{please}, \\ $$$$\mathrm{help}\:\mathrm{me}. \\ $$$$ \\ $$$$\mathrm{Thank}\:\mathrm{you}. \\ $$$$ \\ $$$$\mathrm{H}.\mathrm{T}. \\ $$

Question Number 45646 Answers: 0 Comments: 2

According to relativistic theory, E^2 = p^2 c^2 +m_0 ^2 c^4 where m_0 is rest mass. For photon E= pc... (m_0 =0 for photon) For electron E=mc^2 ... Unlike photon ,Why p^2 c^2 is neglected in case of electron ?

$${According}\:{to}\:{relativistic}\:{theory}, \\ $$$${E}^{\mathrm{2}} =\:{p}^{\mathrm{2}} {c}^{\mathrm{2}} +{m}_{\mathrm{0}} ^{\mathrm{2}} {c}^{\mathrm{4}} \:{where}\:{m}_{\mathrm{0}} \:{is}\:{rest}\:{mass}. \\ $$$${For}\:{photon}\:{E}=\:{pc}... \\ $$$$\left({m}_{\mathrm{0}} =\mathrm{0}\:{for}\:{photon}\right) \\ $$$${For}\:{electron}\:{E}={mc}^{\mathrm{2}} ... \\ $$$${Unlike}\:{photon}\:,{Why}\:{p}^{\mathrm{2}} {c}^{\mathrm{2}} \:{is}\:{neglected}\:\: \\ $$$${in}\:{case}\:{of}\:{electron}\:? \\ $$

Question Number 45645 Answers: 1 Comments: 0

Question Number 45641 Answers: 2 Comments: 1

Question Number 45639 Answers: 3 Comments: 0

Question Number 45638 Answers: 1 Comments: 0

Question Number 45635 Answers: 0 Comments: 2

1)find f(x)=∫_0 ^1 ln(1+xt^3 )dt with ∣x∣<1 2) calculate ∫_0 ^1 ln(2+t^3 )dt .

$$\left.\mathrm{1}\right){find}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{1}+{xt}^{\mathrm{3}} \right){dt}\:\:{with}\:\:\mid{x}\mid<\mathrm{1} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{2}+{t}^{\mathrm{3}} \right){dt}\:. \\ $$

Question Number 45634 Answers: 0 Comments: 0

1)find ∫ ln(1−x^6 )dx 2) calculate ∫_0 ^1 ln(1−x^6 )dx

$$\left.\mathrm{1}\right){find}\:\int\:{ln}\left(\mathrm{1}−{x}^{\mathrm{6}} \right){dx} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{1}−{x}^{\mathrm{6}} \right){dx} \\ $$

Question Number 45632 Answers: 0 Comments: 2

1)find ∫ ln(1+x^3 )dx 2) calculate ∫_0 ^1 ln(1+x^3 )ex

$$\left.\mathrm{1}\right){find}\:\int\:{ln}\left(\mathrm{1}+{x}^{\mathrm{3}} \right){dx} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{1}+{x}^{\mathrm{3}} \right){ex} \\ $$

Question Number 45630 Answers: 2 Comments: 0

Question Number 45610 Answers: 0 Comments: 1

Question Number 45609 Answers: 0 Comments: 3

Question Number 45608 Answers: 1 Comments: 0

Question Number 45602 Answers: 2 Comments: 0

Question Number 45601 Answers: 0 Comments: 0

Question Number 45600 Answers: 0 Comments: 2

find f(x,y) =∫_0 ^(π/2) ln(x+y sinθ)dθ with ∣y∣<∣x∣ 2) find f(2,3) 3)find f((√2),(√3)) .

$${find}\:{f}\left({x},{y}\right)\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {ln}\left({x}+{y}\:{sin}\theta\right){d}\theta\:\:{with}\:\:\mid{y}\mid<\mid{x}\mid \\ $$$$\left.\mathrm{2}\right)\:{find}\:{f}\left(\mathrm{2},\mathrm{3}\right) \\ $$$$\left.\mathrm{3}\right){find}\:{f}\left(\sqrt{\mathrm{2}},\sqrt{\mathrm{3}}\right)\:. \\ $$

Question Number 45599 Answers: 0 Comments: 1

1) calculate A_n = ∫_0 ^n (((−1)^([x]) )/(2x+1−[x]))dx 2) find lim_(n→+∞) A_n 3) study the serie Σ A_n

$$\left.\mathrm{1}\right)\:{calculate}\:{A}_{{n}} =\:\int_{\mathrm{0}} ^{{n}} \:\:\:\:\frac{\left(−\mathrm{1}\right)^{\left[{x}\right]} }{\mathrm{2}{x}+\mathrm{1}−\left[{x}\right]}{dx} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{n}\rightarrow+\infty} {A}_{{n}} \\ $$$$\left.\mathrm{3}\right)\:{study}\:{the}\:{serie}\:\:\Sigma\:{A}_{{n}} \\ $$

Question Number 45598 Answers: 0 Comments: 1

calculate Σ_(n=1) ^∞ ((2n+1)/(n^2 (4n^2 −1)))

$${calculate}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\:\frac{\mathrm{2}{n}+\mathrm{1}}{{n}^{\mathrm{2}} \left(\mathrm{4}{n}^{\mathrm{2}} −\mathrm{1}\right)} \\ $$

Question Number 45594 Answers: 0 Comments: 1

calculate Σ_(n=1) ^∞ (1/((4n^2 −1)^2 ))

$${calculate}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\frac{\mathrm{1}}{\left(\mathrm{4}{n}^{\mathrm{2}} −\mathrm{1}\right)^{\mathrm{2}} } \\ $$

Question Number 45592 Answers: 0 Comments: 0

Question Number 45589 Answers: 0 Comments: 0

pl help me sir mathtype 6c seril key

$$\mathrm{pl}\:\mathrm{help}\:\mathrm{me}\:\mathrm{sir}\:\mathrm{mathtype}\:\mathrm{6c}\:\:\mathrm{seril}\:\mathrm{key} \\ $$

Question Number 45585 Answers: 1 Comments: 6

prove that1^3 +2^3 +3^3 +...+n^3 =((n^2 (n+1)^2 )/4) for every natural number n

$$\mathrm{prove}\:\mathrm{that1}^{\mathrm{3}} +\mathrm{2}^{\mathrm{3}} +\mathrm{3}^{\mathrm{3}} +...+\mathrm{n}^{\mathrm{3}} =\frac{\mathrm{n}^{\mathrm{2}} \left(\mathrm{n}+\mathrm{1}\right)^{\mathrm{2}} }{\mathrm{4}}\:\mathrm{for}\:\mathrm{every} \\ $$$$\mathrm{natural}\:\mathrm{number}\:\boldsymbol{\mathrm{n}} \\ $$

Question Number 45575 Answers: 2 Comments: 4

Question Number 45565 Answers: 2 Comments: 0

If ax^2 +by^2 +2hxy+2gx+2fy+c=0 be the equation of an ellipse, find coordinates of center of ellipse. Q.45506 (another solution)

$${If}\:\:\boldsymbol{{ax}}^{\mathrm{2}} +\boldsymbol{{by}}^{\mathrm{2}} +\mathrm{2}\boldsymbol{{hxy}}+\mathrm{2}\boldsymbol{{gx}}+\mathrm{2}\boldsymbol{{fy}}+\boldsymbol{{c}}=\mathrm{0} \\ $$$${be}\:{the}\:{equation}\:{of}\:{an}\:{ellipse},\:{find} \\ $$$${coordinates}\:{of}\:{center}\:{of}\:{ellipse}. \\ $$$${Q}.\mathrm{45506}\:\:\left({another}\:{solution}\right) \\ $$

Question Number 45563 Answers: 0 Comments: 0

Question Number 45561 Answers: 0 Comments: 2

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