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Question Number 61142    Answers: 0   Comments: 0

Question Number 61140    Answers: 1   Comments: 1

can we find an exact solution? t^6 +4t^4 −12t^3 +24t^2 −24t+8=0

$$\mathrm{can}\:\mathrm{we}\:\mathrm{find}\:\mathrm{an}\:\mathrm{exact}\:\mathrm{solution}? \\ $$$${t}^{\mathrm{6}} +\mathrm{4}{t}^{\mathrm{4}} −\mathrm{12}{t}^{\mathrm{3}} +\mathrm{24}{t}^{\mathrm{2}} −\mathrm{24}{t}+\mathrm{8}=\mathrm{0} \\ $$

Question Number 61137    Answers: 1   Comments: 0

What is the sum of first 3n term of an AP , if the sunm of first n term is 2n and sum of first 2n term is 5n

$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{first}\:\mathrm{3n}\:\mathrm{term}\:\mathrm{of}\:\mathrm{an}\:\mathrm{AP}\:,\:\mathrm{if}\:\mathrm{the}\:\mathrm{sunm}\:\mathrm{of}\:\mathrm{first}\:\mathrm{n}\:\mathrm{term}\:\mathrm{is} \\ $$$$\mathrm{2n}\:\:\mathrm{and}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{first}\:\mathrm{2n}\:\mathrm{term}\:\mathrm{is}\:\:\mathrm{5n} \\ $$

Question Number 61117    Answers: 2   Comments: 0

The 2nd, 4th and 8th term of an AP are the consecutive term of a GP. If the sum of the 3rd and 4th term of the AP is 20. Find the sum of the first four terms of the AP.

$$\mathrm{The}\:\mathrm{2nd},\:\mathrm{4th}\:\mathrm{and}\:\mathrm{8th}\:\mathrm{term}\:\mathrm{of}\:\mathrm{an}\:\mathrm{AP}\:\mathrm{are}\:\mathrm{the}\:\mathrm{consecutive}\:\mathrm{term}\:\mathrm{of}\:\mathrm{a}\:\mathrm{GP}. \\ $$$$\mathrm{If}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{3rd}\:\mathrm{and}\:\mathrm{4th}\:\mathrm{term}\:\mathrm{of}\:\mathrm{the}\:\mathrm{AP}\:\mathrm{is}\:\mathrm{20}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{first}\:\mathrm{four}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{the}\:\mathrm{AP}. \\ $$

Question Number 61116    Answers: 1   Comments: 0

Question Number 61186    Answers: 0   Comments: 3

Question Number 61112    Answers: 2   Comments: 4

Question Number 61107    Answers: 0   Comments: 0

solve Cauchy′s problem x′= t + (((μx)^2 )/(1+(μx)^2 )), μ∈R x(0)=0

$${solve}\:{Cauchy}'{s}\:{problem} \\ $$$${x}'=\:{t}\:+\:\frac{\left(\mu{x}\right)^{\mathrm{2}} }{\mathrm{1}+\left(\mu{x}\right)^{\mathrm{2}} },\:\mu\in\mathbb{R} \\ $$$${x}\left(\mathrm{0}\right)=\mathrm{0} \\ $$

Question Number 61096    Answers: 0   Comments: 0

∀ a, n ∈ N : ∣a−n∣=1 pour a, n ≥3 a^m ≡1modn (∗) posons : m=n−1 (∗′) subtituons cette valeur dans (∗). on a: a^(n−1) ≡1modn. Mais n n′est pas forcement premier. Test de primalite ∀ n ∈ N, n ≥3. (n−2)^(n−1) ≡1modn ⇒ n est premier.

$$\forall\:{a},\:{n}\:\in\:{N}\::\:\mid{a}−{n}\mid=\mathrm{1}\:{pour}\:{a},\:{n}\:\geqslant\mathrm{3} \\ $$$${a}^{{m}} \equiv\mathrm{1}{modn}\:\left(\ast\right) \\ $$$${posons}\::\:{m}={n}−\mathrm{1}\:\left(\ast'\right) \\ $$$${subtituons}\:{cette}\:{valeur}\:{dans}\:\left(\ast\right). \\ $$$${on}\:{a}:\:{a}^{{n}−\mathrm{1}} \equiv\mathrm{1}{modn}.\:{Mais}\:{n}\:{n}'{est}\:{pas}\:{forcement}\:{premier}. \\ $$$${Test}\:{de}\:{primalite} \\ $$$$\forall\:{n}\:\in\:{N},\:{n}\:\geqslant\mathrm{3}. \\ $$$$\left({n}−\mathrm{2}\right)^{{n}−\mathrm{1}} \equiv\mathrm{1}{modn}\:\Rightarrow\:{n}\:{est}\:{premier}. \\ $$

Question Number 61178    Answers: 0   Comments: 1

solve (1+x^2 )y^′ +(1−x^2 )y =x e^(−3x)

$${solve}\:\left(\mathrm{1}+{x}^{\mathrm{2}} \right){y}^{'} \:+\left(\mathrm{1}−{x}^{\mathrm{2}} \right){y}\:={x}\:{e}^{−\mathrm{3}{x}} \\ $$

Question Number 61177    Answers: 0   Comments: 0

Question Number 61069    Answers: 0   Comments: 2

Question Number 61061    Answers: 1   Comments: 1

Question Number 61056    Answers: 2   Comments: 0

∫ ((x + sin(x))/(1 + cos(x))) dx

$$\int\:\frac{\mathrm{x}\:+\:\mathrm{sin}\left(\mathrm{x}\right)}{\mathrm{1}\:+\:\mathrm{cos}\left(\mathrm{x}\right)}\:\mathrm{dx} \\ $$

Question Number 61045    Answers: 2   Comments: 2

calculate I =∫_0 ^1 cos(2arctanx)dx and J =∫_0 ^1 sin(2arctanx)dx

$${calculate}\:{I}\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:{cos}\left(\mathrm{2}{arctanx}\right){dx} \\ $$$${and}\:{J}\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:{sin}\left(\mathrm{2}{arctanx}\right){dx} \\ $$

Question Number 61042    Answers: 1   Comments: 0

Question Number 61041    Answers: 1   Comments: 1

calculate ∫_1 ^(+∞) (([2x]−[x])/x^4 ) dx

$${calculate}\:\int_{\mathrm{1}} ^{+\infty} \:\frac{\left[\mathrm{2}{x}\right]−\left[{x}\right]}{{x}^{\mathrm{4}} }\:{dx}\: \\ $$

Question Number 61030    Answers: 0   Comments: 2

Question Number 61027    Answers: 0   Comments: 3

Question Number 61039    Answers: 0   Comments: 1

find ∫_0 ^1 arctan((2/(1+x)))dx

$${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{arctan}\left(\frac{\mathrm{2}}{\mathrm{1}+{x}}\right){dx} \\ $$

Question Number 61048    Answers: 1   Comments: 1

Question Number 61003    Answers: 3   Comments: 0

Question Number 61111    Answers: 1   Comments: 2

Please what does the 2 on the C mean. C_1 ^2 + 2 C_2 ^2 + 3 C_3 ^2 + ... + n C_n ^2 = (((2n − 1)!)/([(n − 1)!]^2 )) Does the 2 on C mean square ?? I mean: (C_1 )^2 + 2(C_2 )^2 + 3(C_3 )^2 + ... + n (C_n )^2 which is also ( ^n C_1 )^2 + 2( ^n C_2 )^2 + 3( ^n C_3 )^2 + ... + n ( ^n C_n )^2 I just want to know what the 2 on C represent . Thanks. C_1 ^2 + 2 C_2 ^2 + 3 C_3 ^2 + ... + n C_n ^2 = (((2n − 1)!)/([(n − 1)!]^2 ))

$$\mathrm{Please}\:\mathrm{what}\:\mathrm{does}\:\mathrm{the}\:\mathrm{2}\:\mathrm{on}\:\mathrm{the}\:\mathrm{C}\:\mathrm{mean}. \\ $$$$ \\ $$$$\:\:\:\:\:\:\mathrm{C}_{\mathrm{1}} ^{\mathrm{2}} \:+\:\mathrm{2}\:\mathrm{C}_{\mathrm{2}} ^{\mathrm{2}} \:+\:\mathrm{3}\:\mathrm{C}_{\mathrm{3}} ^{\mathrm{2}} \:+\:...\:+\:\mathrm{n}\:\mathrm{C}_{\mathrm{n}} ^{\mathrm{2}} \:\:\:\:=\:\:\:\frac{\left(\mathrm{2n}\:−\:\mathrm{1}\right)!}{\left[\left(\mathrm{n}\:−\:\mathrm{1}\right)!\right]^{\mathrm{2}} } \\ $$$$\mathrm{Does}\:\mathrm{the}\:\mathrm{2}\:\mathrm{on}\:\mathrm{C}\:\mathrm{mean}\:\mathrm{square}\:?? \\ $$$$\:\:\:\:\mathrm{I}\:\mathrm{mean}:\:\:\:\:\:\:\left(\mathrm{C}_{\mathrm{1}} \right)^{\mathrm{2}} \:+\:\mathrm{2}\left(\mathrm{C}_{\mathrm{2}} \right)^{\mathrm{2}} \:+\:\mathrm{3}\left(\mathrm{C}_{\mathrm{3}} \right)^{\mathrm{2}} \:+\:...\:+\:\mathrm{n}\:\left(\mathrm{C}_{\mathrm{n}} \right)^{\mathrm{2}} \\ $$$$\mathrm{which}\:\mathrm{is}\:\mathrm{also} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\left(\overset{\mathrm{n}} {\:}\mathrm{C}_{\mathrm{1}} \right)^{\mathrm{2}} \:+\:\mathrm{2}\left(\overset{\mathrm{n}} {\:}\mathrm{C}_{\mathrm{2}} \right)^{\mathrm{2}} \:+\:\mathrm{3}\left(\overset{\mathrm{n}} {\:}\mathrm{C}_{\mathrm{3}} \right)^{\mathrm{2}} \:+\:...\:+\:\mathrm{n}\:\left(\overset{\mathrm{n}} {\:}\mathrm{C}_{\mathrm{n}} \right)^{\mathrm{2}} \\ $$$$ \\ $$$$\mathrm{I}\:\mathrm{just}\:\mathrm{want}\:\mathrm{to}\:\mathrm{know}\:\mathrm{what}\:\mathrm{the}\:\mathrm{2}\:\mathrm{on}\:\mathrm{C}\:\mathrm{represent}\:.\:\:\mathrm{Thanks}. \\ $$$$\:\:\:\:\:\:\mathrm{C}_{\mathrm{1}} ^{\mathrm{2}} \:+\:\mathrm{2}\:\mathrm{C}_{\mathrm{2}} ^{\mathrm{2}} \:+\:\mathrm{3}\:\mathrm{C}_{\mathrm{3}} ^{\mathrm{2}} \:+\:...\:+\:\mathrm{n}\:\mathrm{C}_{\mathrm{n}} ^{\mathrm{2}} \:\:\:\:=\:\:\:\frac{\left(\mathrm{2n}\:−\:\mathrm{1}\right)!}{\left[\left(\mathrm{n}\:−\:\mathrm{1}\right)!\right]^{\mathrm{2}} } \\ $$

Question Number 61202    Answers: 1   Comments: 1

Question Number 61205    Answers: 0   Comments: 4

A cubical block of ice of mass m and edge L is placed in a large tray of mass M.If the ice block melts,how far does the centre of mass of the system “ice + tray” come down ? a)((ml)/(m+M)) b)((2ml)/(m+M)) c)((ml)/(2(m+M))) d)none

$${A}\:{cubical}\:{block}\:{of}\:{ice}\:{of}\:{mass}\:{m}\:{and} \\ $$$${edge}\:{L}\:{is}\:{placed}\:{in}\:{a}\:{large}\:{tray}\:{of}\:{mass} \\ $$$${M}.{If}\:{the}\:{ice}\:{block}\:{melts},{how}\:{far}\:{does} \\ $$$${the}\:{centre}\:{of}\:{mass}\:{of}\:{the}\:{system}\:``{ice}\:+\:{tray}'' \\ $$$${come}\:{down}\:? \\ $$$$ \\ $$$$\left.{a}\left.\right)\left.\frac{{ml}}{{m}+{M}}\left.\:\:{b}\right)\frac{\mathrm{2}{ml}}{{m}+{M}}\:\:{c}\right)\frac{{ml}}{\mathrm{2}\left({m}+{M}\right)}\:\:{d}\right){none} \\ $$

Question Number 60987    Answers: 1   Comments: 1

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