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Question Number 28260    Answers: 0   Comments: 0

let give ( 1 1 −1) A= ( 1 1 1 ) ( −1 1 1 ) and the matrices I= ( 1 0 0 ) ( 0 1 1 ) ( 0 0 1 ) and J= ( 0 1 −1) ( 1 0 1). ( −1 1 0) 1) find J^2 and J^(−1) . 2) let put J^n = x_n I +y_n J .prove that x_(n+2 ) +x_(n+1) −2x_n =0 3) calculate J^n and A^n .

$${let}\:{give}\:\:\:\:\left(\:\:\:\:\mathrm{1}\:\:\:\:\:\mathrm{1}\:\:\:\:\:−\mathrm{1}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:{A}=\:\:\:\:\:\left(\:\:\:\mathrm{1}\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\:\:\:\mathrm{1}\:\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\:−\mathrm{1}\:\:\:\mathrm{1}\:\:\:\:\:\:\:\:\:\mathrm{1}\:\:\right) \\ $$$${and}\:{the}\:{matrices}\:\:{I}=\:\:\left(\:\:\mathrm{1}\:\:\:\:\:\mathrm{0}\:\:\:\:\:\:\mathrm{0}\:\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\:\mathrm{0}\:\:\:\:\mathrm{1}\:\:\:\:\:\:\:\mathrm{1}\:\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\:\mathrm{0}\:\:\:\:\mathrm{0}\:\:\:\:\:\:\mathrm{1}\:\right) \\ $$$${and}\:\:{J}=\:\:\:\left(\:\:\mathrm{0}\:\:\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:−\mathrm{1}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\:\:\mathrm{1}\:\:\:\:\:\:\:\mathrm{0}\:\:\:\:\:\:\:\:\:\mathrm{1}\right).\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\:−\mathrm{1}\:\:\:\:\mathrm{1}\:\:\:\:\:\:\:\:\:\mathrm{0}\right) \\ $$$$\left.\mathrm{1}\right)\:{find}\:\:{J}^{\mathrm{2}} \:{and}\:{J}^{−\mathrm{1}} .\: \\ $$$$\left.\mathrm{2}\right)\:\:{let}\:{put}\:\:{J}^{{n}} =\:{x}_{{n}} {I}\:+{y}_{{n}} {J}\:\:\:\:.{prove}\:{that}\: \\ $$$${x}_{{n}+\mathrm{2}\:} +{x}_{{n}+\mathrm{1}} \:−\mathrm{2}{x}_{{n}} \:=\mathrm{0}\: \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\:{J}^{{n}} {and}\:{A}^{{n}} . \\ $$$$ \\ $$

Question Number 28259    Answers: 0   Comments: 0

let give A= ( cosθ −sinθ ) ( sinθ cosθ ) 1) calculate^t A. A .prove that A is inversible and find A^(−1) 2) find A^n for n∈ N 3) developp (A +A^(−1) )^n then prove that 2^n cos^n θ = Σ_(k=0) ^n C_n ^k (n−2k)θ and Σ_(k=0) ^n C_n ^n sin(n−2k)θ =0 .

$${let}\:{give}\:\:\:\:{A}=\:\:\:\left(\:\:{cos}\theta\:\:\:\:\:\:\:−{sin}\theta\:\:\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\:{sin}\theta\:\:\:\:\:\:\:\:\:{cos}\theta\:\:\:\:\right) \\ $$$$\left.\mathrm{1}\right)\:\:{calculate}\:^{{t}} {A}.\:{A}\:\:.{prove}\:{that}\:{A}\:{is}\:{inversible}\:{and}\:{find} \\ $$$${A}^{−\mathrm{1}} \\ $$$$\left.\mathrm{2}\right)\:\:{find}\:\:{A}^{{n}} \:\:\:{for}\:{n}\in\:{N} \\ $$$$\left.\mathrm{3}\right)\:{developp}\:\left({A}\:+{A}^{−\mathrm{1}} \right)^{{n}} \:\:{then}\:{prove}\:{that} \\ $$$$\mathrm{2}^{{n}} \:{cos}^{{n}} \theta\:\:=\:\:\sum_{{k}=\mathrm{0}} ^{{n}} \:{C}_{{n}} ^{{k}} \left({n}−\mathrm{2}{k}\right)\theta\:\:{and} \\ $$$$\sum_{{k}=\mathrm{0}} ^{{n}} \:{C}_{{n}} ^{{n}} \:{sin}\left({n}−\mathrm{2}{k}\right)\theta\:\:=\mathrm{0}\:\:. \\ $$

Question Number 28258    Answers: 0   Comments: 1

let give A = ( 1 1 ) ( 2 −1) find e^(A ) and e^(−tA) .

$${let}\:{give}\:{A}\:\:=\:\left(\:\mathrm{1}\:\:\:\:\:\:\:\mathrm{1}\:\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\:\:\mathrm{2}\:\:\:\:−\mathrm{1}\right) \\ $$$$\:\:{find}\:\:{e}^{{A}\:} \:\:\:\:{and}\:\:\:{e}^{−{tA}} .\:\: \\ $$

Question Number 28257    Answers: 0   Comments: 0

let give ( 2 3 −3) A = ( −1 0 1) ( −1 1 0 ) find a diagoal matrice D and a inversible matrice P wich verify A = P.D.P^(−1) and calculate A^n .

$${let}\:{give}\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\:\:\:\:\:\mathrm{2}\:\:\:\:\:\:\:\mathrm{3}\:\:\:\:\:\:\:−\mathrm{3}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:{A}\:\:\:\:\:=\:\:\:\:\:\:\left(\:\:−\mathrm{1}\:\:\:\:\:\mathrm{0}\:\:\:\:\:\:\:\:\:\:\:\mathrm{1}\right)\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\:\:\:−\mathrm{1}\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\mathrm{0}\:\:\right)\: \\ $$$${find}\:{a}\:{diagoal}\:{matrice}\:{D}\:{and}\:{a}\:{inversible}\:{matrice}\:{P}\:\:{wich} \\ $$$${verify}\:{A}\:\:=\:{P}.{D}.{P}^{−\mathrm{1}} \:\:\:{and}\:\:{calculate}\:{A}^{{n}} . \\ $$

Question Number 28248    Answers: 1   Comments: 0

Question Number 28247    Answers: 0   Comments: 1

find the value of ∫_1 ^(+∞) ((lnx)/(1+x^2 ))dx .

$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{1}} ^{+\infty} \:\:\frac{{lnx}}{\mathrm{1}+{x}^{\mathrm{2}} }{dx}\:. \\ $$

Question Number 28242    Answers: 0   Comments: 1

find the value of ∫_0 ^∞ e^(−x) lnxdx .

$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} {e}^{−{x}} {lnxdx}\:\:. \\ $$

Question Number 28241    Answers: 1   Comments: 1

Question Number 28240    Answers: 1   Comments: 0

Question Number 28255    Answers: 0   Comments: 0

let give p=(_(1 −2) ^(1 −1) ) and D= (_(3 −6) ^(2 −2) ) calculate A= p.D.p^(−1) .

$${let}\:{give}\:{p}=\left(_{\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:−\mathrm{2}} ^{\mathrm{1}\:\:\:\:\:\:\:\:\:\:−\mathrm{1}} \:\:\right)\:\:{and}\:\:{D}=\:\:\left(_{\mathrm{3}\:\:\:\:\:\:\:\:\:\:−\mathrm{6}} ^{\mathrm{2}\:\:\:\:\:\:\:\:−\mathrm{2}} \:\right)\:{calculate} \\ $$$${A}=\:{p}.{D}.{p}^{−\mathrm{1}} \:. \\ $$

Question Number 28256    Answers: 0   Comments: 0

let give A= _( () −1 1 1) ( 1 −1 1) find A^n for n integr. ( 1 1 −1)

$$\left.{let}\:{give}\:\:\:{A}=\:\:_{\:\:\:\:\left(\right.} \:−\mathrm{1}\:\:\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\mathrm{1}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\:\:\:\:\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\:−\mathrm{1}\:\:\:\:\mathrm{1}\right)\:\:\:\:\:\:\:{find}\:\:{A}^{{n}} \:\:{for}\:{n}\:{integr}. \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\:\:\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:−\mathrm{1}\right) \\ $$$$\:\:\:\: \\ $$

Question Number 28219    Answers: 0   Comments: 4

Question Number 28211    Answers: 0   Comments: 6

Question Number 28199    Answers: 1   Comments: 0

suppose one of the side of any box that can be carried onto an airplane must be less than 8m. Find the maximum value of such a box if the sum of the three sides can not exceed 46m.

$${suppose}\:{one}\:{of}\:{the}\:{side}\:{of}\:{any} \\ $$$${box}\:{that}\:{can}\:{be}\:{carried}\:{onto}\:{an} \\ $$$${airplane}\:{must}\:{be}\:{less}\:{than}\:\mathrm{8}{m}. \\ $$$${Find}\:{the}\:{maximum}\:{value}\:{of}\:{such} \\ $$$${a}\:{box}\:{if}\:{the}\:{sum}\:{of}\:{the}\:{three}\:{sides} \\ $$$${can}\:{not}\:{exceed}\:\mathrm{46}{m}. \\ $$

Question Number 28198    Answers: 1   Comments: 0

Find the shortest distance from the origin to the curve xy=3

$${Find}\:{the}\:{shortest}\:{distance}\:{from} \\ $$$${the}\:{origin}\:{to}\:{the}\:{curve}\:{xy}=\mathrm{3} \\ $$

Question Number 28190    Answers: 1   Comments: 2

Question Number 28189    Answers: 0   Comments: 2

Question Number 28188    Answers: 0   Comments: 0

Question Number 28186    Answers: 0   Comments: 1

Find the number of positive integers x such that [(x/(m−1))]=[(x/(m+1))], for a particular integer m≥2. [ ] means G.I.F.

$${Find}\:{the}\:{number}\:{of}\:{positive}\:{integers} \\ $$$${x}\:{such}\:{that}\:\left[\frac{{x}}{{m}−\mathrm{1}}\right]=\left[\frac{{x}}{{m}+\mathrm{1}}\right],\:{for}\:{a} \\ $$$${particular}\:{integer}\:{m}\geqslant\mathrm{2}. \\ $$$$\left[\:\right]\:{means}\:{G}.{I}.{F}. \\ $$

Question Number 28179    Answers: 0   Comments: 1

Question Number 28177    Answers: 0   Comments: 1

Question Number 28174    Answers: 0   Comments: 2

if the sum of root 7x+px−q=0 is 7 then p= ??

$$\mathrm{if}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{root}\:\mathrm{7x}+\mathrm{px}−\mathrm{q}=\mathrm{0}\:\mathrm{is}\:\mathrm{7}\:\mathrm{then}\:\mathrm{p}= \\ $$$$?? \\ $$

Question Number 28171    Answers: 1   Comments: 1

Question Number 28170    Answers: 0   Comments: 0

find ∫_0 ^1 ((1 −e^(−x) − e^(−(1/x)) )/x)dx .

$${find}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{\mathrm{1}\:−{e}^{−{x}} −\:{e}^{−\frac{\mathrm{1}}{{x}}} }{{x}}{dx}\:\:. \\ $$

Question Number 28169    Answers: 0   Comments: 1

prove that ∫_0 ^1 (((lnx)^k )/(1−x))dx=(−1)^k (k!)ξ(k+1) .

$${prove}\:{that}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{\left({lnx}\right)^{{k}} }{\mathrm{1}−{x}}{dx}=\left(−\mathrm{1}\right)^{{k}} \:\left({k}!\right)\xi\left({k}+\mathrm{1}\right)\:\:. \\ $$

Question Number 28168    Answers: 0   Comments: 0

find the nature of Σ_(n=1) ^∞ arcos(1−(1/n^a ))z^n with a>1.

$${find}\:{the}\:{nature}\:{of}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} {arcos}\left(\mathrm{1}−\frac{\mathrm{1}}{{n}^{{a}} }\right){z}^{{n}} \:\:{with}\:{a}>\mathrm{1}. \\ $$

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