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Question Number 37232    Answers: 0   Comments: 0

let A = (((1 −1 0)),((−1 1 1)) ) (0 0 3 ) calculate A^n .

$${let}\:{A}\:=\:\begin{pmatrix}{\mathrm{1}\:\:\:\:\:−\mathrm{1}\:\:\:\:\:\:\:\mathrm{0}}\\{−\mathrm{1}\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\:\:\mathrm{1}}\end{pmatrix} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{0}\:\:\:\:\:\:\:\:\mathrm{0}\:\:\:\:\:\:\:\:\:\mathrm{3}\:\right) \\ $$$${calculate}\:{A}^{{n}} \:. \\ $$

Question Number 37231    Answers: 0   Comments: 0

let A = (((0 1 1)),((1 0 1)) ) (1 1 0 ) 1) calculate p_c (A) the caracteristic polunom of A 2) calculate A^n with n integr natural 3) calcypulate e^(tA) t∈ R

$${let}\:{A}\:=\:\begin{pmatrix}{\mathrm{0}\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\mathrm{1}}\\{\mathrm{1}\:\:\:\:\:\:\mathrm{0}\:\:\:\:\:\:\:\mathrm{1}}\end{pmatrix} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{1}\:\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\:\mathrm{0}\:\right) \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{p}_{{c}} \left({A}\right)\:{the}\:{caracteristic}\: \\ $$$${polunom}\:{of}\:{A} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{A}^{{n}} \:\:{with}\:{n}\:{integr}\:{natural} \\ $$$$\left.\mathrm{3}\right)\:{calcypulate}\:{e}^{{tA}} \:\:\:\:\:{t}\in\:{R}\:\: \\ $$

Question Number 37230    Answers: 0   Comments: 0

let A = (((1 −2)),((1 4)) ) calculate A^n 2) find e^A , e^(−A) 3) find e^(iA) , e^(−iA) and e^(iA) +e^(−iA) .

$${let}\:{A}\:=\begin{pmatrix}{\mathrm{1}\:\:\:\:\:\:\:−\mathrm{2}}\\{\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\mathrm{4}}\end{pmatrix} \\ $$$${calculate}\:\:{A}^{{n}} \: \\ $$$$\left.\mathrm{2}\right)\:{find}\:\:{e}^{{A}} \:\:,\:{e}^{−{A}} \\ $$$$\left.\mathrm{3}\right)\:{find}\:\:{e}^{{iA}} ,\:{e}^{−{iA}} \:\:\:{and}\:{e}^{{iA}} \:+{e}^{−{iA}} \:\:. \\ $$

Question Number 37228    Answers: 0   Comments: 0

E id k vectorial space and f∈L(E) 1)prove that if f is nilpotent with indice p≥1 ,I −f is bijective and (I−f)^(−1) =Σ_(i=0) ^(p−1) f^i 2)let E=R_n [x] and f∈L(E) / f(p) =p−p^′ prove that f is inversible and find f^(−1) .

$${E}\:{id}\:{k}\:{vectorial}\:{space}\:{and}\:{f}\in{L}\left({E}\right) \\ $$$$\left.\mathrm{1}\right){prove}\:{that}\:{if}\:{f}\:{is}\:{nilpotent}\:{with}\:{indice} \\ $$$${p}\geqslant\mathrm{1}\:,{I}\:−{f}\:{is}\:{bijective}\:{and} \\ $$$$\left({I}−{f}\right)^{−\mathrm{1}} =\sum_{{i}=\mathrm{0}} ^{{p}−\mathrm{1}} {f}^{{i}} \\ $$$$\left.\mathrm{2}\right){let}\:{E}={R}_{{n}} \left[{x}\right]\:{and}\:{f}\in{L}\left({E}\right)\:/ \\ $$$${f}\left({p}\right)\:={p}−{p}^{'} \:\:{prove}\:{that}\:{f}\:{is}\:{inversible} \\ $$$${and}\:{find}\:{f}^{−\mathrm{1}} \:. \\ $$

Question Number 37166    Answers: 0   Comments: 2

if 3x^2 +2αxy+2y^2 +2ax−4y+1 can be resolved into two linear factors, prove that ′α′ is a root of the equation x^2 +4ax+2a^2 +6=0

$${if}\:\:\mathrm{3}{x}^{\mathrm{2}} +\mathrm{2}\alpha{xy}+\mathrm{2}{y}^{\mathrm{2}} +\mathrm{2}{ax}−\mathrm{4}{y}+\mathrm{1} \\ $$$${can}\:{be}\:{resolved}\:\:{into}\:\:{two}\:\:{linear} \\ $$$${factors},\:\:{prove}\:\:{that}\:\:'\alpha'\:\:{is}\:{a}\:{root}\: \\ $$$${of}\:{the}\:{equation}\:{x}^{\mathrm{2}} +\mathrm{4}{ax}+\mathrm{2}{a}^{\mathrm{2}} +\mathrm{6}=\mathrm{0} \\ $$

Question Number 37139    Answers: 2   Comments: 0

if α , β are the roots of the quadratic equation ax^2 +bx+c =0 then find the quadratic equation whose roots are α^(2 ) , β^2

$${if}\:\alpha\:,\:\beta\:\:{are}\:{the}\:{roots}\:{of}\:{the}\:{quadratic} \\ $$$${equation}\:{ax}^{\mathrm{2}} +{bx}+{c}\:=\mathrm{0}\:{then}\:\:{find} \\ $$$${the}\:{quadratic}\:{equation}\:{whose}\:{roots} \\ $$$${are}\:\:\alpha^{\mathrm{2}\:\:\:} ,\:\beta^{\mathrm{2}} \\ $$$$ \\ $$$$ \\ $$

Question Number 36926    Answers: 1   Comments: 0

let f(x) = e^(−x^2 ) 1) prove that f^((n)) (x)=p_n (x)e^(−x^2 ) with p_n is a polynom 2) find a relation of recurrence between the p_n 3) calculate p_1 ,p_2 ,p_3 ,p_4

$${let}\:{f}\left({x}\right)\:=\:{e}^{−{x}^{\mathrm{2}} } \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{f}^{\left({n}\right)} \left({x}\right)={p}_{{n}} \left({x}\right){e}^{−{x}^{\mathrm{2}} } \:\:{with}\:{p}_{{n}} \:{is}\:{a}\:{polynom} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{a}\:{relation}\:{of}\:{recurrence}\:{between}\:{the}\:{p}_{{n}} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:{p}_{\mathrm{1}} ,{p}_{\mathrm{2}} ,{p}_{\mathrm{3}} ,{p}_{\mathrm{4}} \\ $$

Question Number 36911    Answers: 0   Comments: 1

p is a polynome having nroots simples x_i (1≤x_i ≤n ) with x_i ^2 ≠1 calculste Σ_(k=1) ^n (1/(1−x_k )) .

$${p}\:{is}\:{a}\:{polynome}\:{having}\:{nroots}\:{simples} \\ $$$${x}_{{i}} \:\left(\mathrm{1}\leqslant{x}_{{i}} \leqslant{n}\:\right)\:{with}\:{x}_{{i}} ^{\mathrm{2}} \:\neq\mathrm{1}\:\:{calculste} \\ $$$$\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{\mathrm{1}}{\mathrm{1}−{x}_{{k}} }\:. \\ $$

Question Number 36909    Answers: 0   Comments: 0

let p(x)=x^3 −2x^2 −1 and α is root of p(x) prove that α∉ Q .

$${let}\:{p}\left({x}\right)={x}^{\mathrm{3}} \:−\mathrm{2}{x}^{\mathrm{2}} \:−\mathrm{1}\:{and}\:\alpha\:{is}\:{root}\:{of}\:{p}\left({x}\right) \\ $$$${prove}\:{that}\:\alpha\notin\:{Q}\:. \\ $$

Question Number 36905    Answers: 0   Comments: 0

p is apolynom with n roots differents let Q = p^2 +p^′ let α the number of roots of Q prove that n−1≤α≤n+1 .

$${p}\:{is}\:{apolynom}\:{with}\:{n}\:{roots}\:{differents} \\ $$$${let}\:{Q}\:=\:{p}^{\mathrm{2}} \:+{p}^{'} \:\:\:\:{let}\:\alpha\:{the}\:{number}\:{of}\:{roots}\:{of} \\ $$$${Q}\:{prove}\:{that}\:\:\:{n}−\mathrm{1}\leqslant\alpha\leqslant{n}+\mathrm{1}\:. \\ $$

Question Number 36904    Answers: 0   Comments: 1

1)decompose inside C[x] p(x)=x^(2n) −2(cosα)x^n +1 2) decopose p(x)inside R[x]

$$\left.\mathrm{1}\right){decompose}\:{inside}\:{C}\left[{x}\right] \\ $$$${p}\left({x}\right)={x}^{\mathrm{2}{n}} \:−\mathrm{2}\left({cos}\alpha\right){x}^{{n}} \:+\mathrm{1} \\ $$$$\left.\mathrm{2}\right)\:{decopose}\:{p}\left({x}\right){inside}\:{R}\left[{x}\right] \\ $$

Question Number 36903    Answers: 0   Comments: 0

prove that 2^(n+1) divide [(1+(√3))^(2n+1) ] [x] mean integr part of x

$${prove}\:{that}\:\:\mathrm{2}^{{n}+\mathrm{1}} \:{divide}\:\left[\left(\mathrm{1}+\sqrt{\mathrm{3}}\right)^{\mathrm{2}{n}+\mathrm{1}} \right]\: \\ $$$$\left[{x}\right]\:{mean}\:{integr}\:{part}\:{of}\:{x} \\ $$

Question Number 36871    Answers: 0   Comments: 0

Question Number 36844    Answers: 1   Comments: 0

find the sum of 4 digit even numbers formed from the digit 1, 2, 3, 4

$$\mathrm{find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{4}\:\mathrm{digit}\:\mathrm{even}\:\mathrm{numbers}\:\mathrm{formed}\:\mathrm{from}\:\mathrm{the}\:\mathrm{digit}\:\:\:\mathrm{1},\:\mathrm{2},\:\mathrm{3},\:\mathrm{4} \\ $$

Question Number 36843    Answers: 1   Comments: 1

lim_(x→∞) (x^x^x^(.....) )

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\left({x}^{{x}^{{x}^{.....} } } \right) \\ $$

Question Number 36828    Answers: 0   Comments: 7

Question Number 36699    Answers: 0   Comments: 0

Question Number 36696    Answers: 0   Comments: 0

if y=tan^(−1) x show that (1+x^2 )y_(n+2) +2(n+1)xy_(n+1) +n(n+1)y_n =0

$${if}\:{y}={tan}^{−\mathrm{1}} {x}\:{show}\:{that} \\ $$$$\left(\mathrm{1}+{x}^{\mathrm{2}} \right){y}_{{n}+\mathrm{2}} +\mathrm{2}\left({n}+\mathrm{1}\right){xy}_{{n}+\mathrm{1}} +{n}\left({n}+\mathrm{1}\right){y}_{{n}} =\mathrm{0} \\ $$

Question Number 36692    Answers: 2   Comments: 1

x^3 +y^3 =5 x^2 +y^2 =3

$${x}^{\mathrm{3}} +{y}^{\mathrm{3}} =\mathrm{5} \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} =\mathrm{3} \\ $$

Question Number 36691    Answers: 0   Comments: 2

Question Number 36676    Answers: 1   Comments: 3

if z = − 27, find all the root of z in complex plain

$$\mathrm{if}\:\:\mathrm{z}\:=\:−\:\mathrm{27},\:\:\mathrm{find}\:\mathrm{all}\:\mathrm{the}\:\mathrm{root}\:\mathrm{of}\:\mathrm{z}\:\mathrm{in}\:\mathrm{complex}\:\mathrm{plain} \\ $$

Question Number 36660    Answers: 1   Comments: 0

Question Number 36437    Answers: 0   Comments: 3

simplify Σ_(k=0) ^n (C_n ^k /(k+1))

$${simplify}\:\:\sum_{{k}=\mathrm{0}} ^{{n}} \:\:\:\:\frac{{C}_{{n}} ^{{k}} }{{k}+\mathrm{1}} \\ $$

Question Number 36359    Answers: 0   Comments: 0

decompose inside C(x) the fraction F(x)= (1/((x^2 +1)^n )) with n integr nstural.

$${decompose}\:{inside}\:{C}\left({x}\right)\:{the}\:{fraction}\: \\ $$$${F}\left({x}\right)=\:\:\frac{\mathrm{1}}{\left({x}^{\mathrm{2}} \:+\mathrm{1}\right)^{{n}} }\:\:{with}\:{n}\:{integr}\:{nstural}. \\ $$

Question Number 36352    Answers: 0   Comments: 1

let p(x) =x^n −e^(inα) with n integr and α fromR 1) find the roots of p(x) 2) factorize p(x) inside C[x] .

$${let}\:{p}\left({x}\right)\:={x}^{{n}} \:−{e}^{{in}\alpha} \:\:\:\:{with}\:{n}\:{integr}\:{and}\:\alpha\:{fromR} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{the}\:{roots}\:{of}\:{p}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{factorize}\:{p}\left({x}\right)\:{inside}\:{C}\left[{x}\right]\:. \\ $$

Question Number 36344    Answers: 0   Comments: 3

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