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Question Number 34608 Answers: 0 Comments: 0

Question Number 34607 Answers: 0 Comments: 0

let p∈C[x] degp=n (x_i )_(1≤k≤n) the roots of p(x) a∈C?/p(a)≠0 1) calculate S_1 = Σ_(k=1) ^n (1/(x_k −a)) interms of p,p^′ and a 2)calculste S_2 =Σ_(k=1) ^n (1/((x_k −a)^2 )) interms of p,p^, p^(′′) and a.

$${let}\:{p}\in{C}\left[{x}\right]\:{degp}={n}\:\:\:\left({x}_{{i}} \right)_{\mathrm{1}\leqslant{k}\leqslant{n}} {the}\:{roots}\:{of}\:{p}\left({x}\right) \\ $$$${a}\in{C}?/{p}\left({a}\right)\neq\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{S}_{\mathrm{1}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{x}_{{k}} −{a}}\:{interms}\:{of}\:{p},{p}^{'} \:{and}\:{a} \\ $$$$\left.\mathrm{2}\right){calculste}\:{S}_{\mathrm{2}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{\mathrm{1}}{\left({x}_{{k}} −{a}\right)^{\mathrm{2}} }\:\:{interms}\:{of}\:{p},{p}^{,} \\ $$$${p}^{''} \:{and}\:{a}. \\ $$

Question Number 34606 Answers: 0 Comments: 0

let give p(x)=(x+1)^n −(x−1)^n 1) factorize p(x) inside C[x] 2) find the value of Π_(k=1) ^p cotan(((kπ)/(2p+1)))

$${let}\:{give}\:{p}\left({x}\right)=\left({x}+\mathrm{1}\right)^{{n}} \:−\left({x}−\mathrm{1}\right)^{{n}} \\ $$$$\left.\mathrm{1}\right)\:{factorize}\:{p}\left({x}\right)\:{inside}\:{C}\left[{x}\right] \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\prod_{{k}=\mathrm{1}} ^{{p}} \:\:{cotan}\left(\frac{{k}\pi}{\mathrm{2}{p}+\mathrm{1}}\right) \\ $$

Question Number 34605 Answers: 0 Comments: 0

decompose inside R(x) thefraction F(x)= ((x^5 +1)/(x^2^ (x−1)^2 )) .

$${decompose}\:{inside}\:{R}\left({x}\right)\:{thefraction} \\ $$$${F}\left({x}\right)=\:\:\frac{{x}^{\mathrm{5}} \:+\mathrm{1}}{{x}^{\mathrm{2}^{} } \left({x}−\mathrm{1}\right)^{\mathrm{2}} }\:. \\ $$

Question Number 34604 Answers: 0 Comments: 0

let p(x)= x^n +x+1 ∈C[x] and z∈C/p(z)=0 prove that ∣z∣<2 .

$${let}\:\:{p}\left({x}\right)=\:{x}^{{n}} \:+{x}+\mathrm{1}\:\in{C}\left[{x}\right]\:{and}\:{z}\in{C}/{p}\left({z}\right)=\mathrm{0} \\ $$$${prove}\:{that}\:\mid{z}\mid<\mathrm{2}\:. \\ $$

Question Number 34603 Answers: 0 Comments: 0

prove that ∀ p∈K[x] p(x) −x divide p(p(x))−x

$${prove}\:{that}\:\forall\:{p}\in{K}\left[{x}\right]\:{p}\left({x}\right)\:−{x}\:{divide}\:{p}\left({p}\left({x}\right)\right)−{x} \\ $$

Question Number 34602 Answers: 0 Comments: 0

simplify Σ_(k=0) ^n ((k/n) −α)^2 C_n ^k x^k (1−x)^(n−k ) α∈C.

$${simplify}\:\sum_{{k}=\mathrm{0}} ^{{n}} \left(\frac{{k}}{{n}}\:−\alpha\right)^{\mathrm{2}} {C}_{{n}} ^{{k}} \:{x}^{{k}} \left(\mathrm{1}−{x}\right)^{{n}−{k}\:} \\ $$$$\alpha\in{C}. \\ $$

Question Number 34596 Answers: 0 Comments: 0

1) prove that Σ_(k=1) ^n H_k =(n+1)H_n −n 2) prove that Σ_(k=1) ^n H_k ^2 =(n+1)H_n ^2 −(3n+1)H_n +2n H_n =Σ_(k=1) ^n (1/k) .

$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\:\sum_{{k}=\mathrm{1}} ^{{n}} \:{H}_{{k}} =\left({n}+\mathrm{1}\right){H}_{{n}} \:−{n} \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\:\sum_{{k}=\mathrm{1}} ^{{n}} \:{H}_{{k}} ^{\mathrm{2}} \:\:=\left({n}+\mathrm{1}\right){H}_{{n}} ^{\mathrm{2}} \:\:−\left(\mathrm{3}{n}+\mathrm{1}\right){H}_{{n}} \:+\mathrm{2}{n} \\ $$$${H}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{\mathrm{1}}{{k}}\:. \\ $$

Question Number 34595 Answers: 0 Comments: 0

simplify Σ_(k=1) ^n (((−1)^(k−1) )/k) C_n ^k

$${simplify}\:\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{\left(−\mathrm{1}\right)^{{k}−\mathrm{1}} }{{k}}\:{C}_{{n}} ^{{k}} \\ $$

Question Number 34594 Answers: 0 Comments: 0

prove that Σ_(k=0) ^p (−1)^k C_n ^k =(−1)^p C_(n−1) ^p

$${prove}\:{that}\:\:\sum_{{k}=\mathrm{0}} ^{{p}} \left(−\mathrm{1}\right)^{{k}} \:{C}_{{n}} ^{{k}} \:\:=\left(−\mathrm{1}\right)^{{p}} \:{C}_{{n}−\mathrm{1}} ^{{p}} \\ $$

Question Number 34593 Answers: 0 Comments: 0

1) calculate ∫_(−∞) ^(+∞) ((cos(αx^n ))/(x^2 +x +1)) dx with n integr natural 2) find the value of ∫_(−∞) ^∞ ((cos( α x^(2n) ))/(x^2 +x +1))dx 3) calculate ∫_(−∞) ^(+∞) ((cos(π x^3 ))/(x^2 +x +1)) dx

$$\left.\mathrm{1}\right)\:{calculate}\:\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{cos}\left(\alpha{x}^{{n}} \right)}{{x}^{\mathrm{2}} \:+{x}\:+\mathrm{1}}\:{dx}\:\:{with}\:{n}\:{integr} \\ $$$${natural} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\:\:\int_{−\infty} ^{\infty} \:\:\:\:\frac{{cos}\left(\:\alpha\:{x}^{\mathrm{2}{n}} \right)}{{x}^{\mathrm{2}} \:+{x}\:+\mathrm{1}}{dx} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\:\int_{−\infty} ^{+\infty} \:\:\:\:\frac{{cos}\left(\pi\:{x}^{\mathrm{3}} \right)}{{x}^{\mathrm{2}} \:+{x}\:+\mathrm{1}}\:{dx} \\ $$

Question Number 34587 Answers: 2 Comments: 1

Question Number 34585 Answers: 1 Comments: 1

Question Number 34571 Answers: 0 Comments: 0

A machine with a velocity ratio of 5 requires 150J of work to raise a 500N load through a vertical distance of 200cm,calculate: a)the efficiency b)the M.A of the machine

$${A}\:{machine}\:{with}\:{a}\:{velocity}\:{ratio} \\ $$$${of}\:\mathrm{5}\:{requires}\:\mathrm{150}{J}\:{of}\:{work}\:{to}\:{raise} \\ $$$${a}\:\mathrm{500}{N}\:{load}\:{through}\:{a}\:{vertical} \\ $$$${distance}\:{of}\:\mathrm{200}{cm},{calculate}: \\ $$$$\left.{a}\right){the}\:{efficiency} \\ $$$$\left.{b}\right){the}\:{M}.{A}\:{of}\:{the}\:{machine} \\ $$

Question Number 34567 Answers: 1 Comments: 0

x determinant ((2),())−2x−15=0

$${x}\begin{vmatrix}{\mathrm{2}}\\{}\end{vmatrix}−\mathrm{2}{x}−\mathrm{15}=\mathrm{0} \\ $$

Question Number 34562 Answers: 1 Comments: 1

find the value of ∫_0 ^1 ((arctanx)/((1+x^2 )^2 )) dx

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

Question Number 34561 Answers: 0 Comments: 1

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

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

Question Number 34614 Answers: 0 Comments: 1

decompose inside R(x) the fraction F(x)= (1/((x−3)^6 (x+2))) .