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AlgebraQuestion and Answers: Page 341

Question Number 28370 Answers: 0 Comments: 1

1) factorizse p(x) =x^n −1 inside C[x] 2) find the value of Π_(k=1) ^(n−1) sin(((kπ)/n)) 3)find also the value of Π_(k=0) ^(n−1) sin(((kπ)/n) +θ).

$$\left.\mathrm{1}\right)\:{factorizse}\:{p}\left({x}\right)\:={x}^{{n}} \:−\mathrm{1}\:\:{inside}\:{C}\left[{x}\right] \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\:\prod_{{k}=\mathrm{1}} ^{{n}−\mathrm{1}} \:{sin}\left(\frac{{k}\pi}{{n}}\right) \\ $$$$\left.\mathrm{3}\right){find}\:{also}\:{the}\:{value}\:{of}\:\:\:\:\prod_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:\:{sin}\left(\frac{{k}\pi}{{n}}\:+\theta\right). \\ $$

Question Number 28369 Answers: 0 Comments: 0

let give the matrice A = (((1 0 0)),((0 1 1)) ) [ (1 0 1 A ∈ M_3 (R) write A at form A= I +J and calculate A^n .

$${let}\:{give}\:{the}\:{matrice}\:\:\:{A}\:=\:\begin{pmatrix}{\mathrm{1}\:\:\:\:\:\:\mathrm{0}\:\:\:\:\:\:\mathrm{0}}\\{\mathrm{0}\:\:\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\mathrm{1}}\end{pmatrix} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left[\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{1}\:\:\:\:\:\:\:\:\mathrm{0}\:\:\:\:\:\:\mathrm{1}\right.\right. \\ $$$${A}\:\in\:{M}_{\mathrm{3}} \left({R}\right)\:\:{write}\:\:{A}\:{at}\:{form}\:\:{A}=\:{I}\:+{J}\:\:\:\:{and}\:{calculate} \\ $$$${A}^{{n}} . \\ $$

Question Number 28368 Answers: 0 Comments: 0

P is apolynomial from C_n [x] having n roots (x_i )_(1≤i≤n ) and x_i # x_j for i#j 1) prove that Σ_(i=1) ^n (1/(p^′ (x_i ))) =0 2) find Σ_(i=1) ^n (x_i ^k /(p^′ (x_i ))) with k∈[[0,n−1]] .

$${P}\:{is}\:{apolynomial}\:{from}\:{C}_{{n}} \left[{x}\right]\:{having}\:{n}\:{roots} \\ $$$$\left({x}_{{i}} \right)_{\mathrm{1}\leqslant{i}\leqslant{n}\:} \:\:\:\:{and}\:{x}_{{i}} #\:{x}_{{j}} \:{for}\:{i}#{j} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\:\:\sum_{{i}=\mathrm{1}} ^{{n}} \:\:\:\frac{\mathrm{1}}{{p}^{'} \left({x}_{{i}} \right)}\:\:=\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:{find}\:\:\:\:\:\sum_{{i}=\mathrm{1}} ^{{n}} \:\:\:\frac{{x}_{{i}} ^{{k}} }{{p}^{'} \left({x}_{{i}} \right)}\:\:\:\:{with}\:{k}\in\left[\left[\mathrm{0},{n}−\mathrm{1}\right]\right]\:\:. \\ $$

Question Number 28367 Answers: 0 Comments: 0

find F∈R(x) wich verify F(x+1) −F(x)= ((x+3)/(x(x−1)(x+1))).

$${find}\:{F}\in{R}\left({x}\right)\:\:{wich}\:{verify}\:\:{F}\left({x}+\mathrm{1}\right)\:−{F}\left({x}\right)=\:\frac{{x}+\mathrm{3}}{{x}\left({x}−\mathrm{1}\right)\left({x}+\mathrm{1}\right)}. \\ $$

Question Number 28366 Answers: 0 Comments: 0

let give P(x)= α(x−x_1 )^m_1 (x−x_2 )^m_2 .....(x−x_n )^m_n give the decomposition of F(x)= ((d(P))/P) .d mean derivative

$${let}\:{give}\:{P}\left({x}\right)=\:\alpha\left({x}−{x}_{\mathrm{1}} \right)^{{m}_{\mathrm{1}} } \left({x}−{x}_{\mathrm{2}} \right)^{{m}_{\mathrm{2}} } .....\left({x}−{x}_{{n}} \right)^{{m}_{{n}} } \\ $$$${give}\:{the}\:{decomposition}\:{of}\:{F}\left({x}\right)=\:\frac{{d}\left({P}\right)}{{P}}\:.{d}\:{mean}\:{derivative} \\ $$

Question Number 28364 Answers: 0 Comments: 0

let give F(x) = (1/(x^2 +1)) prove that ∃ P_n ∈ Z_n [x] / F^((n)) (x)= ((P_n (x))/((1+x^2 )^n )) find a relation of recurence between the P_n .prove that all roots of P_n are reals and smples.

$${let}\:{give}\:{F}\left({x}\right)\:=\:\frac{\mathrm{1}}{{x}^{\mathrm{2}} +\mathrm{1}}\:{prove}\:{that}\:\exists\:{P}_{{n}} \in\:{Z}_{{n}} \left[{x}\right]\:/ \\ $$$${F}^{\left({n}\right)} \left({x}\right)=\:\:\frac{{P}_{{n}} \left({x}\right)}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{{n}} }\:\:\:{find}\:{a}\:{relation}\:{of}\:{recurence}\:{between}\: \\ $$$${the}\:\:{P}_{{n}} \:.{prove}\:{that}\:{all}\:{roots}\:{of}\:{P}_{{n}} \:{are}\:{reals}\:{and}\:{smples}. \\ $$

Question Number 28363 Answers: 1 Comments: 0

simlify the sum S= Σ_(k=0) ^(n−1) ((x+ e^(i2kπ) )/(x −e^(i2kπ) )) .

$${simlify}\:{the}\:{sum}\:\:\:{S}=\:\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:\:\frac{{x}+\:{e}^{{i}\mathrm{2}{k}\pi} }{{x}\:−{e}^{{i}\mathrm{2}{k}\pi} }\:\:. \\ $$

Question Number 28327 Answers: 1 Comments: 0

Given that ((a^(n+1) +b^(n+1) )/(a^n +b^n )) is AM between a and b ,where a≠b ∧ a,b≠0; find out the value of n.

$$\mathrm{Given}\:\mathrm{that}\:\frac{{a}^{{n}+\mathrm{1}} +{b}^{{n}+\mathrm{1}} }{{a}^{{n}} +{b}^{{n}} }\:\mathrm{is}\:\mathrm{AM}\:\mathrm{between}\:{a} \\ $$$$\mathrm{and}\:{b}\:,\mathrm{where}\:{a}\neq{b}\:\wedge\:{a},{b}\neq\mathrm{0};\:\mathrm{find}\:\mathrm{out}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{n}. \\ $$

Question Number 28320 Answers: 0 Comments: 3

If the arithmetic mean of a and b is ((a^(n+1) +b^(n+1) )/2), show that n=0

$${If}\:{the}\:{arithmetic}\:{mean}\:{of}\:{a}\:{and} \\ $$$${b}\:{is}\:\frac{{a}^{{n}+\mathrm{1}} +{b}^{{n}+\mathrm{1}} }{\mathrm{2}},\:{show}\:{that}\:{n}=\mathrm{0} \\ $$

Question Number 28319 Answers: 2 Comments: 0

If the roots of x^2 +px+q=0, q≠0 are α and β.Find the roots of qx^2 +(2q−p^2 )x+q=0 in terms of α and β.

$${If}\:{the}\:{roots}\:{of}\:{x}^{\mathrm{2}} +{px}+{q}=\mathrm{0},\:{q}\neq\mathrm{0} \\ $$$${are}\:\alpha\:{and}\:\beta.{Find}\:{the}\:{roots}\:{of} \\ $$$${qx}^{\mathrm{2}} +\left(\mathrm{2}{q}−{p}^{\mathrm{2}} \right){x}+{q}=\mathrm{0}\:{in}\:{terms}\:{of} \\ $$$$\alpha\:{and}\:\beta. \\ $$

Question Number 28312 Answers: 1 Comments: 0

let give P_n (x)=(x+1)^(2n) +(x+2)^n −1 and Q(x)= x^2 +3x +2 find R(x) /P_n (x)=R(x) Q(x) .

$${let}\:{give}\:\:{P}_{{n}} \left({x}\right)=\left({x}+\mathrm{1}\right)^{\mathrm{2}{n}} \:+\left({x}+\mathrm{2}\right)^{{n}} −\mathrm{1}\:{and} \\ $$$${Q}\left({x}\right)=\:{x}^{\mathrm{2}} \:+\mathrm{3}{x}\:+\mathrm{2}\:\:{find}\:{R}\left({x}\right)\:/{P}_{{n}} \left({x}\right)={R}\left({x}\right)\:{Q}\left({x}\right)\:. \\ $$

Question Number 28311 Answers: 0 Comments: 0

let give P_n (x)= Σ_(k=0) ^(2n) (1+(1/2) +...+(1/(k+1)))x^k and Q_n (x)= 1+(x/2)+(x^2 /3) +...(x^n /(n+1)) .prove that Q_(n ) divide P_n .

$${let}\:{give}\:{P}_{{n}} \left({x}\right)=\:\sum_{{k}=\mathrm{0}} ^{\mathrm{2}{n}} \:\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}\:+...+\frac{\mathrm{1}}{{k}+\mathrm{1}}\right){x}^{{k}} \:\:{and} \\ $$$${Q}_{{n}} \left({x}\right)=\:\mathrm{1}+\frac{{x}}{\mathrm{2}}+\frac{{x}^{\mathrm{2}} }{\mathrm{3}}\:+...\frac{{x}^{{n}} }{{n}+\mathrm{1}}\:\:.{prove}\:{that}\:{Q}_{{n}\:} \:{divide}\:{P}_{{n}} . \\ $$

Question Number 28313 Answers: 0 Comments: 0

let give P_n (x)= 1−x^2^(n+1) and Q_n (x)= Π_(k=0) ^n (1+x^2^k ) prove that Q_(n ) divide P_n .

$${let}\:{give}\:\:{P}_{{n}} \left({x}\right)=\:\mathrm{1}−{x}^{\mathrm{2}^{{n}+\mathrm{1}} } \:\:\:{and}\:\:{Q}_{{n}} \left({x}\right)=\:\prod_{{k}=\mathrm{0}} ^{{n}} \left(\mathrm{1}+{x}^{\mathrm{2}^{{k}} } \right) \\ $$$${prove}\:{that}\:{Q}_{{n}\:} \:{divide}\:{P}_{{n}} . \\ $$

Question Number 28288 Answers: 2 Comments: 0

Question Number 28275 Answers: 0 Comments: 2

Find area of the region [y]=[x] for x∈[2, 5] . [x] is greatest integer less than or equal to x .

$${Find}\:{area}\:{of}\:{the}\:{region} \\ $$$$\left[{y}\right]=\left[{x}\right]\:\:{for}\:\:{x}\in\left[\mathrm{2},\:\mathrm{5}\right]\:. \\ $$$$\left[{x}\right]\:{is}\:{greatest}\:{integer}\:{less}\:{than}\:{or} \\ $$$${equal}\:{to}\:{x}\:. \\ $$

Question Number 28267 Answers: 1 Comments: 1

let give the polynomial P(x)= (1/(2i))( (1+ix)^n −(1−ix)^n ) .find the roots of P(x) and factorize P(x).

$${let}\:{give}\:{the}\:{polynomial} \\ $$$${P}\left({x}\right)=\:\frac{\mathrm{1}}{\mathrm{2}{i}}\left(\:\left(\mathrm{1}+{ix}\right)^{{n}} \:−\left(\mathrm{1}−{ix}\right)^{{n}} \right)\:.{find}\:{the}\:{roots}\:{of}\:{P}\left({x}\right) \\ $$$${and}\:{factorize}\:{P}\left({x}\right). \\ $$

Question Number 28265 Answers: 0 Comments: 0

1) find P∈R[x] / P(sinx) =sin(2n+1)x 2) find the roots of P and degP 3) decompose (1/P) and prove that ((2n+1)/(sin(2n+1)x)) = Σ_(k=0) ^(2n) (((−1)^k cos(((kπ)/(2n+1))))/(sinx−sin (((kπ)/(2n+1)))))) .

$$\left.\mathrm{1}\right)\:\:{find}\:{P}\in{R}\left[{x}\right]\:/\:{P}\left({sinx}\right)\:={sin}\left(\mathrm{2}{n}+\mathrm{1}\right){x} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{roots}\:{of}\:{P}\:{and}\:{degP} \\ $$$$\left.\mathrm{3}\right)\:{decompose}\:\:\frac{\mathrm{1}}{{P}}\:\:{and}\:{prove}\:{that} \\ $$$$\frac{\mathrm{2}{n}+\mathrm{1}}{{sin}\left(\mathrm{2}{n}+\mathrm{1}\right){x}}\:=\:\sum_{{k}=\mathrm{0}} ^{\mathrm{2}{n}} \:\:\:\:\frac{\left(−\mathrm{1}\right)^{{k}} \:{cos}\left(\frac{{k}\pi}{\mathrm{2}{n}+\mathrm{1}}\right)}{{sinx}−{sin}\:\left(\frac{{k}\pi}{\left.\mathrm{2}{n}+\mathrm{1}\right)}\right)}\:\:. \\ $$

Question Number 28264 Answers: 0 Comments: 0

give the decomposition of F(x) = ((1 )/(Π_(k=1) ^n (x−k^2 ))) .

$${give}\:{the}\:{decomposition}\:{of}\: \\ $$$${F}\left({x}\right)\:\:\:=\:\:\:\:\:\:\frac{\mathrm{1}\:}{\prod_{{k}=\mathrm{1}} ^{{n}} \:\left({x}−{k}^{\mathrm{2}} \right)}\:. \\ $$

Question Number 28219 Answers: 0 Comments: 4