Question and Answers Forum

If one line of the equation : ax^3 +bx^2 y+cxy^2 +dy^3 =0 bisects the angle between the the other two then prove (3a+c)^2 (bc+2cd−3ad)= (b+3d)^2 (bc+2ab−3ad) .

$${If}\:{one}\:{line}\:{of}\:{the}\:{equation}\:: \\ $$$$\boldsymbol{{ax}}^{\mathrm{3}} +\boldsymbol{{bx}}^{\mathrm{2}} \boldsymbol{{y}}+\boldsymbol{{cxy}}^{\mathrm{2}} +\boldsymbol{{dy}}^{\mathrm{3}} =\mathrm{0} \\ $$$${bisects}\:{the}\:{angle}\:{between}\:{the} \\ $$$${the}\:{other}\:{two}\:{then}\:{prove} \\ $$$$\left(\mathrm{3}{a}+{c}\right)^{\mathrm{2}} \left({bc}+\mathrm{2}{cd}−\mathrm{3}{ad}\right)= \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\left({b}+\mathrm{3}{d}\right)^{\mathrm{2}} \left({bc}+\mathrm{2}{ab}−\mathrm{3}{ad}\right)\:. \\ $$

Question Number 28302 Answers: 0 Comments: 1

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 .

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

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Question Number 28289 Answers: 0 Comments: 0

find the loss amt when cost price and % of loss given

$${find}\:{the}\:{loss}\:{amt}\:{when}\:{cost}\:{price} \\ $$$${and}\:\%\:{of}\:{loss}\:{given} \\ $$

Question Number 28288 Answers: 2 Comments: 0

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Find dy/dx x^(2/3) (6−x)^(1/(3 )) to it simplest form

$${Find}\:{dy}/{dx} \\ $$$${x}^{\frac{\mathrm{2}}{\mathrm{3}}} \left(\mathrm{6}−{x}\right)^{\frac{\mathrm{1}}{\mathrm{3}\:}} \:{to}\:{it}\:{simplest}\:{form} \\ $$

Question Number 28278 Answers: 2 Comments: 1

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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 28268 Answers: 0 Comments: 1

find in terms of n the value of A_n = ∫_0 ^1 (1+x^2 )^(n/2) sin(narctanx)dx . ( n∈ N).

$$\:{find}\:{in}\:{terms}\:{of}\:{n}\:{the}\:{value}\:{of} \\ $$$${A}_{{n}} =\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{\frac{{n}}{\mathrm{2}}} \:{sin}\left({narctanx}\right){dx}\:.\:\:\left(\:{n}\in\:{N}\right). \\ $$

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 28263 Answers: 0 Comments: 0

decompose F(x)= (1/((x−1)^2 (x^3 −1))) then calculate ∫_2 ^(+∝) F(x)dx.

$${decompose}\:{F}\left({x}\right)=\:\:\frac{\mathrm{1}}{\left({x}−\mathrm{1}\right)^{\mathrm{2}} \left({x}^{\mathrm{3}} −\mathrm{1}\right)}\:\:{then}\:{calculate} \\ $$$$\int_{\mathrm{2}} ^{+\propto} \:{F}\left({x}\right){dx}. \\ $$

Question Number 28262 Answers: 0 Comments: 1

find ∫_0 ^∞ (dx/(1+x^5 )) .

$${find}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{\mathrm{1}+{x}^{\mathrm{5}} }\:. \\ $$

Question Number 28261 Answers: 0 Comments: 0

let give the matrice ( 0 cosθ cos(2θ)) A= ( cosθ 0 cos(2θ) ) ( cos(θ) cos(2θ 0 ) and D_θ =det A solve inside R D_θ =0

$${let}\:{give}\:{the}\:{matrice} \\ $$$$\:\:\:\:\:\:\:\:\:\:\left(\:\:\:\mathrm{0}\:\:\:\:\:\:\:{cos}\theta\:\:\:\:\:\:{cos}\left(\mathrm{2}\theta\right)\right) \\ $$$${A}=\:\:\:\left(\:{cos}\theta\:\:\:\:\:\mathrm{0}\:\:\:\:\:\:\:\:\:{cos}\left(\mathrm{2}\theta\right)\:\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\left(\:{cos}\left(\theta\right)\:{cos}\left(\mathrm{2}\theta\:\:\:\:\:\:\mathrm{0}\:\:\:\:\:\:\right)\right. \\ $$$${and}\:{D}_{\theta} \:\:={det}\:{A}\:\:{solve}\:{inside}\:{R}\:\:{D}_{\theta} =\mathrm{0} \\ $$$$\:\:\:\: \\ $$$$\: \\ $$

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 .

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