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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 28359 Answers: 0 Comments: 1

Question Number 28349 Answers: 1 Comments: 1

Question Number 28341 Answers: 1 Comments: 4

Find lim_(x→0) ((5x−tan (5x))/x^3 )

$${Find}\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{5}{x}−\mathrm{tan}\:\left(\mathrm{5}{x}\right)}{{x}^{\mathrm{3}} } \\ $$

Question Number 28339 Answers: 1 Comments: 0

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 28305 Answers: 0 Comments: 3

if y=sin^(−1) x^2 +cos^(−1) x^2 find dy/dx

$${if}\:{y}=\mathrm{sin}^{−\mathrm{1}} {x}^{\mathrm{2}} +\mathrm{cos}^{−\mathrm{1}} {x}^{\mathrm{2}} \\ $$$${find}\:{dy}/{dx} \\ $$

Question Number 28304 Answers: 1 Comments: 0

if y=(sin^(−1) x)^2 +(cos^(−1) x)^2 find dy/dx

$${if}\:{y}=\left(\mathrm{sin}^{−\mathrm{1}} {x}\right)^{\mathrm{2}} +\left(\mathrm{cos}^{−\mathrm{1}} {x}\right)^{\mathrm{2}} \\ $$$$ \\ $$$${find}\:{dy}/{dx} \\ $$

Question Number 28303 Answers: 1 Comments: 0

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 .

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

Question Number 28293 Answers: 0 Comments: 0

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

Question Number 28285 Answers: 0 Comments: 1

Question Number 28280 Answers: 1 Comments: 4

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

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 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)))))) .

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

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