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AlgebraQuestion and Answers: Page 341

Question Number 32334    Answers: 0   Comments: 1

p is apolynomial having n roots (x_i ) with x_i ≠xj for i≠j calculate Σ_(k=1) ^n (1/(1−x_k )) .

$${p}\:{is}\:{apolynomial}\:{having}\:{n}\:{roots}\:\:\left({x}_{{i}} \right)\:{with}\:{x}_{{i}} \neq{xj}\:{for} \\ $$$${i}\neq{j}\:\:{calculate}\:\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\:\frac{\mathrm{1}}{\mathrm{1}−{x}_{{k}} }\:. \\ $$

Question Number 32333    Answers: 0   Comments: 1

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

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

Question Number 32332    Answers: 0   Comments: 0

calculate Σ_(p=1) ^n (p/(1+p^2 +p^4 )) .

$${calculate}\:\:\sum_{{p}=\mathrm{1}} ^{{n}} \:\:\:\frac{{p}}{\mathrm{1}+{p}^{\mathrm{2}} \:+{p}^{\mathrm{4}} }\:\:. \\ $$

Question Number 32331    Answers: 0   Comments: 0

p is a polynomial having n simples roots (x_i )_(1≤i≤n) prove that Σ_(k=1) ^n (1/(p^′ (x_k ))) =0

$${p}\:{is}\:{a}\:{polynomial}\:{having}\:{n}\:{simples}\:{roots}\:\left({x}_{{i}} \right)_{\mathrm{1}\leqslant{i}\leqslant{n}} \\ $$$${prove}\:{that}\:\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{\mathrm{1}}{{p}^{'} \left({x}_{{k}} \right)}\:=\mathrm{0} \\ $$

Question Number 32330    Answers: 0   Comments: 1

let p_n (x)=(x+1)^(6n+1) −x^(6n+1) −1 with n integr prove that ∀n (x^2 +x+1)^2 divide p_n (x).

$${let}\:{p}_{{n}} \left({x}\right)=\left({x}+\mathrm{1}\right)^{\mathrm{6}{n}+\mathrm{1}} \:−{x}^{\mathrm{6}{n}+\mathrm{1}} \:−\mathrm{1}\:{with}\:{n}\:{integr} \\ $$$${prove}\:{that}\:\forall{n}\:\:\left({x}^{\mathrm{2}} +{x}+\mathrm{1}\right)^{\mathrm{2}} \:{divide}\:{p}_{{n}} \left({x}\right). \\ $$

Question Number 32326    Answers: 0   Comments: 0

simplify Σ_(k=p) ^(2p) (C_k ^p /2^k ) .

$${simplify}\:\:\sum_{{k}={p}} ^{\mathrm{2}{p}} \:\:\:\:\:\frac{{C}_{{k}} ^{{p}} }{\mathrm{2}^{{k}} }\:. \\ $$

Question Number 32242    Answers: 1   Comments: 0

A company manufactures two types of products; X ($4.50 profit per item x) and Y ($3.00 profit per item y). These items are built using both machine time and manual labour. The X product requires 3 hours of machine time and two hours of manual labour. The Y product requires 3 hours of machine time and no manual labour. If the week′s supply of manual labour is limited to 8 hours and machine time to 15 hours, write down all inequalities involving x and y.

$$\mathrm{A}\:\mathrm{company}\:\mathrm{manufactures}\:\mathrm{two}\:\mathrm{types}\:\mathrm{of}\:\mathrm{products}; \\ $$$$\mathrm{X}\:\left(\$\mathrm{4}.\mathrm{50}\:\mathrm{profit}\:\mathrm{per}\:\mathrm{item}\:\mathrm{x}\right)\:\mathrm{and}\:\mathrm{Y}\:\left(\$\mathrm{3}.\mathrm{00}\:\mathrm{profit}\:\mathrm{per}\right. \\ $$$$\left.\mathrm{item}\:\mathrm{y}\right).\:\mathrm{These}\:\mathrm{items}\:\mathrm{are}\:\mathrm{built}\:\mathrm{using}\:\mathrm{both}\:\mathrm{machine} \\ $$$$\mathrm{time}\:\mathrm{and}\:\mathrm{manual}\:\mathrm{labour}.\:\mathrm{The}\:\mathrm{X}\:\mathrm{product}\:\mathrm{requires} \\ $$$$\mathrm{3}\:\mathrm{hours}\:\mathrm{of}\:\mathrm{machine}\:\mathrm{time}\:\mathrm{and}\:\mathrm{two}\:\mathrm{hours}\:\mathrm{of}\:\mathrm{manual} \\ $$$$\mathrm{labour}.\:\mathrm{The}\:\mathrm{Y}\:\mathrm{product}\:\mathrm{requires}\:\mathrm{3}\:\mathrm{hours}\:\mathrm{of}\:\mathrm{machine} \\ $$$$\mathrm{time}\:\mathrm{and}\:\mathrm{no}\:\mathrm{manual}\:\mathrm{labour}.\:\mathrm{If}\:\mathrm{the}\:\mathrm{week}'\mathrm{s}\:\mathrm{supply}\:\mathrm{of}\: \\ $$$$\mathrm{manual}\:\mathrm{labour}\:\mathrm{is}\:\mathrm{limited}\:\mathrm{to}\:\mathrm{8}\:\mathrm{hours}\:\mathrm{and}\:\mathrm{machine} \\ $$$$\mathrm{time}\:\mathrm{to}\:\mathrm{15}\:\mathrm{hours},\:\mathrm{write}\:\mathrm{down}\:\mathrm{all}\:\mathrm{inequalities}\: \\ $$$$\mathrm{involving}\:\mathrm{x}\:\mathrm{and}\:\mathrm{y}. \\ $$

Question Number 32239    Answers: 1   Comments: 0

Find the sum of the coefficients of all the integral power of x in the expansion of (1+2(√x))^(40) .

$$\boldsymbol{{F}}{ind}\:{the}\:{sum}\:{of}\:{the}\:{coefficients} \\ $$$${of}\:{all}\:{the}\:{integral}\:{power}\:{of}\:{x}\:{in}\:{the} \\ $$$${expansion}\:{of}\:\left(\mathrm{1}+\mathrm{2}\sqrt{{x}}\right)^{\mathrm{40}} . \\ $$

Question Number 32208    Answers: 0   Comments: 0

Question Number 32203    Answers: 0   Comments: 5

Number of solutions of the equation z^3 +(([3(z^− )^2 ])/(∣z∣))=0 where z is a complex no.

$$\boldsymbol{{N}}{umber}\:{of}\:{solutions}\:{of}\:{the}\:{equation} \\ $$$${z}^{\mathrm{3}} +\frac{\left[\mathrm{3}\left(\overset{−} {{z}}\right)^{\mathrm{2}} \right]}{\mid{z}\mid}=\mathrm{0}\:{where}\:{z}\:{is}\:{a}\:{complex}\:{no}. \\ $$

Question Number 32184    Answers: 1   Comments: 0

If one vertex of the triangle having maximum area that can be inscribed in the circle ∣z−i∣=5 is 3−3i, then find other vertices of triangle.

$$\boldsymbol{{I}}{f}\:{one}\:{vertex}\:{of}\:{the}\:{triangle}\:{having} \\ $$$${maximum}\:{area}\:{that}\:{can}\:{be}\:{inscribed} \\ $$$${in}\:{the}\:{circle}\:\mid\boldsymbol{{z}}−\boldsymbol{{i}}\mid=\mathrm{5}\:{is}\:\mathrm{3}−\mathrm{3}\boldsymbol{{i}},\:{then} \\ $$$${find}\:{other}\:{vertices}\:{of}\:{triangle}. \\ $$

Question Number 32181    Answers: 1   Comments: 1

Intercept made by the circle zz^− +a^− z+az^− +r=0 on the real axis on complex plane is :−

$$\boldsymbol{{I}}{ntercept}\:{made}\:{by}\:{the}\:{circle}\: \\ $$$$\boldsymbol{{z}}\overset{−} {\boldsymbol{{z}}}+\overset{−} {\boldsymbol{{a}z}}+\boldsymbol{{a}}\overset{−} {\boldsymbol{{z}}}+\boldsymbol{{r}}=\mathrm{0}\:\boldsymbol{{o}}{n}\:{the}\:{real}\:{axis}\:{on} \\ $$$${complex}\:{plane}\:{is}\::− \\ $$

Question Number 32160    Answers: 1   Comments: 0

If z=cosθ+isinθ is a root of equation a_0 z^n +a_1 z^(n−1) +a_2 z^(n−2) +.....+a_(n−1) z+a_n =0 then prove that: i) a_0 +a_1 cos θ+a_2 cos 2θ+.....+a_n cos nθ=0 ii) a_1 sin θ + a_2 sin 2θ+....+a_n sin nθ=0.

$$\boldsymbol{{I}}{f}\:{z}={cos}\theta+{isin}\theta\:{is}\:{a}\:{root}\:{of}\:{equation} \\ $$$${a}_{\mathrm{0}} {z}^{{n}} +{a}_{\mathrm{1}} {z}^{{n}−\mathrm{1}} +{a}_{\mathrm{2}} {z}^{{n}−\mathrm{2}} +.....+{a}_{{n}−\mathrm{1}} {z}+{a}_{{n}} =\mathrm{0} \\ $$$${then}\:{prove}\:{that}: \\ $$$$\left.{i}\right)\:{a}_{\mathrm{0}} +{a}_{\mathrm{1}} \mathrm{cos}\:\theta+{a}_{\mathrm{2}} \mathrm{cos}\:\mathrm{2}\theta+.....+{a}_{{n}} \mathrm{cos}\:{n}\theta=\mathrm{0} \\ $$$$\left.{ii}\right)\:{a}_{\mathrm{1}} \mathrm{sin}\:\theta\:+\:{a}_{\mathrm{2}} \mathrm{sin}\:\mathrm{2}\theta+....+{a}_{{n}} \mathrm{sin}\:{n}\theta=\mathrm{0}. \\ $$

Question Number 32159    Answers: 1   Comments: 2

Express the following in a+ib form: (((cos x+isin x)(cos y+isin y))/((cosa+isin a)(cosb+isinb))).

$$\boldsymbol{{E}}{xpress}\:{the}\:{following}\:{in}\:{a}+{ib}\:{form}: \\ $$$$\frac{\left(\mathrm{cos}\:{x}+{i}\mathrm{sin}\:{x}\right)\left(\mathrm{cos}\:{y}+{i}\mathrm{sin}\:{y}\right)}{\left({cosa}+{i}\mathrm{sin}\:{a}\right)\left({cosb}+{isinb}\right)}. \\ $$

Question Number 32142    Answers: 0   Comments: 0

Question Number 32132    Answers: 0   Comments: 0

∫(1/((x+1)ln(x)))dx=?

$$\int\frac{\mathrm{1}}{\left({x}+\mathrm{1}\right){ln}\left({x}\right)}{dx}=? \\ $$

Question Number 32110    Answers: 1   Comments: 0

If y=1+x^2 +x^3 and x=1+α, where α is small, show that y≈3+5α. Hence, find the increase in y when x is increased from 1 to 1.02

$$\mathrm{If}\:\mathrm{y}=\mathrm{1}+\mathrm{x}^{\mathrm{2}} +\mathrm{x}^{\mathrm{3}} \:\mathrm{and}\:\mathrm{x}=\mathrm{1}+\alpha,\:\mathrm{where}\:\alpha\:\mathrm{is}\:\mathrm{small},\:\mathrm{show} \\ $$$$\mathrm{that}\:\mathrm{y}\approx\mathrm{3}+\mathrm{5}\alpha.\:\mathrm{Hence},\:\mathrm{find}\:\mathrm{the}\:\mathrm{increase}\:\mathrm{in}\:\mathrm{y}\:\mathrm{when} \\ $$$$\mathrm{x}\:\mathrm{is}\:\mathrm{increased}\:\mathrm{from}\:\mathrm{1}\:\mathrm{to}\:\mathrm{1}.\mathrm{02} \\ $$

Question Number 32099    Answers: 1   Comments: 0

Question Number 32094    Answers: 0   Comments: 0

Find the ordinary argument (arg z) and the principal argument (Arg z) of z=(i/(−2−2i))

$${Find}\:{the}\:{ordinary}\:{argument} \\ $$$$\left({arg}\:{z}\right)\:{and}\:{the}\:{principal}\:{argument} \\ $$$$\left({Arg}\:{z}\right)\:{of}\:{z}=\frac{{i}}{−\mathrm{2}−\mathrm{2}{i}} \\ $$

Question Number 32049    Answers: 2   Comments: 0

If ∣z−6−8i∣≤4 then Minimum value of ∣z∣ is A) 4 B) 5 C) 6 D) 8.

$$\boldsymbol{{I}}{f}\:\mid\boldsymbol{{z}}−\mathrm{6}−\mathrm{8}\boldsymbol{{i}}\mid\leqslant\mathrm{4}\:{then}\:\boldsymbol{{M}}{inimum}\:{value} \\ $$$${of}\:\mid\boldsymbol{{z}}\mid\:{is}\: \\ $$$$\left.{A}\right)\:\mathrm{4}\: \\ $$$$\left.{B}\right)\:\mathrm{5} \\ $$$$\left.{C}\right)\:\mathrm{6} \\ $$$$\left.{D}\right)\:\mathrm{8}. \\ $$

Question Number 32047    Answers: 0   Comments: 1

Question Number 32002    Answers: 1   Comments: 0

If z^3 =z^ prove then ∣z∣=1.

$${If}\:\:\boldsymbol{{z}}^{\mathrm{3}} =\bar {\boldsymbol{{z}}}\:{prove}\: \\ $$$${then}\:\mid\boldsymbol{{z}}\mid=\mathrm{1}. \\ $$

Question Number 31991    Answers: 0   Comments: 1

g_n =(√(g_(n−1) +g_(n−2) )) g_1 =1 g_2 =3 g_n =..

$${g}_{{n}} =\sqrt{{g}_{{n}−\mathrm{1}} +{g}_{{n}−\mathrm{2}} } \\ $$$${g}_{\mathrm{1}} =\mathrm{1} \\ $$$${g}_{\mathrm{2}} =\mathrm{3} \\ $$$${g}_{{n}} =.. \\ $$

Question Number 31990    Answers: 1   Comments: 3

a_n =2a_(n−1) +3a_(n−2) a_0 =1 a_1 =2 a_n =...

$${a}_{{n}} =\mathrm{2}{a}_{{n}−\mathrm{1}} +\mathrm{3}{a}_{{n}−\mathrm{2}} \\ $$$${a}_{\mathrm{0}} =\mathrm{1} \\ $$$${a}_{\mathrm{1}} =\mathrm{2} \\ $$$${a}_{{n}} =... \\ $$

Question Number 31963    Answers: 0   Comments: 0

find Re (((1+e^(iα) )/(1+e^(iβ) ))) and Im ( ((1+e^(iα) )/(1+e^(iβ) )) ) .

$${find}\:{Re}\:\left(\frac{\mathrm{1}+{e}^{{i}\alpha} }{\mathrm{1}+{e}^{{i}\beta} }\right)\:{and}\:{Im}\:\left(\:\frac{\mathrm{1}+{e}^{{i}\alpha} }{\mathrm{1}+{e}^{{i}\beta} }\:\right)\:. \\ $$

Question Number 31927    Answers: 1   Comments: 0

The number of distinct real roots of equation x^4 −4x^3 +12x^2 +x−1=0.

$${The}\:{number}\:{of}\:{distinct}\:{real}\:{roots} \\ $$$${of}\:{equation}\:\boldsymbol{{x}}^{\mathrm{4}} −\mathrm{4}\boldsymbol{{x}}^{\mathrm{3}} +\mathrm{12}\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{{x}}−\mathrm{1}=\mathrm{0}. \\ $$

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