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AlgebraQuestion and Answers: Page 327

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.

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

Question Number 31915 Answers: 1 Comments: 3

If : (x^2 +x+2)^2 −(a−3)(x^2 +x+1)(x^2 +x+2) + (a−4)(x^2 +x+1)^2 =0 has at least one root , then find complete set of values of a.

$$\boldsymbol{{If}}\::\: \\ $$$$\left({x}^{\mathrm{2}} +{x}+\mathrm{2}\right)^{\mathrm{2}} −\left({a}−\mathrm{3}\right)\left({x}^{\mathrm{2}} +{x}+\mathrm{1}\right)\left({x}^{\mathrm{2}} +{x}+\mathrm{2}\right) \\ $$$$+\:\left({a}−\mathrm{4}\right)\left({x}^{\mathrm{2}} +{x}+\mathrm{1}\right)^{\mathrm{2}} =\mathrm{0}\:{has}\:{at}\:{least}\: \\ $$$${one}\:{root}\:,\:{then}\:{find}\:{complete}\:{set}\:{of}\: \\ $$$${values}\:{of}\:{a}. \\ $$

Question Number 31895 Answers: 0 Comments: 3

Question Number 32372 Answers: 1 Comments: 0

Question Number 31868 Answers: 0 Comments: 0

Let a>b>1 be positive integers with b odd. Let n be a positive integer as well. If b^n divides a^n −1, prove that a^b > (3^n /n). Solution please. Thanks in advance!!

$${Let}\:{a}>{b}>\mathrm{1}\:{be}\:{positive}\:{integers}\:{with}\:{b}\:{odd}. \\ $$$${Let}\:{n}\:{be}\:{a}\:{positive}\:{integer}\:{as}\:{well}.\:{If}\:\:{b}^{{n}} \:{divides} \\ $$$${a}^{{n}} −\mathrm{1},\:{prove}\:{that}\:{a}^{{b}} \:>\:\frac{\mathrm{3}^{{n}} }{{n}}. \\ $$$${Solution}\:{please}.\:{Thanks}\:{in}\:{advance}!! \\ $$

Question Number 31865 Answers: 1 Comments: 6

Range of function : f(x)= 6^x +3^x +6^(−x) +3^(−x) +2.

$${Range}\:{of}\:{function}\:: \\ $$$${f}\left({x}\right)=\:\mathrm{6}^{{x}} +\mathrm{3}^{{x}} +\mathrm{6}^{−{x}} +\mathrm{3}^{−{x}} +\mathrm{2}. \\ $$

Question Number 31864 Answers: 1 Comments: 0

Let S_n = Σ_(k=1) ^(4n) (−1)^((k(k+1))/2) k^2 . Then S_n can take the value(s) 1) 1056 2) 1088 3) 1120 4) 1332.

$${Let}\:{S}_{{n}} =\:\underset{{k}=\mathrm{1}} {\overset{\mathrm{4}{n}} {\sum}}\left(−\mathrm{1}\right)^{\frac{{k}\left({k}+\mathrm{1}\right)}{\mathrm{2}}} {k}^{\mathrm{2}} . \\ $$$${Then}\:{S}_{{n}} \:{can}\:{take}\:{the}\:{value}\left({s}\right) \\ $$$$\left.\mathrm{1}\right)\:\mathrm{1056} \\ $$$$\left.\mathrm{2}\right)\:\mathrm{1088} \\ $$$$\left.\mathrm{3}\right)\:\mathrm{1120} \\ $$$$\left.\mathrm{4}\right)\:\mathrm{1332}. \\ $$

Question Number 31804 Answers: 1 Comments: 0

A quadratic equation p(x)=0 having coefficient of x^2 unity is such that p(x)=0 and p(p(p(x)))=0 have a common root then, prove that : p(0)×p(1)=0.

$${A}\:{quadratic}\:{equation}\:{p}\left({x}\right)=\mathrm{0}\:{having} \\ $$$${coefficient}\:{of}\:{x}^{\mathrm{2}} \:{unity}\:{is}\:{such}\:{that} \\ $$$${p}\left({x}\right)=\mathrm{0}\:{and}\:{p}\left({p}\left({p}\left({x}\right)\right)\right)=\mathrm{0}\:{have}\:{a}\: \\ $$$${common}\:{root}\:{then}, \\ $$$${prove}\:{that}\::\:\:{p}\left(\mathrm{0}\right)×{p}\left(\mathrm{1}\right)=\mathrm{0}. \\ $$

Question Number 31771 Answers: 1 Comments: 1

Consider a sequence in the form of groups (1),(2,2),(3,3,3),(4,4,4,4), (5,5,5,5,5),............ then the 2000th term of the above sequence is : ?

$${Consider}\:{a}\:{sequence}\:{in}\:{the}\:{form}\:{of} \\ $$$${groups}\:\left(\mathrm{1}\right),\left(\mathrm{2},\mathrm{2}\right),\left(\mathrm{3},\mathrm{3},\mathrm{3}\right),\left(\mathrm{4},\mathrm{4},\mathrm{4},\mathrm{4}\right), \\ $$$$\left(\mathrm{5},\mathrm{5},\mathrm{5},\mathrm{5},\mathrm{5}\right),............ \\ $$$${then}\:{the}\:\mathrm{2000}{th}\:{term}\:{of}\:{the}\:{above}\: \\ $$$${sequence}\:{is}\::\:? \\ $$

Question Number 31763 Answers: 3 Comments: 0

please find the integral solutions (x and y) (xy−7)^2 =x^2 +y^2

$${please}\:{find}\:{the}\:{integral}\:{solutions}\:\left({x}\:{and}\:{y}\right)\: \\ $$$$\left({xy}−\mathrm{7}\right)^{\mathrm{2}} \:={x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \\ $$

Question Number 31726 Answers: 1 Comments: 3

Let a_1 = (1/2) , a_(k+1) =a_k ^2 +a_k ∀ k≥ 1. then a_(101) is greater than a) 1 b) 2 c) 3 d) 4 .

$${Let}\:{a}_{\mathrm{1}} =\:\frac{\mathrm{1}}{\mathrm{2}}\:,\:{a}_{{k}+\mathrm{1}} ={a}_{{k}} ^{\mathrm{2}} +{a}_{{k}} \forall\:{k}\geqslant\:\mathrm{1}. \\ $$$${then}\:{a}_{\mathrm{101}} \:\:{is}\:{greater}\:{than} \\ $$$$\left.{a}\right)\:\mathrm{1}\: \\ $$$$\left.{b}\right)\:\mathrm{2} \\ $$$$\left.{c}\right)\:\mathrm{3} \\ $$$$\left.{d}\right)\:\mathrm{4}\:. \\ $$

Question Number 31677 Answers: 1 Comments: 0

24x^3 −26x^2 +9x−1=0(solve)

$$ \\ $$$$\mathrm{24}{x}^{\mathrm{3}} −\mathrm{26}{x}^{\mathrm{2}} +\mathrm{9}{x}−\mathrm{1}=\mathrm{0}\left({solve}\right) \\ $$

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