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Question Number 32206 Answers: 0 Comments: 0

Find Σ_(k=1) ^∞ (∫_(k−1) ^k x^(−x) dx) .

$$\mathrm{Find}\:\underset{\mathrm{k}=\mathrm{1}} {\overset{\infty} {\sum}}\left(\underset{\mathrm{k}−\mathrm{1}} {\overset{\mathrm{k}} {\int}}\mathrm{x}^{−\mathrm{x}} \:\mathrm{dx}\right)\:. \\ $$$$ \\ $$

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 32191 Answers: 1 Comments: 1

Question Number 32211 Answers: 1 Comments: 1

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 32163 Answers: 0 Comments: 0

If the corrdinater of the verticle of an eqvilateral triangle with length x are (x_(1+) y_1 ),(y_1 +y_2 ) and (x_3 ,y_3 ) then ( determinant (((x_1 y_1 2)),((x_2 y_2 2)),((x_3 y_3 2))))^2 =3a^4 ?

$${If}\:{the}\:{corrdinater}\:{of}\:{the}\:{verticle}\:{of}\:{an} \\ $$$${eqvilateral}\:{triangle}\:{with}\:{length}\:{x}\:{are} \\ $$$$\left({x}_{\mathrm{1}+} {y}_{\mathrm{1}} \right),\left({y}_{\mathrm{1}} +{y}_{\mathrm{2}} \right)\:{and}\:\left({x}_{\mathrm{3}} ,{y}_{\mathrm{3}} \right)\:{then} \\ $$$$\left(\begin{vmatrix}{{x}_{\mathrm{1}} \:\:\:{y}_{\mathrm{1}} \:\:\:\mathrm{2}}\\{{x}_{\mathrm{2}} \:\:\:{y}_{\mathrm{2}} \:\:\:\mathrm{2}}\\{{x}_{\mathrm{3}} \:\:\:{y}_{\mathrm{3}} \:\:\:\mathrm{2}}\end{vmatrix}\right)^{\mathrm{2}} =\mathrm{3}{a}^{\mathrm{4}} ? \\ $$

Question Number 32161 Answers: 1 Comments: 0

Let a function F :R→R be defined by f(x)=1+ax,α≠ 0 for all X ∈ R. Show that f is invertible and find its inverse function.Also find the value (s) of α if inverse of f is itself

$${Let}\:{a}\:{function}\:{F}\::{R}\rightarrow{R}\:{be}\:{defined}\:{by} \\ $$$${f}\left({x}\right)=\mathrm{1}+{ax},\alpha\neq\:\mathrm{0}\:{for}\:{all}\:{X}\:\in\:{R}.\:{Show} \\ $$$${that}\:{f}\:{is}\:{invertible}\:{and}\:{find}\:{its}\:{inverse} \\ $$$${function}.{Also}\:{find}\:{the}\:{value}\:\left({s}\right)\:{of}\:\alpha \\ $$$${if}\:{inverse}\:{of}\:{f}\:{is}\:{itself} \\ $$

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 32158 Answers: 0 Comments: 0

an elevator of mass 250kg is carrying 3 person whose masses 60kg, 80kg, 100kg and the force exacted by the motion is 5000N. a) With what acceleration will the elevator ascend b) Starting from rest,how far will it go in 5s. (acceleration to gravity G=9.8ms^(−2) ).

$${an}\:{elevator}\:{of}\:{mass}\:\mathrm{250}{kg}\:{is}\:{carrying}\:\mathrm{3}\:{person}\:{whose}\:{masses}\:\mathrm{60}{kg},\:\mathrm{80}{kg},\:\mathrm{100}{kg}\:{and}\:{the}\:{force}\:{exacted}\:{by}\:{the}\:{motion}\:{is}\:\mathrm{5000}{N}. \\ $$$$\left.{a}\right)\:{With}\:{what}\:{acceleration}\:{will}\:{the}\:{elevator}\:{ascend} \\ $$$$\left.{b}\right)\:{Starting}\:{from}\:{rest},{how}\:{far}\:{will}\:{it}\:{go}\:{in}\:\mathrm{5}{s}. \\ $$$$\left({acceleration}\:{to}\:{gravity}\:{G}=\mathrm{9}.\mathrm{8}{ms}^{−\mathrm{2}} \right). \\ $$

Question Number 32164 Answers: 0 Comments: 0

Determine wether the relation on the set R of all real number as R={(a,b):a,b∈R and a−b +(√3) ∈ s where s is the set of all irrational no) is reflexive, symmetric and transitive?

$${Determine}\:{wether}\:{the}\:{relation}\:{on}\:{the} \\ $$$${set}\:{R}\:{of}\:{all}\:{real}\:{number}\:{as} \\ $$$${R}=\left\{\left({a},{b}\right):{a},{b}\in{R}\:{and}\:{a}−{b}\:+\sqrt{\mathrm{3}}\:\in\:{s}\:\right. \\ $$$$\left.{where}\:{s}\:{is}\:{the}\:{set}\:{of}\:{all}\:{irrational}\:{no}\right) \\ $$$${is}\:{reflexive},\:{symmetric}\:{and}\:{transitive}? \\ $$

Question Number 32156 Answers: 0 Comments: 2

evaluate∫((√(cos 2x))/(sin x))dx

$${evaluate}\int\frac{\sqrt{{cos}\:\mathrm{2}{x}}}{{sin}\:{x}}{dx} \\ $$

Question Number 32142 Answers: 0 Comments: 0