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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

Question Number 32150 Answers: 0 Comments: 0

Question Number 32139 Answers: 0 Comments: 4

Find the ∫ ((x+1)/(x^2 +x+1))dx

$${F}\boldsymbol{{ind}}\:\boldsymbol{{the}} \\ $$$$\int\:\frac{{x}+\mathrm{1}}{{x}^{\mathrm{2}} +{x}+\mathrm{1}}{dx} \\ $$

Question Number 32137 Answers: 0 Comments: 0

lim_(x→5) ((f(x)g(x) − 3g(x) − 3)/(f(x) − 3(x − 5))) = 0 Find the value of g′(5)

$$\underset{{x}\rightarrow\mathrm{5}} {\mathrm{lim}}\:\frac{{f}\left({x}\right){g}\left({x}\right)\:−\:\mathrm{3}{g}\left({x}\right)\:−\:\mathrm{3}}{{f}\left({x}\right)\:−\:\mathrm{3}\left({x}\:−\:\mathrm{5}\right)}\:=\:\mathrm{0} \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{g}'\left(\mathrm{5}\right) \\ $$

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 32151 Answers: 1 Comments: 0

Question Number 32126 Answers: 1 Comments: 1

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 32109 Answers: 0 Comments: 0

Question Number 32100 Answers: 0 Comments: 0

Question Number 32099 Answers: 1 Comments: 0

Question Number 32098 Answers: 0 Comments: 14

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 32093 Answers: 1 Comments: 0

Find the number of ways of selecting 9 balls from 6 red balls, 5 white balls and 5 blue balls if each selection consists of 3 balls of each colour.

$${Find}\:{the}\:{number}\:{of}\:{ways}\:{of} \\ $$$${selecting}\:\mathrm{9}\:{balls}\:{from}\:\mathrm{6}\:{red}\:{balls}, \\ $$$$\mathrm{5}\:{white}\:{balls}\:{and}\:\mathrm{5}\:{blue}\:{balls}\:{if} \\ $$$${each}\:{selection}\:{consists}\:{of}\:\mathrm{3}\:{balls} \\ $$$${of}\:{each}\:{colour}. \\ $$

Question Number 32076 Answers: 1 Comments: 2

Question Number 32075 Answers: 0 Comments: 1

A rigid uniform bar of length 2.4m is pivoted horizontally at its mid-point. Weights are hung from two points of the bar such that a 200N is at the 0.4m mark and a 300N is at 1.6m mark.To maintain horizontal equilibrum a couple is applied to the bar. What is the torque and direction of the couple?

$${A}\:{rigid}\:{uniform}\:{bar}\:{of}\:{length}\:\mathrm{2}.\mathrm{4}{m} \\ $$$${is}\:{pivoted}\:{horizontally}\:{at}\:{its}\:{mid}-{point}. \\ $$$${Weights}\:{are}\:{hung}\:{from}\:{two}\:{points} \\ $$$${of}\:{the}\:{bar}\:{such}\:{that}\:{a}\:\mathrm{200}{N}\:{is}\:{at} \\ $$$${the}\:\mathrm{0}.\mathrm{4}{m}\:{mark}\:{and}\:{a}\:\mathrm{300}{N}\:{is}\:{at} \\ $$$$\mathrm{1}.\mathrm{6}{m}\:{mark}.{To}\:{maintain} \\ $$$${horizontal}\:{equilibrum}\:{a}\:{couple}\:{is} \\ $$$${applied}\:{to}\:{the}\:{bar}.\:{What}\:{is}\:{the} \\ $$$${torque}\:{and}\:{direction}\:{of}\:{the}\:{couple}? \\ $$

Question Number 32074 Answers: 0 Comments: 1

A machine with a velocity ratio of 5 requires 150J of work to raise a 500N load through a vertical distance of 200cm.Calculate a)the efficiency of the machine b)the mechanical advantage of the machine.

$${A}\:{machine}\:{with}\:{a}\:{velocity}\:{ratio} \\ $$$${of}\:\mathrm{5}\:{requires}\:\mathrm{150}{J}\:{of}\:{work}\:{to}\:{raise} \\ $$$${a}\:\mathrm{500}{N}\:{load}\:{through}\:{a}\:{vertical} \\ $$$${distance}\:{of}\:\mathrm{200}{cm}.{Calculate} \\ $$$$\left.{a}\right){the}\:{efficiency}\:{of}\:{the}\:{machine} \\ $$$$\left.{b}\right){the}\:{mechanical}\:{advantage}\:{of} \\ $$$${the}\:{machine}. \\ $$

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