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Question Number 171213 Answers: 0 Comments: 0

In electricity, the electrostatic field is defined as: E = ∫_0 ^π [((a^2 σ sin θ)/(2ε(√(a^2 −x^2 −2ax cosθ))))]dθ where a,σ and ε are constants. Consider that x>a and show that E= ((a^2 σ)/(εx))

$$\mathrm{In}\:\mathrm{electricity},\:\mathrm{the}\:\mathrm{electrostatic}\:\mathrm{field} \\ $$$$\mathrm{is}\:\mathrm{defined}\:\mathrm{as}: \\ $$$${E}\:=\:\int_{\mathrm{0}} ^{\pi} \left[\frac{{a}^{\mathrm{2}} \sigma\:\mathrm{sin}\:\theta}{\mathrm{2}\epsilon\sqrt{{a}^{\mathrm{2}} −{x}^{\mathrm{2}} −\mathrm{2}{ax}\:\mathrm{cos}\theta}}\right]{d}\theta \\ $$$$\mathrm{where}\:{a},\sigma\:\mathrm{and}\:\epsilon\:\mathrm{are}\:\mathrm{constants}.\:\mathrm{Consider} \\ $$$$\mathrm{that}\:{x}>{a}\:\mathrm{and}\:\mathrm{show}\:\mathrm{that}\:{E}=\:\frac{{a}^{\mathrm{2}} \sigma}{\epsilon{x}} \\ $$

Question Number 171199 Answers: 2 Comments: 1

solve x^2 +y^2 −xy=9 y^2 +z^2 −yz=30 z^2 +x^2 −zx=50

$${solve} \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} −{xy}=\mathrm{9} \\ $$$${y}^{\mathrm{2}} +{z}^{\mathrm{2}} −{yz}=\mathrm{30} \\ $$$${z}^{\mathrm{2}} +{x}^{\mathrm{2}} −{zx}=\mathrm{50} \\ $$

Question Number 171198 Answers: 1 Comments: 0

evaluate ∫_0 ^( π) log (a+cos x)dx

$$ \\ $$$$\:\:{evaluate} \\ $$$$\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\pi} \mathrm{log}\:\left({a}+\mathrm{cos}\:{x}\right){dx} \\ $$

Question Number 171194 Answers: 0 Comments: 0

Question Number 171179 Answers: 1 Comments: 0

Question Number 171171 Answers: 1 Comments: 2

Question Number 171170 Answers: 0 Comments: 0

Question Number 171165 Answers: 1 Comments: 0

A triangle QRS is to be constructed from a line segment of lenght 15cm. Construct the triangle using the division of the line segment into the ratio 5:4:3 such that QS and RS are laegest and smallest ratio respectively. circumscribe the triangle by locating the circumcenter.

$${A}\:{triangle}\:{QRS}\:{is}\:{to}\:{be}\:{constructed}\:{from} \\ $$$${a}\:{line}\:{segment}\:{of}\:{lenght}\:\mathrm{15}{cm}.\: \\ $$$${Construct}\:{the}\:{triangle}\:{using}\:{the} \\ $$$${division}\:{of}\:{the}\:{line}\:{segment}\:{into}\:{the} \\ $$$${ratio}\:\mathrm{5}:\mathrm{4}:\mathrm{3}\:{such}\:{that}\:{QS}\:{and}\:{RS}\:{are} \\ $$$${laegest}\:{and}\:{smallest}\:{ratio}\:{respectively}. \\ $$$${circumscribe}\:{the}\:{triangle}\:{by}\:{locating} \\ $$$${the}\:{circumcenter}. \\ $$

Question Number 171164 Answers: 2 Comments: 0

show that the common chord of the circle x^2 +y^2 =4 and x^2 +y^2 −4x−2y−4=0 passes through the origin.

Question Number 171161 Answers: 1 Comments: 0

Question Number 171160 Answers: 1 Comments: 0

Question Number 171156 Answers: 0 Comments: 0

show that the equation 3x^2 +3y^2 −24x+12y+12=0 is a circle

$$\boldsymbol{\mathrm{show}}\:\boldsymbol{\mathrm{that}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{equation}}\: \\ $$$$\:\mathrm{3}\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\mathrm{3}\boldsymbol{\mathrm{y}}^{\mathrm{2}} −\mathrm{24}\boldsymbol{\mathrm{x}}+\mathrm{12}\boldsymbol{\mathrm{y}}+\mathrm{12}=\mathrm{0} \\ $$$$\:\boldsymbol{\mathrm{is}}\:\boldsymbol{\mathrm{a}}\:\boldsymbol{\mathrm{circle}} \\ $$

Question Number 171153 Answers: 1 Comments: 0

An aeroplane covers a certain distance at a speed of 240kmph in 5 hours. to cover the same distance in 1(2/3) hrs, it must travel at a speed of: A) 300kmph B) 360kmph C) 600kmph D) 700kmph Mastermind

$${An}\:{aeroplane}\:{covers}\:{a}\:{certain}\:{distance} \\ $$$${at}\:{a}\:{speed}\:{of}\:\mathrm{240}{kmph}\:{in}\:\mathrm{5}\:{hours}. \\ $$$${to}\:{cover}\:{the}\:{same}\:{distance}\:{in}\:\mathrm{1}\frac{\mathrm{2}}{\mathrm{3}}\:{hrs}, \\ $$$${it}\:{must}\:{travel}\:{at}\:{a}\:{speed}\:{of}: \\ $$$$\left.{A}\right)\:\mathrm{300}{kmph} \\ $$$$\left.{B}\right)\:\mathrm{360}{kmph} \\ $$$$\left.{C}\right)\:\mathrm{600}{kmph} \\ $$$$\left.{D}\right)\:\mathrm{700}{kmph} \\ $$$$ \\ $$$${Mastermind} \\ $$

Question Number 171152 Answers: 1 Comments: 0

The average ages of three person is 27 years. Their ages are in the propor tion of 1:3:5. What is the age in years of the youngest one among them? Mastermind

$${The}\:{average}\:{ages}\:{of}\:{three}\:{person}\:{is} \\ $$$$\mathrm{27}\:{years}.\:{Their}\:{ages}\:{are}\:{in}\:{the}\:{propor} \\ $$$${tion}\:{of}\:\mathrm{1}:\mathrm{3}:\mathrm{5}.\:{What}\:{is}\:{the}\:{age}\:{in}\:{years} \\ $$$${of}\:{the}\:{youngest}\:{one}\:{among}\:{them}? \\ $$$$ \\ $$$${Mastermind} \\ $$

Question Number 171189 Answers: 2 Comments: 0

f(x)=((16^x )/(4+16^x )) faind volue of f((1/(20)))+f((2/(20)))+........+f(((19)/(20)))

$${f}\left({x}\right)=\frac{\mathrm{16}^{{x}} }{\mathrm{4}+\mathrm{16}^{{x}} }\:\:\:\:\:{faind}\:{volue}\:{of} \\ $$$${f}\left(\frac{\mathrm{1}}{\mathrm{20}}\right)+{f}\left(\frac{\mathrm{2}}{\mathrm{20}}\right)+........+{f}\left(\frac{\mathrm{19}}{\mathrm{20}}\right) \\ $$

Question Number 171147 Answers: 0 Comments: 2

Question Number 171146 Answers: 0 Comments: 2

Question Number 171145 Answers: 0 Comments: 0

let p be a non constant polynomial with degree n such that: p(z)=a_n z^n +...+a_0 show that if z∈C with∣z∣≥max{1,(2/(∣a_n ∣))Σ_(i=1) ^(n−1) ∣a_i ∣} then (1/2)∣a_n ∣∣z∣^n ≤∣p(z)∣≤(3/2)∣a_n ∣∣z∣^n (i already did the right side of the inequality so try to show ∣p(z)∣≥(1/2)∣a_n ∣∣z∣^n )

$$\mathrm{let}\:\mathrm{p}\:\mathrm{be}\:\mathrm{a}\:\mathrm{non}\:\mathrm{constant}\:\mathrm{polynomial}\:\mathrm{with}\: \\ $$$$\mathrm{degree}\:\mathrm{n}\:\mathrm{such}\:\mathrm{that}:\:\mathrm{p}\left(\mathrm{z}\right)=\mathrm{a}_{\mathrm{n}} \mathrm{z}^{\mathrm{n}} +...+\mathrm{a}_{\mathrm{0}} \\ $$$$\mathrm{show}\:\mathrm{that}\:\mathrm{if}\:\mathrm{z}\in\mathbb{C}\:\mathrm{with}\mid\mathrm{z}\mid\geqslant\mathrm{max}\left\{\mathrm{1},\frac{\mathrm{2}}{\mid\mathrm{a}_{\mathrm{n}} \mid}\underset{\mathrm{i}=\mathrm{1}} {\overset{\mathrm{n}−\mathrm{1}} {\sum}}\mid\mathrm{a}_{\mathrm{i}} \mid\right\} \\ $$$$\mathrm{then}\:\frac{\mathrm{1}}{\mathrm{2}}\mid\mathrm{a}_{\mathrm{n}} \mid\mid\mathrm{z}\mid^{\mathrm{n}} \leqslant\mid\mathrm{p}\left(\mathrm{z}\right)\mid\leqslant\frac{\mathrm{3}}{\mathrm{2}}\mid\mathrm{a}_{\mathrm{n}} \mid\mid\mathrm{z}\mid^{\mathrm{n}} \\ $$$$\left(\mathrm{i}\:\mathrm{already}\:\mathrm{did}\:\mathrm{the}\:\mathrm{right}\:\mathrm{side}\:\mathrm{of}\:\mathrm{the}\:\mathrm{inequality}\right. \\ $$$$\left.\mathrm{so}\:\mathrm{try}\:\mathrm{to}\:\mathrm{show}\:\mid\mathrm{p}\left(\mathrm{z}\right)\mid\geqslant\frac{\mathrm{1}}{\mathrm{2}}\mid\mathrm{a}_{\mathrm{n}} \mid\mid\mathrm{z}\mid^{\mathrm{n}} \right) \\ $$

Question Number 171136 Answers: 0 Comments: 4

Using Taylor′s theorem, prove that x−(x^3 /6)<sin x<x−(x^3 /6)+(x^5 /(120)) for x>0

$$\mathrm{Using}\:\mathrm{Taylor}'\mathrm{s}\:\mathrm{theorem},\:\mathrm{prove}\:\mathrm{that} \\ $$$${x}−\frac{{x}^{\mathrm{3}} }{\mathrm{6}}<\mathrm{sin}\:{x}<{x}−\frac{{x}^{\mathrm{3}} }{\mathrm{6}}+\frac{{x}^{\mathrm{5}} }{\mathrm{120}}\:\:\:\mathrm{for}\:{x}>\mathrm{0} \\ $$

Question Number 171134 Answers: 0 Comments: 0