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Question Number 54607    Answers: 1   Comments: 2

If ((u^5 +v^5 )/((u+v)^5 )) = −(1/5) , find ((u^3 +v^3 )/((u+v)^3 )) = ?

$${If}\:\:\frac{{u}^{\mathrm{5}} +{v}^{\mathrm{5}} }{\left({u}+{v}\right)^{\mathrm{5}} }\:=\:−\frac{\mathrm{1}}{\mathrm{5}}\:, \\ $$$${find}\:\:\:\frac{{u}^{\mathrm{3}} +{v}^{\mathrm{3}} }{\left({u}+{v}\right)^{\mathrm{3}} }\:\:=\:? \\ $$

Question Number 54603    Answers: 1   Comments: 0

a,b,c ,are nonnegative real numbers and: a+b+c=1 . show that: 0≤ ab+bc+ca−2abc ≤(7/(27)) .

$${a},{b},{c}\:,{are}\:{nonnegative}\:{real}\:{numbers} \\ $$$${and}:\:\:\:{a}+{b}+{c}=\mathrm{1}\:\:. \\ $$$${show}\:{that}: \\ $$$$\:\:\:\:\:\:\mathrm{0}\leqslant\:\:\boldsymbol{\mathrm{ab}}+\boldsymbol{\mathrm{bc}}+\boldsymbol{\mathrm{ca}}−\mathrm{2}\boldsymbol{\mathrm{abc}}\:\:\:\leqslant\frac{\mathrm{7}}{\mathrm{27}}\:. \\ $$

Question Number 54602    Answers: 0   Comments: 1

in a given triangle: tg(C/2)=((a.tgA+b.tgB)/(a+b)) . define the kind of triangle.

$${in}\:{a}\:{given}\:{triangle}: \\ $$$$\:\:\:\boldsymbol{\mathrm{tg}}\frac{\boldsymbol{\mathrm{C}}}{\mathrm{2}}=\frac{\boldsymbol{\mathrm{a}}.\boldsymbol{\mathrm{tgA}}+\boldsymbol{\mathrm{b}}.\boldsymbol{\mathrm{tgB}}}{\boldsymbol{\mathrm{a}}+\boldsymbol{\mathrm{b}}}\:. \\ $$$$\boldsymbol{\mathrm{define}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{kind}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{triangle}}. \\ $$

Question Number 54600    Answers: 2   Comments: 2

solve for: x 1) (√(3−x))+(√(x+1))>(1/2) 2) cos^2 x+cos^2 2x+cos^2 3x=1 3)(√(x^2 −p))+2(√(x^2 −1))=x [p∈R]

$$\boldsymbol{\mathrm{solve}}\:\boldsymbol{\mathrm{for}}:\:\boldsymbol{\mathrm{x}} \\ $$$$\left.\:\:\:\mathrm{1}\right)\:\sqrt{\mathrm{3}−\boldsymbol{\mathrm{x}}}+\sqrt{\boldsymbol{\mathrm{x}}+\mathrm{1}}>\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\left.\:\:\:\:\mathrm{2}\right)\:\boldsymbol{\mathrm{cos}}^{\mathrm{2}} \boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{cos}}^{\mathrm{2}} \mathrm{2}\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{cos}}^{\mathrm{2}} \mathrm{3}\boldsymbol{\mathrm{x}}=\mathrm{1} \\ $$$$\left.\:\:\:\:\mathrm{3}\right)\sqrt{\boldsymbol{\mathrm{x}}^{\mathrm{2}} −\boldsymbol{\mathrm{p}}}+\mathrm{2}\sqrt{\boldsymbol{\mathrm{x}}^{\mathrm{2}} −\mathrm{1}}=\boldsymbol{\mathrm{x}}\:\:\:\:\:\left[\boldsymbol{\mathrm{p}}\in\boldsymbol{\mathrm{R}}\right] \\ $$

Question Number 54595    Answers: 0   Comments: 3

x+y=a z+bx=c bz+xy=d Find yz in terms of a,b,c,d.

$$\:\:\:\:\:{x}+{y}={a} \\ $$$$\:\:\:\:\:{z}+{bx}={c} \\ $$$$\:\:\:\:\:{bz}+{xy}={d} \\ $$$$\:\:\:\:\:{Find}\:{yz}\:{in}\:{terms}\:{of}\:{a},{b},{c},{d}. \\ $$

Question Number 54588    Answers: 1   Comments: 0

What is : (d/dx) [ u(x) × v(x) × w(x) ] = ... and more generally, what is : (d/dx) [ Π_(i=1) ^n u_i (x)] = ... Thank you

$$\mathrm{What}\:\mathrm{is}\:: \\ $$$$\:\:\:\frac{\mathrm{d}}{\mathrm{dx}}\:\left[\:{u}\left({x}\right)\:×\:{v}\left({x}\right)\:×\:{w}\left({x}\right)\:\right]\:=\:... \\ $$$$ \\ $$$$\mathrm{and}\:\mathrm{more}\:\mathrm{generally},\:\mathrm{what}\:\mathrm{is}\:: \\ $$$$\:\:\:\frac{\mathrm{d}}{\mathrm{dx}}\:\left[\:\underset{{i}=\mathrm{1}} {\overset{{n}} {\prod}}\:{u}_{{i}} \left({x}\right)\right]\:=\:... \\ $$$$ \\ $$$$\mathrm{Thank}\:\mathrm{you} \\ $$

Question Number 54586    Answers: 1   Comments: 0

(√(1−x^2 )) + (√(1−y^2 )) = a(x−y) prove that (dy/dx)=(√((1−y^2 )/(1−x^2 )))

$$\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }\:+\:\sqrt{\mathrm{1}−{y}^{\mathrm{2}} }\:=\:{a}\left({x}−{y}\right) \\ $$$${prove}\:{that} \\ $$$$\frac{{dy}}{{dx}}=\sqrt{\frac{\mathrm{1}−{y}^{\mathrm{2}} }{\mathrm{1}−{x}^{\mathrm{2}} }} \\ $$

Question Number 54585    Answers: 1   Comments: 1

show that ∫_0 ^∞ (x/(1+x^6 ))dx=(π/(3(√3)))

$${show}\:{that}\:\int_{\mathrm{0}} ^{\infty} \frac{{x}}{\mathrm{1}+{x}^{\mathrm{6}} }{dx}=\frac{\pi}{\mathrm{3}\sqrt{\mathrm{3}}} \\ $$$$ \\ $$

Question Number 54584    Answers: 2   Comments: 0

Question Number 54582    Answers: 0   Comments: 1

Solve for x: (1/(√(x + (√x) + 1))) + (2/(√(x + (√x) − 2))) = (√(x + 1))

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{x}:\:\:\:\:\:\frac{\mathrm{1}}{\sqrt{\mathrm{x}\:+\:\sqrt{\mathrm{x}}\:+\:\mathrm{1}}}\:+\:\frac{\mathrm{2}}{\sqrt{\mathrm{x}\:+\:\sqrt{\mathrm{x}}\:−\:\mathrm{2}}}\:\:=\:\sqrt{\mathrm{x}\:+\:\mathrm{1}} \\ $$

Question Number 54578    Answers: 1   Comments: 1

A particle of mass 1.5kg rests on a rough plane inclined at 45° to the horizontal. It is maintained in equilibrium by a horizontal force of p newtons. Given that the coefficient of friction between the particle and the plane is (1/4), calculate the value of p when the particle is on the point of moving i. down the plane ii. up the plane [take g=10ms^(−2) ].

$$\mathrm{A}\:\mathrm{particle}\:\mathrm{of}\:\mathrm{mass}\:\mathrm{1}.\mathrm{5kg}\:\mathrm{rests}\:\mathrm{on}\:\mathrm{a}\:\mathrm{rough}\: \\ $$$$\mathrm{plane}\:\mathrm{inclined}\:\mathrm{at}\:\mathrm{45}°\:\mathrm{to}\:\mathrm{the}\:\mathrm{horizontal}. \\ $$$$\mathrm{It}\:\mathrm{is}\:\mathrm{maintained}\:\mathrm{in}\:\mathrm{equilibrium}\:\mathrm{by}\:\mathrm{a}\: \\ $$$$\mathrm{horizontal}\:\mathrm{force}\:\mathrm{of}\:{p}\:\mathrm{newtons}.\:\mathrm{Given} \\ $$$$\mathrm{that}\:\mathrm{the}\:\mathrm{coefficient}\:\mathrm{of}\:\mathrm{friction}\:\mathrm{between} \\ $$$$\mathrm{the}\:\mathrm{particle}\:\mathrm{and}\:\mathrm{the}\:\mathrm{plane}\:\mathrm{is}\:\frac{\mathrm{1}}{\mathrm{4}},\:\mathrm{calculate} \\ $$$$\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{p}\:\mathrm{when}\:\mathrm{the}\:\mathrm{particle}\:\mathrm{is}\:\mathrm{on} \\ $$$$\mathrm{the}\:\mathrm{point}\:\mathrm{of}\:\mathrm{moving} \\ $$$$\mathrm{i}.\:\mathrm{down}\:\mathrm{the}\:\mathrm{plane} \\ $$$$\mathrm{ii}.\:\mathrm{up}\:\mathrm{the}\:\mathrm{plane} \\ $$$$\left[\mathrm{take}\:\mathrm{g}=\mathrm{10ms}^{−\mathrm{2}} \right]. \\ $$

Question Number 54573    Answers: 0   Comments: 1

calculate ∫_0 ^∞ ((cos(e^(−x^2 ) ))/(1+x^2 ))dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{cos}\left({e}^{−{x}^{\mathrm{2}} } \right)}{\mathrm{1}+{x}^{\mathrm{2}} }{dx}\: \\ $$

Question Number 54566    Answers: 0   Comments: 3

lim_(n→∞) (((1+c)/(1+μ^(1/n) )))^n =? with 0≤c≤1, μ>0

$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\left(\frac{\mathrm{1}+{c}}{\mathrm{1}+\mu^{\frac{\mathrm{1}}{{n}}} }\right)^{{n}} =? \\ $$$${with}\:\mathrm{0}\leqslant{c}\leqslant\mathrm{1},\:\mu>\mathrm{0} \\ $$

Question Number 54563    Answers: 1   Comments: 1

If ((sin^3 α)/(sin β)) + ((cos^3 α)/(cos β)) = 1 show that sin 2α+2sin (α+β)=0

$$\mathrm{If}\:\frac{\mathrm{sin}\:^{\mathrm{3}} \alpha}{\mathrm{sin}\:\beta}\:+\:\frac{\mathrm{cos}\:^{\mathrm{3}} \alpha}{\mathrm{cos}\:\beta}\:=\:\mathrm{1}\:\mathrm{show}\:\mathrm{that} \\ $$$$\mathrm{sin}\:\mathrm{2}\alpha+\mathrm{2sin}\:\left(\alpha+\beta\right)=\mathrm{0} \\ $$

Question Number 54562    Answers: 1   Comments: 0

Question Number 54556    Answers: 0   Comments: 4

∫_(−2π^2 ) ^(2π^2 ) ((sin (x^2 ))/x^2 ) dx

$$\int_{−\mathrm{2}\pi^{\mathrm{2}} } ^{\mathrm{2}\pi^{\mathrm{2}} } \frac{\mathrm{sin}\:\left({x}^{\mathrm{2}} \right)}{{x}^{\mathrm{2}} }\:{dx} \\ $$

Question Number 54543    Answers: 1   Comments: 1

Prove that: ((z^2 − 1)/(z^2 + 1)) = i tan(θ) where z = cos(θ) + i sin(θ)

$$\mathrm{Prove}\:\mathrm{that}:\:\:\:\frac{\mathrm{z}^{\mathrm{2}} \:−\:\mathrm{1}}{\mathrm{z}^{\mathrm{2}} \:+\:\mathrm{1}}\:\:=\:\:\mathrm{i}\:\mathrm{tan}\left(\theta\right) \\ $$$$\mathrm{where}\:\:\:\:\:\mathrm{z}\:\:=\:\:\mathrm{cos}\left(\theta\right)\:+\:\mathrm{i}\:\mathrm{sin}\left(\theta\right) \\ $$

Question Number 54541    Answers: 1   Comments: 0

The value of x between 0 and 2π which satisfy the equation sin x (√(8 cos^2 x)) = 1 are in AP Find the common difference.

$$\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:{x}\:\mathrm{between}\:\:\:\mathrm{0}\:\:\mathrm{and}\:\:\:\mathrm{2}\pi\: \\ $$$$\mathrm{which}\:\mathrm{satisfy}\:\mathrm{the}\:\mathrm{equation}\: \\ $$$$\mathrm{sin}\:{x}\:\sqrt{\mathrm{8}\:\mathrm{cos}\:^{\mathrm{2}} {x}}\:=\:\mathrm{1}\:\:\mathrm{are}\:\mathrm{in}\:\mathrm{AP}\: \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{common}\:\mathrm{difference}. \\ $$

Question Number 54537    Answers: 1   Comments: 2

tan^2 20+tan^2 40+tan^2 80=33 (please solve this and i want to know that which standard that question belongs?) help needed

$${tan}^{\mathrm{2}} \mathrm{20}+{tan}^{\mathrm{2}} \mathrm{40}+{tan}^{\mathrm{2}} \mathrm{80}=\mathrm{33}\:\:\: \\ $$$$\left({please}\:{solve}\:{this}\:{and}\:{i}\:{want}\:{to}\:{know}\:\right. \\ $$$$\left.{that}\:{which}\:{standard}\:{that}\:{question}\:{belongs}?\right) \\ $$$${help}\:{needed} \\ $$

Question Number 54536    Answers: 1   Comments: 0

If A,B,C are angles of a triangle show that tan^(−1) (cot Acot B)+tan^(−1) (cot Bcot C)+tan^(−1) (cot Ccot A) = tan^(−1) {1+((8cos Acos Bcos C)/(sin^2 2A+sin^2 2B+sin^2 2C))}

$$\mathrm{If}\:\mathrm{A},\mathrm{B},\mathrm{C}\:\mathrm{are}\:\mathrm{angles}\:\mathrm{of}\:\mathrm{a}\:\mathrm{triangle}\:\mathrm{show}\:\mathrm{that} \\ $$$$\mathrm{tan}^{−\mathrm{1}} \left(\mathrm{cot}\:\mathrm{Acot}\:\mathrm{B}\right)+\mathrm{tan}^{−\mathrm{1}} \left(\mathrm{cot}\:\mathrm{Bcot}\:\mathrm{C}\right)+\mathrm{tan}^{−\mathrm{1}} \left(\mathrm{cot}\:\mathrm{Ccot}\:\mathrm{A}\right) \\ $$$$=\:\mathrm{tan}^{−\mathrm{1}} \left\{\mathrm{1}+\frac{\mathrm{8cos}\:\mathrm{Acos}\:\mathrm{Bcos}\:\mathrm{C}}{\mathrm{sin}\:^{\mathrm{2}} \mathrm{2A}+\mathrm{sin}\:^{\mathrm{2}} \mathrm{2B}+\mathrm{sin}\:^{\mathrm{2}} \mathrm{2C}}\right\} \\ $$

Question Number 54526    Answers: 1   Comments: 1

Question Number 54517    Answers: 2   Comments: 0

Prove that (((√(h+1))−1)/h) = (1/((√(h+1))+1)) please...

$$\mathrm{Prove}\:\mathrm{that} \\ $$$$ \\ $$$$\frac{\sqrt{{h}+\mathrm{1}}−\mathrm{1}}{{h}}\:=\:\frac{\mathrm{1}}{\sqrt{{h}+\mathrm{1}}+\mathrm{1}} \\ $$$$ \\ $$$$\mathrm{please}... \\ $$

Question Number 54529    Answers: 1   Comments: 2

∫5cos x−4sin x/2cos x+sin x dx

$$\int\mathrm{5cos}\:{x}−\mathrm{4sin}\:{x}/\mathrm{2cos}\:{x}+\mathrm{sin}\:{x}\:{dx} \\ $$

Question Number 54513    Answers: 1   Comments: 0

Calculate the interquartile range: 20,26,27,30,34,41,41,64,65,65,72,85

$$\mathrm{Calculate}\:\mathrm{the}\:\mathrm{interquartile}\:\mathrm{range}: \\ $$$$\mathrm{20},\mathrm{26},\mathrm{27},\mathrm{30},\mathrm{34},\mathrm{41},\mathrm{41},\mathrm{64},\mathrm{65},\mathrm{65},\mathrm{72},\mathrm{85} \\ $$

Question Number 54506    Answers: 3   Comments: 0

L′Hopital rule lim_(x→α) ((f(x))/(g(x)))=lim_(x→α) ((f ′(x))/(g′(x)))= ((f ′(α))/(g′(α))) f ′(x)=(d/dx)f(x) , g′(x)=(d/dx)g(x) differential What is it? Proof of the rule.. plz :)

$${L}'{Hopital}\:{rule} \\ $$$$\underset{{x}\rightarrow\alpha} {\mathrm{lim}}\:\frac{{f}\left({x}\right)}{{g}\left({x}\right)}=\underset{{x}\rightarrow\alpha} {\mathrm{lim}}\:\frac{{f}\:'\left({x}\right)}{{g}'\left({x}\right)}=\:\frac{{f}\:'\left(\alpha\right)}{{g}'\left(\alpha\right)} \\ $$$${f}\:'\left({x}\right)=\frac{{d}}{{dx}}{f}\left({x}\right)\:,\:{g}'\left({x}\right)=\frac{{d}}{{dx}}{g}\left({x}\right)\:{differential} \\ $$$$\left.{What}\:{is}\:{it}?\:{Proof}\:{of}\:{the}\:{rule}..\:\mathrm{plz}\::\right) \\ $$

Question Number 54504    Answers: 1   Comments: 0

please,Sir. Would u explain to me how ? if f((x/(x − 1))) + 2f(((x − 1)/x)) = (x/(x − 1)) , then f(x) is ....

$$\boldsymbol{\mathrm{please}},\boldsymbol{\mathrm{Sir}}.\:\:\:\boldsymbol{\mathrm{Would}}\:\boldsymbol{\mathrm{u}}\:\boldsymbol{\mathrm{explain}}\:\boldsymbol{\mathrm{to}}\:\boldsymbol{\mathrm{me}}\:\boldsymbol{\mathrm{how}}\:? \\ $$$$\boldsymbol{\mathrm{if}}\:\:\boldsymbol{{f}}\left(\frac{\boldsymbol{{x}}}{\boldsymbol{{x}}\:−\:\mathrm{1}}\right)\:\:+\:\:\mathrm{2}\boldsymbol{{f}}\left(\frac{\boldsymbol{{x}}\:−\:\mathrm{1}}{\boldsymbol{{x}}}\right)\:\:=\:\:\frac{\boldsymbol{{x}}}{\boldsymbol{{x}}\:−\:\mathrm{1}}\:\:\:, \\ $$$$\boldsymbol{\mathrm{then}}\:\:\boldsymbol{{f}}\left(\boldsymbol{{x}}\right)\:\:\boldsymbol{\mathrm{is}}\:.... \\ $$

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