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Question Number 171563    Answers: 1   Comments: 0

evaluate ((√(((a−b)^7 + (b−c)^7 + (c−a)^7 )/((a−b)^3 + (b−c)^3 + (c−a)^3 )))/(a^2 +b^2 +c^2 −ab−bc−ca)) = ??

$$\:\:\:\:\:\:\:\:{evaluate}\:\:\: \\ $$$$\:\:\:\:\:\frac{\sqrt{\frac{\left({a}−{b}\right)^{\mathrm{7}} \:+\:\left({b}−{c}\right)^{\mathrm{7}} \:+\:\left({c}−{a}\right)^{\mathrm{7}} }{\left({a}−{b}\right)^{\mathrm{3}} \:+\:\left({b}−{c}\right)^{\mathrm{3}} \:+\:\left({c}−{a}\right)^{\mathrm{3}} }}}{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} −{ab}−{bc}−{ca}}\:=\:\:?? \\ $$

Question Number 171558    Answers: 0   Comments: 0

In △ABC , I-incenter ID⊥BC , IE⊥CA , IF⊥AB D∈(BC) , E∈(CA) , F∈(AB) I_a , I_b , I_c -excenters. Prove that: Σ_(cyc) ((EF)/(sin (A/2))) + Π_(cyc) ((EF)/(sin (A/2))) = ((1 + 4r^2 )/R) ∙ [I_a I_b I_c ]

$$\mathrm{In}\:\:\bigtriangleup\mathrm{ABC}\:,\:\mathrm{I}-\mathrm{incenter} \\ $$$$\mathrm{ID}\bot\mathrm{BC}\:,\:\mathrm{IE}\bot\mathrm{CA}\:,\:\mathrm{IF}\bot\mathrm{AB} \\ $$$$\mathrm{D}\in\left(\mathrm{BC}\right)\:,\:\mathrm{E}\in\left(\mathrm{CA}\right)\:,\:\mathrm{F}\in\left(\mathrm{AB}\right) \\ $$$$\mathrm{I}_{\boldsymbol{\mathrm{a}}} \:,\:\mathrm{I}_{\boldsymbol{\mathrm{b}}} \:,\:\mathrm{I}_{\boldsymbol{\mathrm{c}}} -\mathrm{excenters}.\:\mathrm{Prove}\:\mathrm{that}: \\ $$$$\underset{\boldsymbol{\mathrm{cyc}}} {\sum}\:\frac{\mathrm{EF}}{\mathrm{sin}\:\frac{\mathrm{A}}{\mathrm{2}}}\:\:+\:\:\underset{\boldsymbol{\mathrm{cyc}}} {\prod}\:\frac{\mathrm{EF}}{\mathrm{sin}\:\frac{\mathrm{A}}{\mathrm{2}}}\:\:=\:\:\frac{\mathrm{1}\:+\:\mathrm{4}\boldsymbol{\mathrm{r}}^{\mathrm{2}} }{\mathrm{R}}\:\centerdot\:\left[\mathrm{I}_{\boldsymbol{\mathrm{a}}} \mathrm{I}_{\boldsymbol{\mathrm{b}}} \mathrm{I}_{\boldsymbol{\mathrm{c}}} \right] \\ $$

Question Number 171552    Answers: 2   Comments: 0

Question Number 171593    Answers: 0   Comments: 2

The maximum value of the expression ∣(√(sin^2 x+2a^2 )) −(√(2a^2 −1−cos^2 x)) ∣ where a and x real numbers is−−−

$$\:{The}\:{maximum}\:{value}\:{of}\:{the} \\ $$$${expression}\:\mid\sqrt{\mathrm{sin}\:^{\mathrm{2}} {x}+\mathrm{2}{a}^{\mathrm{2}} }\:−\sqrt{\mathrm{2}{a}^{\mathrm{2}} −\mathrm{1}−\mathrm{cos}\:^{\mathrm{2}} {x}}\:\mid\: \\ $$$${where}\:{a}\:{and}\:{x}\:{real}\:{numbers}\:{is}−−− \\ $$

Question Number 171549    Answers: 1   Comments: 0

Solve for real numbers: { ((2x^2 + 3y^2 + z^2 = 7)),((x^2 + y^2 + z^2 = (√2) z (x + y))) :}

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{real}\:\mathrm{numbers}: \\ $$$$\begin{cases}{\mathrm{2x}^{\mathrm{2}} \:+\:\mathrm{3y}^{\mathrm{2}} \:+\:\mathrm{z}^{\mathrm{2}} \:=\:\mathrm{7}}\\{\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{y}^{\mathrm{2}} \:+\:\mathrm{z}^{\mathrm{2}} \:=\:\sqrt{\mathrm{2}}\:\mathrm{z}\:\left(\mathrm{x}\:+\:\mathrm{y}\right)}\end{cases} \\ $$

Question Number 171546    Answers: 0   Comments: 1

f(x)=((−ln∣x∣)/x)+x−2 , g(x)=−x^2 +1−ln∣x∣ Calculate the derivative of f(x) as a function of g(x)

$${f}\left({x}\right)=\frac{−{ln}\mid{x}\mid}{{x}}+{x}−\mathrm{2}\:\:,\:\:\:{g}\left({x}\right)=−{x}^{\mathrm{2}} +\mathrm{1}−{ln}\mid{x}\mid \\ $$$$ \\ $$Calculate the derivative of f(x) as a function of g(x)

Question Number 171530    Answers: 0   Comments: 0

Question Number 171529    Answers: 4   Comments: 1

Question Number 171528    Answers: 1   Comments: 0

Question Number 171527    Answers: 0   Comments: 0

Question Number 171519    Answers: 0   Comments: 0

Question Number 171518    Answers: 0   Comments: 2

Question Number 176847    Answers: 3   Comments: 0

If x, y ∈Z 23!=2^x 5^y k and k is an even number, then what is max(x+y) ?

$$\mathrm{If}\:{x},\:{y}\:\in\mathbb{Z} \\ $$$$\mathrm{23}!=\mathrm{2}^{{x}} \mathrm{5}^{{y}} {k} \\ $$$$\mathrm{and}\:{k}\:\mathrm{is}\:\mathrm{an}\:\mathrm{even}\:\mathrm{number},\:\mathrm{then}\:\mathrm{what}\:\mathrm{is}\: \\ $$$$\mathrm{max}\left({x}+{y}\right)\:? \\ $$

Question Number 176846    Answers: 1   Comments: 0

m∈Z What is the leading coefficient of the polynomial P(x)=4x^((13)/(m−5)) −6x^(25−2m) +4x^(13) +5x^(10) −4 ?

$${m}\in\mathbb{Z} \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{leading}\:\mathrm{coefficient}\:\mathrm{of}\:\mathrm{the}\:\mathrm{polynomial} \\ $$$$\mathrm{P}\left({x}\right)=\mathrm{4}{x}^{\frac{\mathrm{13}}{{m}−\mathrm{5}}} −\mathrm{6}{x}^{\mathrm{25}−\mathrm{2}{m}} +\mathrm{4}{x}^{\mathrm{13}} +\mathrm{5}{x}^{\mathrm{10}} −\mathrm{4}\:? \\ $$

Question Number 171514    Answers: 0   Comments: 0

compute by betta function ∫_0 ^( 4π) (√(cscx)) dx

$$\:{compute}\:{by}\:{betta}\:{function}\:\int_{\mathrm{0}} ^{\:\mathrm{4}\pi} \sqrt{{cscx}}\:{dx} \\ $$

Question Number 171512    Answers: 1   Comments: 0

Question Number 171507    Answers: 0   Comments: 0

Question Number 171502    Answers: 0   Comments: 1

Solve (dy/dx)(xcos y+asin 2y)=−1 (dy/dx)=(((y+2)/(x+y−1)))^2

$${Solve} \\ $$$$\frac{{dy}}{{dx}}\left({x}\mathrm{cos}\:{y}+{a}\mathrm{sin}\:\mathrm{2}{y}\right)=−\mathrm{1} \\ $$$$\frac{{dy}}{{dx}}=\left(\frac{{y}+\mathrm{2}}{{x}+{y}−\mathrm{1}}\right)^{\mathrm{2}} \\ $$

Question Number 171499    Answers: 1   Comments: 0

The sum of a sample of 20 numbers is 320 and the sum of their squares is 5840. Calculate the mean of the first nineteen numbers if the 20^(th) observation is 25.

$$\boldsymbol{\mathrm{The}}\:\boldsymbol{\mathrm{sum}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{a}}\:\boldsymbol{\mathrm{sample}}\:\boldsymbol{\mathrm{of}}\:\mathrm{20}\:\boldsymbol{\mathrm{numbers}}\:\boldsymbol{\mathrm{is}} \\ $$$$\mathrm{320}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{sum}}\:\boldsymbol{\mathrm{of}}\:\:\boldsymbol{\mathrm{their}}\:\boldsymbol{\mathrm{squares}}\: \\ $$$$\:\boldsymbol{\mathrm{is}}\:\mathrm{5840}.\:\boldsymbol{\mathrm{Calculate}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{mean}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{the}}\: \\ $$$$\:\boldsymbol{\mathrm{first}}\:\boldsymbol{\mathrm{nineteen}}\:\boldsymbol{\mathrm{numbers}}\:\boldsymbol{\mathrm{if}}\:\boldsymbol{\mathrm{the}}\:\mathrm{20}^{\boldsymbol{\mathrm{th}}} \: \\ $$$$\boldsymbol{\mathrm{observation}}\:\boldsymbol{\mathrm{is}}\:\mathrm{25}. \\ $$

Question Number 171497    Answers: 0   Comments: 3

Question Number 171493    Answers: 1   Comments: 0

Question Number 171491    Answers: 0   Comments: 0

Question Number 171490    Answers: 0   Comments: 0

Question Number 171479    Answers: 1   Comments: 5

tan^2 (𝛑/7) +tan^2 ((3𝛑)/7) +tan^2 ((5𝛑)/7)=?

$$\:\boldsymbol{{tan}}^{\mathrm{2}} \frac{\boldsymbol{\pi}}{\mathrm{7}}\:+\boldsymbol{{tan}}^{\mathrm{2}} \frac{\mathrm{3}\boldsymbol{\pi}}{\mathrm{7}}\:+\boldsymbol{{tan}}^{\mathrm{2}} \frac{\mathrm{5}\boldsymbol{\pi}}{\mathrm{7}}=? \\ $$

Question Number 171477    Answers: 0   Comments: 0

Question Number 171475    Answers: 1   Comments: 0

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