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Question Number 174568 Answers: 1 Comments: 0

If (1/a) + (1/b) + (1/c) = (1/(a + b + c)) then (1/a^3 ) + (1/(b^3 )) + (1/c^3 ) = ?

$$\mathrm{If}\:\frac{\mathrm{1}}{{a}}\:+\:\frac{\mathrm{1}}{{b}}\:+\:\frac{\mathrm{1}}{{c}}\:=\:\frac{\mathrm{1}}{{a}\:+\:{b}\:+\:{c}}\:\mathrm{then}\: \\ $$$$\frac{\mathrm{1}}{{a}^{\mathrm{3}} }\:+\:\frac{\mathrm{1}}{{b}^{\mathrm{3}} \:}\:+\:\frac{\mathrm{1}}{{c}^{\mathrm{3}} }\:=\:? \\ $$

Question Number 174567 Answers: 0 Comments: 0

Question Number 174566 Answers: 1 Comments: 0

Question Number 174560 Answers: 2 Comments: 0

Ω=∫_0 ^( 1) (( dx)/( (√(8+3x−(√(1+( x^( 2) +3x +2)(x^( 2) +7x+12)))))))

$$ \\ $$$$\:\:\:\Omega=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:{dx}}{\:\sqrt{\mathrm{8}+\mathrm{3}{x}−\sqrt{\mathrm{1}+\left(\:{x}^{\:\mathrm{2}} +\mathrm{3}{x}\:+\mathrm{2}\right)\left({x}^{\:\mathrm{2}} +\mathrm{7}{x}+\mathrm{12}\right)}}} \\ $$$$ \\ $$

Question Number 174556 Answers: 2 Comments: 1

Question Number 174552 Answers: 2 Comments: 3

Question Number 174548 Answers: 1 Comments: 0

Evaluate . 𝛀 = ∫_0 ^( 2) (( e^( x) )/(e^( 1−x) + e^( x−1) )) dx= ?

$$ \\ $$$$\boldsymbol{{Evaluate}}\:. \\ $$$$\:\:\boldsymbol{\Omega}\:=\:\int_{\mathrm{0}} ^{\:\mathrm{2}} \frac{\:\:\boldsymbol{{e}}^{\:\boldsymbol{{x}}} }{\boldsymbol{{e}}^{\:\mathrm{1}−\boldsymbol{{x}}} +\:\boldsymbol{{e}}^{\:\boldsymbol{{x}}−\mathrm{1}} }\:\boldsymbol{{dx}}=\:? \\ $$$$\:\:\:\: \\ $$

Question Number 174538 Answers: 1 Comments: 3

what is the maximum and minimum value of “x” if x+y+z = 6, x^2 +y^2 +z^2 = 18, for real values of “x,y and z”

$$\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{maximum}\:\mathrm{and} \\ $$$$\mathrm{minimum}\:\mathrm{value}\:\mathrm{of}\:``\mathrm{x}''\:\mathrm{if} \\ $$$$\mathrm{x}+\mathrm{y}+\mathrm{z}\:=\:\mathrm{6},\:\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} +\mathrm{z}^{\mathrm{2}} \:=\:\mathrm{18}, \\ $$$$\mathrm{for}\:\mathrm{real}\:\mathrm{values}\:\mathrm{of}\:``\mathrm{x},\mathrm{y}\:\mathrm{and}\:\mathrm{z}'' \\ $$

Question Number 174534 Answers: 0 Comments: 0

Question Number 174533 Answers: 1 Comments: 0

Question Number 174532 Answers: 0 Comments: 0

Question Number 174528 Answers: 1 Comments: 0

Question Number 174519 Answers: 1 Comments: 2

Question Number 174522 Answers: 0 Comments: 0

Let σ(n) be the sum of all positive divisors of the integer n and let p be any prime number. Show that σ(n)<2n holds true for all n of the form n=p^2 . Mastermind

$$\mathrm{Let}\:\sigma\left(\mathrm{n}\right)\:\mathrm{be}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{all}\:\mathrm{positive}\:\mathrm{divisors} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{integer}\:\mathrm{n}\:\mathrm{and}\:\mathrm{let}\:\mathrm{p}\:\mathrm{be}\:\mathrm{any}\:\mathrm{prime} \\ $$$$\mathrm{number}.\:\mathrm{Show}\:\mathrm{that}\:\sigma\left(\mathrm{n}\right)<\mathrm{2n}\:\mathrm{holds}\:\mathrm{true} \\ $$$$\mathrm{for}\:\mathrm{all}\:\mathrm{n}\:\mathrm{of}\:\mathrm{the}\:\mathrm{form}\:\mathrm{n}=\mathrm{p}^{\mathrm{2}} . \\ $$$$ \\ $$$$\mathrm{Mastermind} \\ $$

Question Number 174514 Answers: 3 Comments: 1

a+b+c=1 a^2 +b^2 +c^2 =2 a^3 +b^3 +c^3 =3 then a^5 +b^5 +c^5 ?

$${a}+{b}+{c}=\mathrm{1} \\ $$$${a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} =\mathrm{2} \\ $$$${a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}} =\mathrm{3} \\ $$$${then}\:{a}^{\mathrm{5}} +{b}^{\mathrm{5}} +{c}^{\mathrm{5}} \:? \\ $$

Question Number 174512 Answers: 1 Comments: 0

how many integer a,b∈z^+ a^5 −b^5 =10(b+1)^2 −9

$${how}\:{many}\:{integer}\:{a},{b}\in{z}^{+} \\ $$$${a}^{\mathrm{5}} −{b}^{\mathrm{5}} =\mathrm{10}\left({b}+\mathrm{1}\right)^{\mathrm{2}} −\mathrm{9} \\ $$$$ \\ $$

Question Number 174507 Answers: 1 Comments: 0

Determine the numerical value of the following expression without the use of a calculator log[log(3)∙(log(2)∙((((√3)−2sin((π/3)))/(π^3 +1))+1))−log(2)log(3)+(−1)^(100) ] Mastermind

$$\mathrm{Determine}\:\mathrm{the}\:\mathrm{numerical}\:\mathrm{value}\:\mathrm{of}\:\mathrm{the}\: \\ $$$$\mathrm{following}\:\mathrm{expression}\:\mathrm{without}\:\mathrm{the}\:\mathrm{use} \\ $$$$\mathrm{of}\:\mathrm{a}\:\mathrm{calculator} \\ $$$$\mathrm{log}\left[\mathrm{log}\left(\mathrm{3}\right)\centerdot\left(\mathrm{log}\left(\mathrm{2}\right)\centerdot\left(\frac{\sqrt{\mathrm{3}}−\mathrm{2sin}\left(\frac{\pi}{\mathrm{3}}\right)}{\pi^{\mathrm{3}} +\mathrm{1}}+\mathrm{1}\right)\right)−\mathrm{log}\left(\mathrm{2}\right)\mathrm{log}\left(\mathrm{3}\right)+\left(−\mathrm{1}\right)^{\mathrm{100}} \right] \\ $$$$ \\ $$$$\mathrm{Mastermind} \\ $$

Question Number 174500 Answers: 1 Comments: 0

Find the values of the following infinite sum: 1+(3/π)+(3/π^2 )+(3/π^3 )+(3/π^4 )+(3/π^5 )+... Mastermind

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{values}\:\mathrm{of}\:\mathrm{the}\:\mathrm{following}\:\mathrm{infinite} \\ $$$$\mathrm{sum}: \\ $$$$\mathrm{1}+\frac{\mathrm{3}}{\pi}+\frac{\mathrm{3}}{\pi^{\mathrm{2}} }+\frac{\mathrm{3}}{\pi^{\mathrm{3}} }+\frac{\mathrm{3}}{\pi^{\mathrm{4}} }+\frac{\mathrm{3}}{\pi^{\mathrm{5}} }+... \\ $$$$ \\ $$$$\mathrm{Mastermind} \\ $$

Question Number 174495 Answers: 2 Comments: 2

What are the roots of the function f(x)=(log(3^x )−2log(3))∙(x^2 −1) with x∈R? Mastermind

$$\mathrm{What}\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the}\:\mathrm{function} \\ $$$$\mathrm{f}\left(\mathrm{x}\right)=\left(\mathrm{log}\left(\mathrm{3}^{\mathrm{x}} \right)−\mathrm{2log}\left(\mathrm{3}\right)\right)\centerdot\left(\mathrm{x}^{\mathrm{2}} −\mathrm{1}\right)\:\mathrm{with} \\ $$$$\mathrm{x}\in\mathrm{R}? \\ $$$$ \\ $$$$\mathrm{Mastermind} \\ $$

Question Number 174492 Answers: 0 Comments: 1

lim_(x→∞) ((3x tan (2/x) − 2x sin (3/x))/(cos (1/x) − cos (2/x))) = ?

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\:\frac{\mathrm{3}{x}\:\mathrm{tan}\:\frac{\mathrm{2}}{{x}}\:−\:\mathrm{2}{x}\:\mathrm{sin}\:\frac{\mathrm{3}}{{x}}}{\mathrm{cos}\:\frac{\mathrm{1}}{{x}}\:−\:\mathrm{cos}\:\frac{\mathrm{2}}{{x}}}\:=\:\:? \\ $$

Question Number 174490 Answers: 0 Comments: 1

A die is rolled 57 times, what is the probability that the sum of its outcome is 100?

$$\mathrm{A}\:\mathrm{die}\:\mathrm{is}\:\mathrm{rolled}\:\mathrm{57}\:\mathrm{times},\:\mathrm{what}\:\mathrm{is}\:\mathrm{the} \\ $$$$\mathrm{probability}\:\mathrm{that}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{its}\:\mathrm{outcome} \\ $$$$\mathrm{is}\:\mathrm{100}? \\ $$

Question Number 174489 Answers: 0 Comments: 0

The points A, B and C have position vectors a, b and c respectively reffrred to an origin O. i. Given that the point X lie on AB produced so that AB : BX=2:1, find x, the position vector of X in terms of b and c. ii. if Y lies on BC, between B and C so that BY : YC = 1:3, find y, the position vector of Y in terms of b and c. iii. Given that Z is the mid point of AC, show that X, Y and Z are collinear.

$$\mathrm{The}\:\mathrm{points}\:\mathrm{A},\:\mathrm{B}\:\mathrm{and}\:\mathrm{C}\:\mathrm{have}\:\mathrm{position}\:\mathrm{vectors} \\ $$$$\boldsymbol{\mathrm{a}},\:\boldsymbol{\mathrm{b}}\:\mathrm{and}\:\boldsymbol{\mathrm{c}}\:\mathrm{respectively}\:\mathrm{reffrred}\:\mathrm{to}\:\mathrm{an}\:\mathrm{origin}\:\mathrm{O}. \\ $$$$\mathrm{i}.\:\mathrm{Given}\:\mathrm{that}\:\mathrm{the}\:\mathrm{point}\:\mathrm{X}\:\mathrm{lie}\:\mathrm{on}\:\mathrm{AB}\:\mathrm{produced} \\ $$$$\mathrm{so}\:\mathrm{that}\:\mathrm{AB}\::\:\mathrm{BX}=\mathrm{2}:\mathrm{1},\:\mathrm{find}\:{x},\:\mathrm{the}\:\mathrm{position} \\ $$$$\mathrm{vector}\:\mathrm{of}\:\mathrm{X}\:\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:\boldsymbol{\mathrm{b}}\:\mathrm{and}\:\boldsymbol{\mathrm{c}}. \\ $$$$\mathrm{ii}.\:\mathrm{if}\:\mathrm{Y}\:\mathrm{lies}\:\mathrm{on}\:\mathrm{BC},\:\mathrm{between}\:\mathrm{B}\:\mathrm{and}\:\mathrm{C}\:\mathrm{so}\:\mathrm{that} \\ $$$$\mathrm{BY}\::\:\mathrm{YC}\:=\:\mathrm{1}:\mathrm{3},\:\mathrm{find}\:{y},\:\mathrm{the}\:\mathrm{position}\:\mathrm{vector} \\ $$$$\mathrm{of}\:\mathrm{Y}\:\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:\boldsymbol{\mathrm{b}}\:\mathrm{and}\:\boldsymbol{\mathrm{c}}. \\ $$$$\mathrm{iii}.\:\mathrm{Given}\:\mathrm{that}\:\mathrm{Z}\:\:\mathrm{is}\:\mathrm{the}\:\mathrm{mid}\:\mathrm{point}\:\mathrm{of}\:\mathrm{AC},\: \\ $$$$\mathrm{show}\:\mathrm{that}\:\mathrm{X},\:\mathrm{Y}\:\mathrm{and}\:\mathrm{Z}\:\mathrm{are}\:\mathrm{collinear}. \\ $$

Question Number 174487 Answers: 1 Comments: 0

Question Number 174486 Answers: 0 Comments: 0

Question Number 174485 Answers: 0 Comments: 0

Question Number 174508 Answers: 1 Comments: 0

L(sin^n (x))=?

$$ \\ $$$$\mathscr{L}\left(\boldsymbol{\mathrm{sin}}^{\boldsymbol{\mathrm{n}}} \left(\boldsymbol{\mathrm{x}}\right)\right)=? \\ $$

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