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Question Number 162371 Answers: 2 Comments: 0

If x ∈R the maximum value of ((3x^2 +9x+17)/(3x^2 +9x+7)) is ...

$$\:\:{If}\:{x}\:\in\mathbb{R}\:{the}\:{maximum}\:{value}\: \\ $$$$\:{of}\:\frac{\mathrm{3}{x}^{\mathrm{2}} +\mathrm{9}{x}+\mathrm{17}}{\mathrm{3}{x}^{\mathrm{2}} +\mathrm{9}{x}+\mathrm{7}}\:{is}\:... \\ $$

Question Number 162368 Answers: 1 Comments: 2

Question Number 162367 Answers: 1 Comments: 0

Let x_1 ,x_2 ,x_3 be the roots of the equation x^3 +3x+5=0 . Then the value of expression (x_1 +(1/x_1 ))(x_2 +(1/x_2 ))(x_3 +(1/x_3 )) is equal to

$$\:\:{Let}\:{x}_{\mathrm{1}} ,{x}_{\mathrm{2}} ,{x}_{\mathrm{3}} \:{be}\:{the}\:{roots}\:{of}\:{the}\: \\ $$$${equation}\:{x}^{\mathrm{3}} +\mathrm{3}{x}+\mathrm{5}=\mathrm{0}\:.\:{Then}\:{the} \\ $$$${value}\:{of}\:{expression}\:\left({x}_{\mathrm{1}} +\frac{\mathrm{1}}{{x}_{\mathrm{1}} }\right)\left({x}_{\mathrm{2}} +\frac{\mathrm{1}}{{x}_{\mathrm{2}} }\right)\left({x}_{\mathrm{3}} +\frac{\mathrm{1}}{{x}_{\mathrm{3}} }\right)\:{is} \\ $$$$\:{equal}\:{to} \\ $$

Question Number 162366 Answers: 1 Comments: 0

Given that the solution set of the quadratic inequality ax^2 +bx+c >0 is (2,3). Then the solution set of the inequality cx^2 +bx+a <0 will be

$$\:{Given}\:{that}\:{the}\:{solution}\:{set}\:{of}\:{the}\: \\ $$$$\:{quadratic}\:{inequality}\:{ax}^{\mathrm{2}} +{bx}+{c}\:>\mathrm{0} \\ $$$$\:{is}\:\left(\mathrm{2},\mathrm{3}\right).\:{Then}\:{the}\:{solution}\:{set}\: \\ $$$$\:{of}\:{the}\:{inequality}\:{cx}^{\mathrm{2}} +{bx}+{a}\:<\mathrm{0}\: \\ $$$$\:{will}\:{be}\: \\ $$

Question Number 162365 Answers: 0 Comments: 0

∫_0 ^1 ∫_0 ^1 ∫_0 ^1 ln^2 (x+y+z)dxdydz=?

$$\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}{ln}^{\mathrm{2}} \left({x}+{y}+{z}\right){dxdydz}=? \\ $$

Question Number 162364 Answers: 2 Comments: 1

lim_(x→0) ((5sin x−sin 3x cos 2x−cos 3x sin 2x)/x^3 ) =?

$$\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{5sin}\:{x}−\mathrm{sin}\:\mathrm{3}{x}\:\mathrm{cos}\:\mathrm{2}{x}−\mathrm{cos}\:\mathrm{3}{x}\:\mathrm{sin}\:\mathrm{2}{x}}{{x}^{\mathrm{3}} }\:=? \\ $$

Question Number 162351 Answers: 1 Comments: 0

how to show f(x)=x^4 +2x^3 +5x^2 −16x−20 in the form of (x^2 +x+a)^2 −4(x+b)^2 .

$$\mathrm{how}\:\mathrm{to}\:\mathrm{show}\: \\ $$$${f}\left({x}\right)={x}^{\mathrm{4}} +\mathrm{2}{x}^{\mathrm{3}} +\mathrm{5}{x}^{\mathrm{2}} −\mathrm{16}{x}−\mathrm{20}\: \\ $$$$\mathrm{in}\:\mathrm{the}\:\mathrm{form}\:\mathrm{of}\:\left({x}^{\mathrm{2}} +{x}+{a}\right)^{\mathrm{2}} −\mathrm{4}\left({x}+{b}\right)^{\mathrm{2}} . \\ $$

Question Number 162348 Answers: 1 Comments: 4

Question Number 162344 Answers: 1 Comments: 1

Question Number 162338 Answers: 1 Comments: 0

(d^2 y/dx^2 ) - 3((dy/dx)) - 4y = tan(x)log(cos(x))

$$\frac{\mathrm{d}^{\mathrm{2}} \mathrm{y}}{\mathrm{dx}^{\mathrm{2}} }\:-\:\mathrm{3}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)\:-\:\mathrm{4y}\:=\:\mathrm{tan}\left(\mathrm{x}\right)\mathrm{log}\left(\mathrm{cos}\left(\mathrm{x}\right)\right) \\ $$

Question Number 162336 Answers: 1 Comments: 0

lim_( n→∞) ((1/(1+n^( 3) )) +(( 4)/(8 +n^( 3) )) + (9/(27 +n^( 3) )) +...+(n^( 2) /(2n^( 3) )) )=?

$$ \\ $$$${lim}_{\:{n}\rightarrow\infty} \:\left(\frac{\mathrm{1}}{\mathrm{1}+{n}^{\:\mathrm{3}} }\:+\frac{\:\mathrm{4}}{\mathrm{8}\:+{n}^{\:\mathrm{3}} }\:+\:\frac{\mathrm{9}}{\mathrm{27}\:+{n}^{\:\mathrm{3}} }\:+...+\frac{{n}^{\:\mathrm{2}} }{\mathrm{2}{n}^{\:\mathrm{3}} }\:\right)=? \\ $$$$ \\ $$

Question Number 162326 Answers: 1 Comments: 0

Hello please show it... a ∈ [0 , (π/4)] a ≤tan a ≤ 2a

$${Hello}\:{please}\:{show}\:{it}... \\ $$$$\:{a}\:\in\:\left[\mathrm{0}\:,\:\frac{\pi}{\mathrm{4}}\right]\:\:\:\:\:\:\:\:{a}\:\leqslant{tan}\:{a}\:\leqslant\:\mathrm{2}{a} \\ $$

Question Number 162309 Answers: 1 Comments: 3

Question Number 162305 Answers: 2 Comments: 0

proof that 2^(n+1) >(n+2)sin n

$${proof}\:{that} \\ $$$$\mathrm{2}^{{n}+\mathrm{1}} >\left({n}+\mathrm{2}\right)\mathrm{sin}\:{n} \\ $$

Question Number 162303 Answers: 0 Comments: 0

∫_0 ^1 ∫_0 ^1 ∫_0 ^1 ln^2 (x+y+z)dxdydz=?

$$\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}{ln}^{\mathrm{2}} \left({x}+{y}+{z}\right){dxdydz}=? \\ $$

Question Number 162280 Answers: 1 Comments: 0

y=x^(sinx) find y′

$${y}={x}^{{sinx}} \\ $$$${find}\:\:{y}' \\ $$

Question Number 162278 Answers: 0 Comments: 0

Question Number 162275 Answers: 0 Comments: 0

Question Number 162265 Answers: 0 Comments: 0

Calculate: Σ_(k=1) ^∞ ((H_(2k) (-1)^(k-1) )/(2k + 1)) where, H_n is the n-th harmonic number

$$\mathrm{Calculate}:\:\:\underset{\boldsymbol{\mathrm{k}}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{H}_{\mathrm{2}\boldsymbol{\mathrm{k}}} \:\left(-\mathrm{1}\right)^{\boldsymbol{\mathrm{k}}-\mathrm{1}} }{\mathrm{2k}\:+\:\mathrm{1}} \\ $$$$\mathrm{where},\:\mathrm{H}_{\boldsymbol{\mathrm{n}}} \:\mathrm{is}\:\mathrm{the}\:\mathrm{n}-\mathrm{th}\:\mathrm{harmonic}\:\mathrm{number} \\ $$

Question Number 162264 Answers: 0 Comments: 0

𝛀 =∫_( 0) ^( 1) ∫_( 0) ^( 1) ∫_( 0) ^( 1) ln^2 (x+y+z) dxdydz = ?

$$\boldsymbol{\Omega}\:=\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\:\mathrm{ln}^{\mathrm{2}} \left(\mathrm{x}+\mathrm{y}+\mathrm{z}\right)\:\mathrm{dxdydz}\:=\:? \\ $$

Question Number 162261 Answers: 3 Comments: 1

find Σ_(n=1) ^∞ (1/(5^n −1))=? or generally Φ(k)= Σ_(n=1) ^∞ (1/(k^n −1))=? with k∈N, k≥2

$${find}\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{\mathrm{5}^{{n}} −\mathrm{1}}=? \\ $$$${or}\:{generally} \\ $$$$\Phi\left({k}\right)=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{{k}^{{n}} −\mathrm{1}}=?\:{with}\:{k}\in{N},\:{k}\geqslant\mathrm{2} \\ $$

Question Number 162301 Answers: 1 Comments: 0

lim _(x→(π/2)) ( 1− sin(x))^( ( tan((x/2))−1 )) =?

$$\: \\ $$$$\mathrm{lim}\:_{{x}\rightarrow\frac{\pi}{\mathrm{2}}} \:\left(\:\mathrm{1}−\:{sin}\left({x}\right)\right)^{\:\left(\:{tan}\left(\frac{{x}}{\mathrm{2}}\right)−\mathrm{1}\:\right)} =? \\ $$$$ \\ $$

Question Number 162299 Answers: 1 Comments: 0

calculta ∫_0 ^∞ ((lnx)/((x^2 +x+1)^2 ))dx

$$\mathrm{calculta}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{lnx}}{\left(\mathrm{x}^{\mathrm{2}} +\mathrm{x}+\mathrm{1}\right)^{\mathrm{2}} }\mathrm{dx} \\ $$

Question Number 162298 Answers: 1 Comments: 0

find ∫_0 ^1 lnx ln(1−x^3 )dx

$$\mathrm{find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{lnx}\:\mathrm{ln}\left(\mathrm{1}−\mathrm{x}^{\mathrm{3}} \right)\mathrm{dx} \\ $$

Question Number 162297 Answers: 1 Comments: 0

find ∫_0 ^1 ln(1−x)ln(1+x)dx

$$\mathrm{find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{ln}\left(\mathrm{1}−\mathrm{x}\right)\mathrm{ln}\left(\mathrm{1}+\mathrm{x}\right)\mathrm{dx} \\ $$

Question Number 162253 Answers: 1 Comments: 0

The tangent of a parabola y^2 =4ax at the point P (ap^2 , 2ap) intersects the line x+a=0 at T . (i) If M is the midpoint of PT , find the coordinates of M in terms of a and p. (ii) Prove that the equation of locus of M is y^2 (2x+a)=a(3x+a)^2

$$\mathrm{The}\:\mathrm{tangent}\:\mathrm{of}\:\mathrm{a}\:\mathrm{parabola}\:{y}^{\mathrm{2}} =\mathrm{4}{ax}\:\mathrm{at}\:\mathrm{the}\:\mathrm{point} \\ $$$${P}\:\left({ap}^{\mathrm{2}} ,\:\mathrm{2}{ap}\right)\:\mathrm{intersects}\:\mathrm{the}\:\mathrm{line}\:{x}+{a}=\mathrm{0}\:\mathrm{at}\:{T}\:. \\ $$$$\left(\mathrm{i}\right)\:\mathrm{If}\:{M}\:\mathrm{is}\:\mathrm{the}\:\mathrm{midpoint}\:\mathrm{of}\:{PT}\:,\:\mathrm{find}\:\mathrm{the}\: \\ $$$$\:\:\:\:\:\:\mathrm{coordinates}\:\mathrm{of}\:{M}\:\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:{a}\:\mathrm{and}\:{p}. \\ $$$$\left(\mathrm{ii}\right)\:\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{locus}\:\mathrm{of}\:{M}\:\mathrm{is} \\ $$$$\:\:\:\:\:\:\:\:{y}^{\mathrm{2}} \left(\mathrm{2}{x}+{a}\right)={a}\left(\mathrm{3}{x}+{a}\right)^{\mathrm{2}} \\ $$

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