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Question Number 118084 Answers: 0 Comments: 1

suppose f: [1,3 ]→ [−1,1 ] such that ∫_1 ^3 f(x) dx = 0 . What the maximum value of ∫_1 ^3 x^(−1) .f(x) dx ?

$$\mathrm{suppose}\:\mathrm{f}:\:\left[\mathrm{1},\mathrm{3}\:\right]\rightarrow\:\left[−\mathrm{1},\mathrm{1}\:\right]\:\mathrm{such} \\ $$$$\mathrm{that}\:\underset{\mathrm{1}} {\overset{\mathrm{3}} {\int}}\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{dx}\:=\:\mathrm{0}\:.\:\mathrm{What}\:\mathrm{the}\:\mathrm{maximum} \\ $$$$\mathrm{value}\:\mathrm{of}\:\underset{\mathrm{1}} {\overset{\mathrm{3}} {\int}}\:\mathrm{x}^{−\mathrm{1}} .\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{dx}\:? \\ $$

Question Number 118083 Answers: 1 Comments: 0

Question Number 118081 Answers: 1 Comments: 3

∫ arc cos (((cos x)/(1+2cos x))) dx =?

$$\:\int\:\mathrm{arc}\:\mathrm{cos}\:\left(\frac{\mathrm{cos}\:\mathrm{x}}{\mathrm{1}+\mathrm{2cos}\:\mathrm{x}}\right)\:\mathrm{dx}\:=? \\ $$

Question Number 118078 Answers: 2 Comments: 0

3(cos x+sin x)^4 +6(sin x−cos x)^2 − 3(sin^6 x+cos^6 x) = ?

$$\mathrm{3}\left(\mathrm{cos}\:\mathrm{x}+\mathrm{sin}\:\mathrm{x}\right)^{\mathrm{4}} +\mathrm{6}\left(\mathrm{sin}\:\mathrm{x}−\mathrm{cos}\:\mathrm{x}\right)^{\mathrm{2}} − \\ $$$$\mathrm{3}\left(\mathrm{sin}\:^{\mathrm{6}} \mathrm{x}+\mathrm{cos}\:^{\mathrm{6}} \mathrm{x}\right)\:=\:?\: \\ $$

Question Number 118073 Answers: 2 Comments: 0

Evaluate ∫_0 ^a b(√(1−(x^2 /a^2 ))) dx

$$\mathrm{Evaluate}\:\underset{\mathrm{0}} {\overset{{a}} {\int}}\:{b}\sqrt{\mathrm{1}−\frac{\mathrm{x}^{\mathrm{2}} }{{a}^{\mathrm{2}} }}\:{dx}\: \\ $$

Question Number 118070 Answers: 0 Comments: 4

Question Number 118069 Answers: 1 Comments: 0

1.)i) A right circular cone is circumscribed about a sphere of radius(r). If d is the distance from the center of the sphere to the vertex of the cone, show that the volume of the cone,V=((𝛑r^2 (r+d)^2 )/(3(d−r))). ii) Find the vertical angle of the cone when it′s volume is minimum.

$$\left.\mathrm{1}\left..\right)\mathrm{i}\right)\:\mathrm{A}\:\mathrm{right}\:\mathrm{circular}\:\mathrm{cone}\:\mathrm{is}\:\mathrm{circumscribed} \\ $$$$\mathrm{about}\:\mathrm{a}\:\mathrm{sphere}\:\mathrm{of}\:\mathrm{radius}\left(\boldsymbol{\mathrm{r}}\right).\:\:\mathrm{If}\:\boldsymbol{\mathrm{d}}\:\mathrm{is}\:\mathrm{the}\: \\ $$$$\mathrm{distance}\:\mathrm{from}\:\mathrm{the}\:\mathrm{center}\:\mathrm{of}\:\mathrm{the}\:\mathrm{sphere} \\ $$$$\mathrm{to}\:\mathrm{the}\:\mathrm{vertex}\:\mathrm{of}\:\mathrm{the}\:\mathrm{cone},\:\mathrm{show}\:\mathrm{that}\:\mathrm{the} \\ $$$$\mathrm{volume}\:\mathrm{of}\:\mathrm{the}\:\mathrm{cone},\boldsymbol{\mathrm{V}}=\frac{\boldsymbol{\pi\mathrm{r}}^{\mathrm{2}} \left(\boldsymbol{\mathrm{r}}+\boldsymbol{\mathrm{d}}\right)^{\mathrm{2}} }{\mathrm{3}\left(\boldsymbol{\mathrm{d}}−\boldsymbol{\mathrm{r}}\right)}. \\ $$$$\left.\mathrm{ii}\right) \\ $$$$\boldsymbol{\mathrm{F}}\mathrm{ind}\:\mathrm{the}\:\mathrm{vertical}\:\mathrm{angle}\:\mathrm{of}\:\mathrm{the}\:\mathrm{cone}\:\mathrm{when} \\ $$$$\mathrm{it}'\mathrm{s}\:\mathrm{volume}\:\mathrm{is}\:\mathrm{minimum}. \\ $$

Question Number 118094 Answers: 1 Comments: 0

∫ (dx/(sec x+2)) =?

$$\int\:\frac{\mathrm{dx}}{\mathrm{sec}\:\mathrm{x}+\mathrm{2}}\:=? \\ $$

Question Number 118065 Answers: 1 Comments: 0

Question Number 118064 Answers: 1 Comments: 0

Question Number 118062 Answers: 0 Comments: 2

Question Number 118030 Answers: 2 Comments: 1

find ∫_0 ^∞ ((x^2 −2)/(x^4 +x^2 +1))dx

$$\mathrm{find}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{x}^{\mathrm{2}} −\mathrm{2}}{\mathrm{x}^{\mathrm{4}} +\mathrm{x}^{\mathrm{2}} \:+\mathrm{1}}\mathrm{dx} \\ $$

Question Number 118043 Answers: 1 Comments: 2

How many numbers in the range [1,10^n −1] are divisible by 9? digit repetitions are not allowed.

$$\mathrm{How}\:\mathrm{many}\:\mathrm{numbers}\:\mathrm{in}\:\mathrm{the}\:\mathrm{range} \\ $$$$\left[\mathrm{1},\mathrm{10}^{{n}} −\mathrm{1}\right]\:\mathrm{are}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{9}? \\ $$$$\mathrm{digit}\:\mathrm{repetitions}\:\mathrm{are}\:\mathrm{not}\:\mathrm{allowed}. \\ $$

Question Number 118040 Answers: 2 Comments: 0

Question Number 118023 Answers: 1 Comments: 0

..calculus.. x,y,z ∈R^+ and x^2 +y^2 +z^2 =1 find min_(x,y,z∈R^(+ ) ) ((((yz)/x)+((xz)/y)+((xy)/z)) )=? m.n.1970..

$$ \\ $$$$\:\:\:\:\:\:\:..{calculus}.. \\ $$$$\:\:{x},{y},{z}\:\in\mathbb{R}^{+} \:\:{and}\:\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} \:=\mathrm{1} \\ $$$$\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:{find}\:\:\:\:\:\: \\ $$$$\:\:\:\:{min}_{{x},{y},{z}\in\mathbb{R}^{+\:\:\:\:} } \left(\left(\frac{{yz}}{{x}}+\frac{{xz}}{{y}}+\frac{{xy}}{{z}}\right)\:\right)=? \\ $$$$ \\ $$$$\:\:\:\:\:\:\:{m}.{n}.\mathrm{1970}.. \\ $$

Question Number 118019 Answers: 2 Comments: 3

How many n digit integers are divisible by 9?

$$\mathrm{How}\:\mathrm{many}\:{n}\:\mathrm{digit}\:\mathrm{integers}\:\mathrm{are} \\ $$$$\mathrm{divisible}\:\mathrm{by}\:\mathrm{9}? \\ $$

Question Number 118017 Answers: 0 Comments: 0

Question Number 118011 Answers: 4 Comments: 0

Question Number 118010 Answers: 3 Comments: 1

∫(dx/(x^4 +x^2 +1))

$$\int\frac{{dx}}{{x}^{\mathrm{4}} +{x}^{\mathrm{2}} +\mathrm{1}} \\ $$

Question Number 118005 Answers: 0 Comments: 1

Question Number 118004 Answers: 2 Comments: 0

find value of tan46^0 using calculus

$${find}\:\:{value}\:{of}\:{tan}\mathrm{46}^{\mathrm{0}} \:{using}\:{calculus} \\ $$

Question Number 118002 Answers: 1 Comments: 0

Question Number 117989 Answers: 1 Comments: 0

Question Number 117990 Answers: 1 Comments: 0

(2.3.5.7.9.11.13.17......∞)×(√(6/(3.8.24.48.80.120.168.288.....∞)))

$$\left(\mathrm{2}.\mathrm{3}.\mathrm{5}.\mathrm{7}.\mathrm{9}.\mathrm{11}.\mathrm{13}.\mathrm{17}......\infty\right)×\sqrt{\frac{\mathrm{6}}{\mathrm{3}.\mathrm{8}.\mathrm{24}.\mathrm{48}.\mathrm{80}.\mathrm{120}.\mathrm{168}.\mathrm{288}.....\infty}} \\ $$

Question Number 117984 Answers: 1 Comments: 0

If f(x) is a polynomial function satisfying the relation f(x)+f((1/x))=f(x)f((1/x)) for all 0≠x∈R and if f(2)=9, then f(6) is (A) 216 (B) 217 (C) 126 (D) 127

$$\mathrm{If}\:{f}\left({x}\right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{polynomial}\:\mathrm{function}\:\mathrm{satisfying}\:\mathrm{the}\:\mathrm{relation} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{f}\left({x}\right)+{f}\left(\frac{\mathrm{1}}{{x}}\right)={f}\left({x}\right){f}\left(\frac{\mathrm{1}}{{x}}\right) \\ $$$$\mathrm{for}\:\mathrm{all}\:\mathrm{0}\neq{x}\in\mathbb{R}\:\mathrm{and}\:\mathrm{if}\:{f}\left(\mathrm{2}\right)=\mathrm{9},\:\mathrm{then}\:\mathrm{f}\left(\mathrm{6}\right)\:\mathrm{is} \\ $$$$\left(\mathrm{A}\right)\:\mathrm{216}\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{B}\right)\:\mathrm{217}\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{C}\right)\:\mathrm{126}\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{D}\right)\:\mathrm{127} \\ $$

Question Number 117980 Answers: 0 Comments: 2

Where is the quiz?

$$\mathrm{Where}\:\mathrm{is}\:\mathrm{the}\:\mathrm{quiz}? \\ $$$$ \\ $$

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