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AllQuestion and Answers: Page 1072

Question Number 109728 Answers: 4 Comments: 1

△((♭ε)/(math))▽ lim_(x→π) ((sin^2 x)/(cos 3x+1))

$$\:\:\:\bigtriangleup\frac{\flat\epsilon}{{math}}\bigtriangledown \\ $$$$\underset{{x}\rightarrow\pi} {\mathrm{lim}}\:\frac{\mathrm{sin}\:^{\mathrm{2}} {x}}{\mathrm{cos}\:\mathrm{3}{x}+\mathrm{1}} \\ $$

Question Number 109699 Answers: 1 Comments: 0

x(x−1)^2 ≥ 12(x−1)

$$\:\:\:\:\:{x}\left({x}−\mathrm{1}\right)^{\mathrm{2}} \:\geqslant\:\mathrm{12}\left({x}−\mathrm{1}\right) \\ $$

Question Number 109674 Answers: 2 Comments: 0

Find the area of the region enclose by the curve y = 1 − x^2 , and the line y = 1 + 3x, x = 1.

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{area}\:\mathrm{of}\:\mathrm{the}\:\mathrm{region}\:\mathrm{enclose}\:\mathrm{by}\:\mathrm{the}\:\mathrm{curve}\:\:\:\mathrm{y}\:\:=\:\:\mathrm{1}\:\:−\:\:\mathrm{x}^{\mathrm{2}} \:, \\ $$$$\mathrm{and}\:\mathrm{the}\:\mathrm{line}\:\:\:\:\mathrm{y}\:\:=\:\:\mathrm{1}\:\:+\:\:\mathrm{3x},\:\:\:\:\mathrm{x}\:\:=\:\:\mathrm{1}. \\ $$

Question Number 109658 Answers: 1 Comments: 0

Question Number 109848 Answers: 0 Comments: 0

Question Number 109655 Answers: 0 Comments: 0

Question Number 109642 Answers: 1 Comments: 0

Question Number 109640 Answers: 0 Comments: 2

Question Number 109639 Answers: 2 Comments: 0

Question Number 109626 Answers: 1 Comments: 1

Question Number 109625 Answers: 0 Comments: 2

((sin 2𝛂+2sin 𝛂∙cos 2𝛂)/(1+cos 𝛂+cos2 𝛂+cos3 𝛂))

$$\frac{\mathrm{sin}\:\mathrm{2}\boldsymbol{\alpha}+\mathrm{2sin}\:\boldsymbol{\alpha}\centerdot\mathrm{cos}\:\mathrm{2}\boldsymbol{\alpha}}{\mathrm{1}+\mathrm{cos}\:\boldsymbol{\alpha}+\mathrm{cos2}\:\boldsymbol{\alpha}+\mathrm{cos3}\:\boldsymbol{\alpha}} \\ $$

Question Number 109619 Answers: 0 Comments: 2

Σ_(n=1) ^∞ ((n!)/n^n )

$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{n}!}{{n}^{{n}} } \\ $$

Question Number 109616 Answers: 1 Comments: 1

find ∫_0 ^1 (√(1+x^4 ))dx

$$\mathrm{find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \sqrt{\mathrm{1}+\mathrm{x}^{\mathrm{4}} }\mathrm{dx} \\ $$

Question Number 109611 Answers: 4 Comments: 1

Question Number 109606 Answers: 0 Comments: 15

Question Number 109605 Answers: 0 Comments: 3

Question Number 109595 Answers: 1 Comments: 1

For any Real numbers x,y and z, if (x+y+z)=2, then prove that xyz≥8(1−x)(1−y)(1−z)

$${For}\:{any}\:{Real}\:{numbers}\:{x},{y}\:{and}\:{z}, \\ $$$$\:{if}\:\:\left({x}+{y}+{z}\right)=\mathrm{2},\:{then}\:{prove}\:{that} \\ $$$$\:\:\:\:\:\:\:\:{xyz}\geqslant\mathrm{8}\left(\mathrm{1}−{x}\right)\left(\mathrm{1}−{y}\right)\left(\mathrm{1}−{z}\right) \\ $$

Question Number 109591 Answers: 1 Comments: 3

An isosceles AEF is inscribed into a square ABCD such that pointE is on side BC,point F is on side CDand AE=EF. Knownthat tanAEF^() =2.Find tanFEC^()

$$\mathrm{An}\:\mathrm{isosceles}\:\mathrm{AEF}\:\mathrm{is}\:\mathrm{inscribed}\:\mathrm{into}\:\mathrm{a} \\ $$$$\mathrm{square}\:\mathrm{ABCD}\:\mathrm{such}\:\mathrm{that}\:\mathrm{pointE}\:\mathrm{is}\:\mathrm{on} \\ $$$$\mathrm{side}\:\mathrm{BC},\mathrm{point}\:\mathrm{F}\:\mathrm{is}\:\mathrm{on}\:\mathrm{side}\:\mathrm{CDand}\:\mathrm{AE}=\mathrm{EF}. \\ $$$$\mathrm{Knownthat}\:\mathrm{tan}\widehat {\mathrm{AEF}}=\mathrm{2}.\mathrm{Find}\:\mathrm{tan}\widehat {\mathrm{FEC}} \\ $$

Question Number 109585 Answers: 1 Comments: 0

If (xy+yz+zx)=1, then prove that (x/(1−x^2 ))+(y/(1−y^2 ))+(z/(1−z^2 ))=((4xyz)/((1−x^2 )(1−y^2 )(1−z^2 )))

$${If}\:\left({xy}+{yz}+{zx}\right)=\mathrm{1},\:{then}\:{prove}\:{that} \\ $$$$\frac{{x}}{\mathrm{1}−{x}^{\mathrm{2}} }+\frac{{y}}{\mathrm{1}−{y}^{\mathrm{2}} }+\frac{{z}}{\mathrm{1}−{z}^{\mathrm{2}} }=\frac{\mathrm{4}{xyz}}{\left(\mathrm{1}−{x}^{\mathrm{2}} \right)\left(\mathrm{1}−{y}^{\mathrm{2}} \right)\left(\mathrm{1}−{z}^{\mathrm{2}} \right)} \\ $$

Question Number 109584 Answers: 3 Comments: 1

((−♭o♭−)/(hans)) (1)(√(x−(√(x−(1/4))))) ≥ (1/4) (2)∣a^→ ∣ = 1, ∣b^→ ∣ = 2 , ∣c^→ ∣=3 , ∠(a^→ ,b^→ )=90° ∠(b^→ ,c^→ )=60° , ∠(a^→ ,c^→ )=120° , then ∣a^→ +b^→ −c^→ ∣=? (3) { ((x^(log _3 (y)) +y^(log _3 (x)) =18)),((log _3 (x)+log _3 (y)=3)) :} . Find the solution

Question Number 109577 Answers: 4 Comments: 1

Given x^4 +x^2 y^2 +y^4 =133 and x^2 −xy+y^2 =7 then what is the value of xy ?

$$\:\:\:\mathrm{G}{iven}\:{x}^{\mathrm{4}} +{x}^{\mathrm{2}} {y}^{\mathrm{2}} +{y}^{\mathrm{4}} =\mathrm{133} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:{and}\:{x}^{\mathrm{2}} −{xy}+{y}^{\mathrm{2}} =\mathrm{7} \\ $$$$\:\:{then}\:{what}\:{is}\:{the}\:{value}\:{of}\:{xy}\:? \\ $$

Question Number 109574 Answers: 0 Comments: 6

Question Number 109569 Answers: 1 Comments: 0

Question Number 109553 Answers: 1 Comments: 2

Question Number 109546 Answers: 1 Comments: 2

If f(x) continue in [ 1,30] and ∫_6 ^(30) f(x)dx = 30, then ∫_1 ^9 f(3y+3)dy = __

$${If}\:{f}\left({x}\right)\:{continue}\:{in}\:\left[\:\mathrm{1},\mathrm{30}\right]\:{and}\: \\ $$$$\underset{\mathrm{6}} {\overset{\mathrm{30}} {\int}}{f}\left({x}\right){dx}\:=\:\mathrm{30},\:{then}\:\underset{\mathrm{1}} {\overset{\mathrm{9}} {\int}}{f}\left(\mathrm{3}{y}+\mathrm{3}\right){dy}\:=\:\_\_ \\ $$

Question Number 109544 Answers: 2 Comments: 1

Exclude m and n from the equalities: a=m+n,b^3 =m^3 +n^3 ,c^5 =m^5 +n^5

$$\mathrm{Exclude}\:\mathrm{m}\:\mathrm{and}\:\mathrm{n}\:\mathrm{from}\:\mathrm{the}\:\mathrm{equalities}: \\ $$$$\mathrm{a}=\mathrm{m}+\mathrm{n},\mathrm{b}^{\mathrm{3}} =\mathrm{m}^{\mathrm{3}} +\mathrm{n}^{\mathrm{3}} ,\mathrm{c}^{\mathrm{5}} =\mathrm{m}^{\mathrm{5}} +\mathrm{n}^{\mathrm{5}} \\ $$

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