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AllQuestion and Answers: Page 1084

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 π›‚βˆ™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 xyzβ‰₯8(1βˆ’x)(1βˆ’y)(1βˆ’z)

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

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/(1βˆ’x^2 ))+(y/(1βˆ’y^2 ))+(z/(1βˆ’z^2 ))=((4xyz)/((1βˆ’x^2 )(1βˆ’y^2 )(1βˆ’z^2 )))

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

Question Number 109584    Answers: 3   Comments: 1

((βˆ’β™­oβ™­βˆ’)/(hans)) (1)(√(xβˆ’(√(xβˆ’(1/4))))) β‰₯ (1/4) (2)∣a^β†’ ∣ = 1, ∣b^β†’ ∣ = 2 , ∣c^β†’ ∣=3 , ∠(a^β†’ ,b^β†’ )=90Β° ∠(b^β†’ ,c^β†’ )=60Β° , ∠(a^β†’ ,c^β†’ )=120Β° , then ∣a^β†’ +b^β†’ βˆ’c^β†’ ∣=? (3) { ((x^(log _3 (y)) +y^(log _3 (x)) =18)),((log _3 (x)+log _3 (y)=3)) :} . Find the solution

$$\:\:\:\frac{βˆ’\flat{o}\flatβˆ’}{{hans}} \\ $$$$\left(\mathrm{1}\right)\sqrt{{x}βˆ’\sqrt{{x}βˆ’\frac{\mathrm{1}}{\mathrm{4}}}}\:\geqslant\:\frac{\mathrm{1}}{\mathrm{4}} \\ $$$$\left(\mathrm{2}\right)\mid\overset{\rightarrow} {{a}}\mid\:=\:\mathrm{1},\:\mid\overset{\rightarrow} {{b}}\mid\:=\:\mathrm{2}\:,\:\mid\overset{\rightarrow} {{c}}\mid=\mathrm{3}\:,\:\angle\left(\overset{\rightarrow} {{a}},\overset{\rightarrow} {{b}}\right)=\mathrm{90}Β° \\ $$$$\:\:\:\:\:\:\:\:\angle\left(\overset{\rightarrow} {{b}},\overset{\rightarrow} {{c}}\right)=\mathrm{60}Β°\:,\:\angle\left(\overset{\rightarrow} {{a}},\overset{\rightarrow} {{c}}\right)=\mathrm{120}Β°\:,\:{then}\: \\ $$$$\:\:\:\:\:\:\mid\overset{\rightarrow} {{a}}+\overset{\rightarrow} {{b}}βˆ’\overset{\rightarrow} {{c}}\mid=? \\ $$$$\left(\mathrm{3}\right)\begin{cases}{{x}^{\mathrm{log}\:_{\mathrm{3}} \left({y}\right)} +{y}^{\mathrm{log}\:_{\mathrm{3}} \left({x}\right)} =\mathrm{18}}\\{\mathrm{log}\:_{\mathrm{3}} \left({x}\right)+\mathrm{log}\:_{\mathrm{3}} \left({y}\right)=\mathrm{3}}\end{cases}\:.\:{Find}\:{the}\:{solution} \\ $$$$\:\:\: \\ $$

Question Number 109577    Answers: 4   Comments: 1

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

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

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

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

Question Number 109540    Answers: 1   Comments: 1

Question Number 109516    Answers: 3   Comments: 0

cos (1βˆ’i)=a+ib Find a, b.

$$\mathrm{cos}\:\left(\mathrm{1}βˆ’{i}\right)={a}+{ib} \\ $$$${Find}\:\:{a},\:{b}. \\ $$

Question Number 109506    Answers: 0   Comments: 0

Question Number 109509    Answers: 4   Comments: 0

((bemath)/(Σ_(i=cooll) ^(nice) (joss)_i )) ∫ ((x^2 dx)/( (√(x^2 +25))))

$$\:\:\frac{{bemath}}{\underset{{i}={cooll}} {\overset{{nice}} {\sum}}\left({joss}\right)_{{i}} }\: \\ $$$$ \\ $$$$\int\:\frac{{x}^{\mathrm{2}} \:{dx}}{\:\sqrt{{x}^{\mathrm{2}} +\mathrm{25}}} \\ $$

Question Number 109508    Answers: 2   Comments: 1

Question Number 109500    Answers: 3   Comments: 0

Given { ((a^2 +ab+bc+ac=a+c)),((b^2 +ab+bc+ac=b+a)),((c^2 +ab+bc+ac=c+b)) :} find the value of a+b+c

$${Given}\:\begin{cases}{{a}^{\mathrm{2}} +{ab}+{bc}+{ac}={a}+{c}}\\{{b}^{\mathrm{2}} +{ab}+{bc}+{ac}={b}+{a}}\\{{c}^{\mathrm{2}} +{ab}+{bc}+{ac}={c}+{b}}\end{cases} \\ $$$${find}\:{the}\:{value}\:{of}\:{a}+{b}+{c}\: \\ $$

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