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Question Number 157884    Answers: 1   Comments: 0

(1/(sin(10°))) - 4 sin(70°) = ?

$$\frac{\mathrm{1}}{\mathrm{sin}\left(\mathrm{10}°\right)}\:-\:\mathrm{4}\:\mathrm{sin}\left(\mathrm{70}°\right)\:=\:? \\ $$

Question Number 157883    Answers: 1   Comments: 0

lim_(n→∞) ((1/(n^2 +1)) + (2/(n^2 +1)) + ... + ((n-1)/(n^2 +1))) = ?

$$\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\left(\frac{\mathrm{1}}{\mathrm{n}^{\mathrm{2}} +\mathrm{1}}\:+\:\frac{\mathrm{2}}{\mathrm{n}^{\mathrm{2}} +\mathrm{1}}\:+\:...\:+\:\frac{\mathrm{n}-\mathrm{1}}{\mathrm{n}^{\mathrm{2}} +\mathrm{1}}\right)\:=\:? \\ $$

Question Number 157880    Answers: 0   Comments: 1

Question Number 157873    Answers: 0   Comments: 0

if: 𝛂 =∫_( 0) ^( ∞) ∫_( 0) ^( ∞) ((x)^(1/3) /(1 + (x)^(1/3) )) e^(-𝛑y(1+x^2 +(1/x^2 ))) dydx find: (√(19683𝛂^6 - 94041𝛂^4 + 105786𝛂^2 ))

$$\mathrm{if}:\:\boldsymbol{\alpha}\:=\underset{\:\mathrm{0}} {\overset{\:\infty} {\int}}\underset{\:\mathrm{0}} {\overset{\:\infty} {\int}}\:\frac{\sqrt[{\mathrm{3}}]{\mathrm{x}}}{\mathrm{1}\:+\:\sqrt[{\mathrm{3}}]{\mathrm{x}}}\:\:\mathrm{e}^{-\boldsymbol{\pi\mathrm{y}}\left(\mathrm{1}+\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\frac{\mathrm{1}}{\boldsymbol{\mathrm{x}}^{\mathrm{2}} }\right)} \:\mathrm{dydx} \\ $$$$\mathrm{find}:\:\sqrt{\mathrm{19683}\boldsymbol{\alpha}^{\mathrm{6}} \:-\:\mathrm{94041}\boldsymbol{\alpha}^{\mathrm{4}} \:+\:\mathrm{105786}\boldsymbol{\alpha}^{\mathrm{2}} } \\ $$$$ \\ $$

Question Number 157871    Answers: 0   Comments: 0

if x;y>0 then prove that: (x/(x^2 -x+1)) + (y/(y^2 -y+1)) + ((xy)/(x^2 y^2 -xy+1)) ≤ ≤ (x^2 /(x^2 -x+1)) + (y^2 /(y^2 -y+1)) + (1/(x^2 y^2 -xy+1))

$$\mathrm{if}\:\:\:\mathrm{x};\mathrm{y}>\mathrm{0}\:\:\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\frac{\mathrm{x}}{\mathrm{x}^{\mathrm{2}} -\mathrm{x}+\mathrm{1}}\:+\:\frac{\mathrm{y}}{\mathrm{y}^{\mathrm{2}} -\mathrm{y}+\mathrm{1}}\:+\:\frac{\mathrm{xy}}{\mathrm{x}^{\mathrm{2}} \mathrm{y}^{\mathrm{2}} -\mathrm{xy}+\mathrm{1}}\:\leqslant \\ $$$$\leqslant\:\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{x}^{\mathrm{2}} -\mathrm{x}+\mathrm{1}}\:+\:\frac{\mathrm{y}^{\mathrm{2}} }{\mathrm{y}^{\mathrm{2}} -\mathrm{y}+\mathrm{1}}\:+\:\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} \mathrm{y}^{\mathrm{2}} -\mathrm{xy}+\mathrm{1}} \\ $$

Question Number 157869    Answers: 1   Comments: 0

partial fraction in below: 1. ((x^2 −15x+41)/((x+2)(x−3)^2 )) 2.((4x^2 −5x+6)/((x+1)(x^2 +4)))

$$\mathrm{partial}\:\mathrm{fraction}\:\mathrm{in}\:\mathrm{below}: \\ $$$$\mathrm{1}.\:\frac{\mathrm{x}^{\mathrm{2}} −\mathrm{15x}+\mathrm{41}}{\left(\mathrm{x}+\mathrm{2}\right)\left(\mathrm{x}−\mathrm{3}\right)^{\mathrm{2}} } \\ $$$$\mathrm{2}.\frac{\mathrm{4x}^{\mathrm{2}} −\mathrm{5x}+\mathrm{6}}{\left(\mathrm{x}+\mathrm{1}\right)\left(\mathrm{x}^{\mathrm{2}} +\mathrm{4}\right)} \\ $$

Question Number 157867    Answers: 1   Comments: 0

Given that point P(a cos θ, b sin θ) is a point on the ellipse (x^2 /a^2 )+(y^2 /b^2 )=1. The tangent to the curve at point P is perpendicular to a straight line which passes through the focus, F (ae,0). If N is the intersection point, show that the equation of the locus of N is x^2 +y^2 =a^2 .

$$\mathrm{Given}\:\mathrm{that}\:\mathrm{point}\:{P}\left({a}\:\mathrm{cos}\:\theta,\:{b}\:\mathrm{sin}\:\theta\right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{point}\:\mathrm{on} \\ $$$$\mathrm{the}\:\mathrm{ellipse}\:\frac{{x}^{\mathrm{2}} }{{a}^{\mathrm{2}} }+\frac{{y}^{\mathrm{2}} }{{b}^{\mathrm{2}} }=\mathrm{1}. \\ $$$$\mathrm{The}\:\mathrm{tangent}\:\mathrm{to}\:\mathrm{the}\:\mathrm{curve}\:\mathrm{at}\:\mathrm{point}\:{P}\:\:\mathrm{is}\:\mathrm{perpendicular} \\ $$$$\mathrm{to}\:\mathrm{a}\:\mathrm{straight}\:\mathrm{line}\:\mathrm{which}\:\mathrm{passes}\:\mathrm{through}\:\mathrm{the}\:\mathrm{focus}, \\ $$$${F}\:\left({ae},\mathrm{0}\right).\:\mathrm{If}\:{N}\:\mathrm{is}\:\mathrm{the}\:\mathrm{intersection}\:\mathrm{point},\:\mathrm{show}\:\mathrm{that} \\ $$$$\mathrm{the}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{the}\:\mathrm{locus}\:\mathrm{of}\:{N}\:\mathrm{is}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} ={a}^{\mathrm{2}} . \\ $$

Question Number 157855    Answers: 0   Comments: 2

Question Number 157853    Answers: 0   Comments: 1

Question Number 157851    Answers: 1   Comments: 2

Question Number 157970    Answers: 2   Comments: 0

Find the neext number for this sequence below 1). 1, 3, 6, 10, 15.... 2). 1, 5 ,14 ,30, 55 .... 3). 1,7,17,31, 49 .... 4). 4, 13,28, 49,74... 5). 1,8,27,64,125....

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{neext}\:\mathrm{number}\:\mathrm{for}\:\mathrm{this} \\ $$$$\mathrm{sequence}\:\mathrm{below}\: \\ $$$$\left.\mathrm{1}\right).\:\mathrm{1},\:\:\mathrm{3},\:\mathrm{6},\:\mathrm{10},\:\mathrm{15}.... \\ $$$$\left.\mathrm{2}\right).\:\mathrm{1},\:\mathrm{5}\:,\mathrm{14}\:,\mathrm{30},\:\:\mathrm{55}\:.... \\ $$$$\left.\mathrm{3}\right).\:\mathrm{1},\mathrm{7},\mathrm{17},\mathrm{31},\:\mathrm{49}\:.... \\ $$$$\left.\mathrm{4}\right).\:\:\mathrm{4},\:\mathrm{13},\mathrm{28},\:\mathrm{49},\mathrm{74}... \\ $$$$\left.\mathrm{5}\right).\:\:\mathrm{1},\mathrm{8},\mathrm{27},\mathrm{64},\mathrm{125}.... \\ $$$$ \\ $$

Question Number 157870    Answers: 1   Comments: 0

find the integral: ∫{(3x+1)/(x^2 +4)}dx

$${find}\:{the}\:{integral}: \\ $$$$\int\left\{\left(\mathrm{3}{x}+\mathrm{1}\right)/\left({x}^{\mathrm{2}} +\mathrm{4}\right)\right\}{dx} \\ $$

Question Number 157902    Answers: 1   Comments: 1

Solve for integers: x∙(x + 4) = 5∙(3^y - 1)

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{integers}: \\ $$$$\mathrm{x}\centerdot\left(\mathrm{x}\:+\:\mathrm{4}\right)\:=\:\mathrm{5}\centerdot\left(\mathrm{3}^{\boldsymbol{\mathrm{y}}} \:-\:\mathrm{1}\right) \\ $$$$ \\ $$

Question Number 157839    Answers: 1   Comments: 0

find the last four digits of 11^(15999) ?

$${find}\:{the}\:{last}\:{four}\:{digits}\:{of}\: \\ $$$$\mathrm{11}^{\mathrm{15999}} ? \\ $$

Question Number 157835    Answers: 1   Comments: 0

Question Number 157836    Answers: 1   Comments: 0

find the indicated higher order derivative of the following function f(x) = (x^3 +4x−5)^4 , f(x)^(iv)

$${find}\:{the}\:{indicated}\:{higher}\:{order}\:{derivative} \\ $$$${of}\:{the}\:{following}\:{function} \\ $$$${f}\left({x}\right)\:=\:\left({x}^{\mathrm{3}} +\mathrm{4}{x}−\mathrm{5}\right)^{\mathrm{4}} ,\:{f}\left({x}\right)^{{iv}} \\ $$

Question Number 157832    Answers: 1   Comments: 0

Question Number 157829    Answers: 1   Comments: 0

(((√2) a_n )/a_(n+1) )=(√(2+(a_n )^2 )) a_1 =(1/2) ⇒ a_(43) =?

$$\frac{\sqrt{\mathrm{2}}\:{a}_{{n}} }{{a}_{{n}+\mathrm{1}} }=\sqrt{\mathrm{2}+\left({a}_{{n}} \right)^{\mathrm{2}} }\:\:\:\:\: \\ $$$${a}_{\mathrm{1}} =\frac{\mathrm{1}}{\mathrm{2}}\:\:\:\:\Rightarrow\:\:{a}_{\mathrm{43}} =? \\ $$

Question Number 157810    Answers: 1   Comments: 0

Calculate: (√(√(144 + (√6)))) = ?

$$\mathrm{Calculate}:\:\:\sqrt{\sqrt{\mathrm{144}\:+\:\sqrt{\mathrm{6}}}}\:=\:? \\ $$$$ \\ $$

Question Number 157815    Answers: 0   Comments: 2

Question Number 157813    Answers: 2   Comments: 0

express 5.13^• 4^• 5^• into fraction

$${express}\:\:\mathrm{5}.\mathrm{1}\overset{\bullet} {\mathrm{3}}\overset{\bullet} {\mathrm{4}}\overset{\bullet} {\mathrm{5}}\:\:\:{into}\:{fraction} \\ $$

Question Number 157805    Answers: 1   Comments: 0

Find x∈R: x^3 −2x+3=(√(2x+5))+4(√(x−1))

$$\mathrm{Find}\:\mathrm{x}\in\mathbb{R}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{x}^{\mathrm{3}} −\mathrm{2x}+\mathrm{3}=\sqrt{\mathrm{2x}+\mathrm{5}}+\mathrm{4}\sqrt{\mathrm{x}−\mathrm{1}} \\ $$

Question Number 157826    Answers: 1   Comments: 0

Question Number 157824    Answers: 0   Comments: 0

Question Number 157823    Answers: 1   Comments: 0

calculate : Ω := Σ_(n=0) ^∞ (( 1)/((4n+1)^( 3) )) = ?

$$ \\ $$$$\:\:\:\:\:\:\:{calculate}\:: \\ $$$$\:\:\:\:\Omega\::=\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\:\mathrm{1}}{\left(\mathrm{4}{n}+\mathrm{1}\right)^{\:\mathrm{3}} }\:\:\:=\:? \\ $$$$\: \\ $$

Question Number 157822    Answers: 0   Comments: 1

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