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Question Number 138206    Answers: 1   Comments: 1

Question Number 138205    Answers: 2   Comments: 3

Question Number 138203    Answers: 1   Comments: 0

Three circles each radius 1, touch one another externally and they lie between two parallel line. The minimum possible distance between the lines is _

$${Three}\:{circles}\:{each}\:{radius}\:\mathrm{1},\:{touch}\:{one} \\ $$$${another}\:{externally}\:{and}\:{they}\:{lie} \\ $$$${between}\:{two}\:{parallel}\:{line}.\:{The}\: \\ $$$${minimum}\:{possible}\:{distance}\:{between}\: \\ $$$${the}\:{lines}\:{is}\:\_\: \\ $$

Question Number 142191    Answers: 1   Comments: 3

Question Number 142185    Answers: 2   Comments: 0

∫(1/(x^4 +1))dx

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

Question Number 138193    Answers: 1   Comments: 0

Given x≠y and x^2 =25x+y, y^2 =x+25y solve for the value of (√(x^2 +y^2 +1)) without using calculators or tools. Show your method.

$${Given}\:{x}\neq{y}\:{and}\:{x}^{\mathrm{2}} =\mathrm{25}{x}+{y},\:{y}^{\mathrm{2}} ={x}+\mathrm{25}{y}\: \\ $$$${solve}\:{for}\:{the}\:{value}\:{of}\:\sqrt{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +\mathrm{1}}\:{without}\: \\ $$$${using}\:{calculators}\:{or}\:{tools}. \\ $$$${Show}\:{your}\:{method}. \\ $$

Question Number 138217    Answers: 0   Comments: 2

A rectangular water tank is being filed at the constant rate of 70lt/s. The base of the tank has width w=9m and length length l=16m if the volume of the tank is v=w×l×h where h is the hight of the tank. what is the rate of change of the hight of water in the tank

$$\mathrm{A}\:\mathrm{rectangular}\:\mathrm{water}\:\mathrm{tank}\:\mathrm{is}\:\mathrm{being}\:\mathrm{filed} \\ $$$$\mathrm{at}\:\mathrm{the}\:\mathrm{constant}\:\mathrm{rate}\:\mathrm{of}\:\mathrm{70lt}/\mathrm{s}.\:\mathrm{The}\: \\ $$$$\mathrm{base}\:\:\mathrm{of}\:\mathrm{the}\:\mathrm{tank}\:\mathrm{has}\:\mathrm{width}\:\mathrm{w}=\mathrm{9m} \\ $$$$\mathrm{and}\:\mathrm{length}\:\mathrm{length}\:\mathrm{l}=\mathrm{16m}\:\mathrm{if}\:\mathrm{the}\:\mathrm{volume} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{tank}\:\mathrm{is}\:\mathrm{v}=\mathrm{w}×\mathrm{l}×\mathrm{h}\:\mathrm{where}\:\mathrm{h}\:\mathrm{is}\: \\ $$$$\mathrm{the}\:\mathrm{hight}\:\mathrm{of}\:\mathrm{the}\:\mathrm{tank}.\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{rate} \\ $$$$\mathrm{of}\:\mathrm{change}\:\mathrm{of}\:\mathrm{the}\:\mathrm{hight}\:\mathrm{of}\:\mathrm{water}\:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{tank} \\ $$

Question Number 138175    Answers: 1   Comments: 1

Question Number 138171    Answers: 0   Comments: 0

.......mathematical....analysis....... if : 𝛗=∫_0 ^( 1) ((arctan(x).ln(1+x^2 ))/x^2 )dx =(1/(48))(aπ^2 −bπln(2)+c ln^2 (2)) ......then .... a^2 +(b−c+1)^2 =???

$$\:\:\:\:\:\:\:\:\:\:\:.......{mathematical}....{analysis}....... \\ $$$$\:\:\:\:\:{if}\::\:\:\boldsymbol{\phi}=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{arctan}\left({x}\right).{ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}{{x}^{\mathrm{2}} }{dx} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\frac{\mathrm{1}}{\mathrm{48}}\left({a}\pi^{\mathrm{2}} −{b}\pi{ln}\left(\mathrm{2}\right)+{c}\:{ln}^{\mathrm{2}} \left(\mathrm{2}\right)\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:......{then}\:\:....\:{a}^{\mathrm{2}} +\left({b}−{c}+\mathrm{1}\right)^{\mathrm{2}} =??? \\ $$

Question Number 138167    Answers: 2   Comments: 0

let f(x) = determinant ((x,x^2 ,x^3 ),(0,(2x ),(3x^2 )),(1,0,x)) find f ′(x)

$$\mathrm{let}\:{f}\left({x}\right)\:=\:\begin{vmatrix}{{x}}&{{x}^{\mathrm{2}} }&{{x}^{\mathrm{3}} }\\{\mathrm{0}}&{\mathrm{2}{x}\:}&{\mathrm{3}{x}^{\mathrm{2}} }\\{\mathrm{1}}&{\mathrm{0}}&{{x}}\end{vmatrix}\:\mathrm{find} \\ $$$$\:{f}\:'\left({x}\right)\: \\ $$

Question Number 138163    Answers: 1   Comments: 0

........nice ... .... .... calculus..... prove that:: Ψ=Σ_(n=1) ^∞ ((1/(n^2 π^2 +1)))=^(???) (1/(e^2 −1)) .............

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:........{nice}\:\:...\:....\:....\:{calculus}..... \\ $$$$\:\:\:\:{prove}\:{that}:: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\Psi=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(\frac{\mathrm{1}}{{n}^{\mathrm{2}} \pi^{\mathrm{2}} +\mathrm{1}}\right)\overset{???} {=}\frac{\mathrm{1}}{{e}^{\mathrm{2}} −\mathrm{1}} \\ $$$$\:\:\:\:\:\:\:\:............. \\ $$

Question Number 138159    Answers: 1   Comments: 0

Question Number 138154    Answers: 1   Comments: 0

Question Number 138153    Answers: 1   Comments: 0

lim_(x→0^+ ) ((x−∣tanx∣)/(∣sinx∣−x))=?

$$\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\frac{{x}−\mid{tanx}\mid}{\mid{sinx}\mid−{x}}=? \\ $$

Question Number 138150    Answers: 2   Comments: 0

Given (x+(√(x^2 +1)))(y+(√(y^2 +4)))=9 find the value of x(√(y^2 +4)) + y(√(x^2 +1)) .

$${Given}\:\left({x}+\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}\right)\left({y}+\sqrt{{y}^{\mathrm{2}} +\mathrm{4}}\right)=\mathrm{9} \\ $$$${find}\:{the}\:{value}\:{of}\: \\ $$$$\:\:\:\:\:\:\:{x}\sqrt{{y}^{\mathrm{2}} +\mathrm{4}}\:+\:{y}\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}\:. \\ $$

Question Number 138145    Answers: 4   Comments: 0

...........advanced ... ... ... calculus......... find the value of:: Θ=Σ_(n=1) ^∞ (((−1)^n H_(2n) )/n)=???

$$\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:...........{advanced}\:...\:...\:...\:{calculus}......... \\ $$$$\:\:\:\:\:\:{find}\:{the}\:{value}\:{of}:: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Theta=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} \mathrm{H}_{\mathrm{2}{n}} }{{n}}=??? \\ $$

Question Number 138140    Answers: 4   Comments: 0

lim_(x→0) ((cos (sin x)−cos x)/x^4 )=?

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{cos}\:\left(\mathrm{sin}\:{x}\right)−\mathrm{cos}\:{x}}{{x}^{\mathrm{4}} }=? \\ $$

Question Number 138135    Answers: 1   Comments: 0

(x^2 +x^3 )=4 find x.

$$\left({x}^{\mathrm{2}} +{x}^{\mathrm{3}} \right)=\mathrm{4} \\ $$$${find}\:{x}. \\ $$

Question Number 138134    Answers: 0   Comments: 0

calculate Σ_(n=1) ^∞ (((−1)^n )/(n^3 (2n+1)^4 ))

$$\mathrm{calculate}\:\sum_{\mathrm{n}=\mathrm{1}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{\mathrm{n}} }{\mathrm{n}^{\mathrm{3}} \left(\mathrm{2n}+\mathrm{1}\right)^{\mathrm{4}} } \\ $$

Question Number 138133    Answers: 0   Comments: 0

let A = (((1 −1)),((2 3)) ) 1)calculate e^A and e^(tA) 2)find cosA ,sinA 3) find log(1+A)

$$\mathrm{let}\:\mathrm{A}\:=\begin{pmatrix}{\mathrm{1}\:\:\:\:\:\:\:\:\:\:−\mathrm{1}}\\{\mathrm{2}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{3}}\end{pmatrix} \\ $$$$\left.\mathrm{1}\right)\mathrm{calculate}\:\mathrm{e}^{\mathrm{A}} \:\:\:\:\mathrm{and}\:\mathrm{e}^{\mathrm{tA}} \\ $$$$\left.\mathrm{2}\right)\mathrm{find}\:\mathrm{cosA}\:\:,\mathrm{sinA} \\ $$$$\left.\mathrm{3}\right)\:\mathrm{find}\:\mathrm{log}\left(\mathrm{1}+\mathrm{A}\right) \\ $$

Question Number 138132    Answers: 1   Comments: 0

calculate ∫_0 ^∞ ((logx)/(x^2 −x+2))dx

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{logx}}{\mathrm{x}^{\mathrm{2}} −\mathrm{x}+\mathrm{2}}\mathrm{dx} \\ $$

Question Number 138131    Answers: 1   Comments: 0

calculate ∫_0 ^∞ (dx/((2x+1)(x^2 −x+3)^2 ))

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{\mathrm{dx}}{\left(\mathrm{2x}+\mathrm{1}\right)\left(\mathrm{x}^{\mathrm{2}} −\mathrm{x}+\mathrm{3}\right)^{\mathrm{2}} } \\ $$

Question Number 138129    Answers: 0   Comments: 0

calculate ∫_0 ^1 ((ln(2+x^2 ))/(x^2 +3))dx

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{\mathrm{ln}\left(\mathrm{2}+\mathrm{x}^{\mathrm{2}} \right)}{\mathrm{x}^{\mathrm{2}} +\mathrm{3}}\mathrm{dx} \\ $$

Question Number 138128    Answers: 1   Comments: 0

calculate ∫ (dx/( (√(2−x^2 ))+(√(3+x^2 ))))

$$\mathrm{calculate}\:\int\:\:\:\frac{\mathrm{dx}}{\:\sqrt{\mathrm{2}−\mathrm{x}^{\mathrm{2}} }+\sqrt{\mathrm{3}+\mathrm{x}^{\mathrm{2}} }} \\ $$

Question Number 138123    Answers: 1   Comments: 0

Question Number 138120    Answers: 2   Comments: 1

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