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Question Number 43969    Answers: 0   Comments: 0

Question Number 43971    Answers: 1   Comments: 0

Question Number 43966    Answers: 0   Comments: 0

Question Number 43956    Answers: 0   Comments: 0

Question Number 43945    Answers: 1   Comments: 1

Question Number 43944    Answers: 1   Comments: 0

If the points (−3, 5) , (4, −2) and (6, 2) are the vertices of a triangle. (i) Find the equation of the perpendicular bisector of the sides (ii) Find the coordinate of the circumcenter. (The circumcenter of a triangle is the point of intersection of the perpendicular bisector of the side (iii) Find the radius of the circumcircle.

$$\mathrm{If}\:\mathrm{the}\:\mathrm{points}\:\left(−\mathrm{3},\:\mathrm{5}\right)\:,\:\left(\mathrm{4},\:−\mathrm{2}\right)\:\mathrm{and}\:\left(\mathrm{6},\:\mathrm{2}\right)\:\mathrm{are}\:\mathrm{the}\:\mathrm{vertices}\:\mathrm{of}\:\mathrm{a}\:\mathrm{triangle}. \\ $$$$\left(\mathrm{i}\right)\:\mathrm{Find}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{the}\:\mathrm{perpendicular}\:\mathrm{bisector}\:\mathrm{of}\:\mathrm{the}\:\mathrm{sides} \\ $$$$\left(\mathrm{ii}\right)\:\mathrm{Find}\:\mathrm{the}\:\mathrm{coordinate}\:\mathrm{of}\:\mathrm{the}\:\mathrm{circumcenter}.\:\left(\mathrm{The}\:\mathrm{circumcenter}\:\mathrm{of}\:\mathrm{a}\right. \\ $$$$\mathrm{triangle}\:\mathrm{is}\:\mathrm{the}\:\mathrm{point}\:\mathrm{of}\:\mathrm{intersection}\:\mathrm{of}\:\mathrm{the}\:\mathrm{perpendicular}\:\mathrm{bisector}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{side} \\ $$$$\left(\mathrm{iii}\right)\:\mathrm{Find}\:\mathrm{the}\:\mathrm{radius}\:\mathrm{of}\:\mathrm{the}\:\mathrm{circumcircle}. \\ $$

Question Number 43939    Answers: 0   Comments: 1

find f(ξ) =∫_0 ^∞ (dt/(1+(t−iξ)^2 )) and calculate f^′ (ξ)

$${find}\:{f}\left(\xi\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dt}}{\mathrm{1}+\left({t}−{i}\xi\right)^{\mathrm{2}} } \\ $$$${and}\:{calculate}\:{f}^{'} \left(\xi\right) \\ $$

Question Number 43938    Answers: 0   Comments: 1

calvulste A_n =∫_0 ^n t^2 [(1/((t+1)^3 ))]dt and lim_(n→+∞) A_n

$${calvulste}\:{A}_{{n}} =\int_{\mathrm{0}} ^{{n}} \:{t}^{\mathrm{2}} \left[\frac{\mathrm{1}}{\left({t}+\mathrm{1}\right)^{\mathrm{3}} }\right]{dt} \\ $$$${and}\:{lim}_{{n}\rightarrow+\infty} \:{A}_{{n}} \\ $$

Question Number 43937    Answers: 1   Comments: 1

calculate ∫_0 ^1 ((2x+1)/(√(x^2 −2x+5)))dx

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{\mathrm{2}{x}+\mathrm{1}}{\sqrt{{x}^{\mathrm{2}} −\mathrm{2}{x}+\mathrm{5}}}{dx} \\ $$

Question Number 43935    Answers: 0   Comments: 1

find f(a) =∫_0 ^∞ ((1−cos^2 (ax))/x^2 )dx 2)calculate f^′ (a).

$${find}\:{f}\left({a}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{1}−{cos}^{\mathrm{2}} \left({ax}\right)}{{x}^{\mathrm{2}} }{dx} \\ $$$$\left.\mathrm{2}\right){calculate}\:{f}^{'} \left({a}\right). \\ $$

Question Number 43934    Answers: 0   Comments: 1

find the value of ∫_0 ^(+∞) (dx/(1+x^4 +x^8 ))

$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{+\infty} \:\frac{{dx}}{\mathrm{1}+{x}^{\mathrm{4}} \:+{x}^{\mathrm{8}} } \\ $$

Question Number 43923    Answers: 1   Comments: 0

A curve passes through the point (1,−11) and its gradient at any point is ax^2 +b, where a and b are constants. The tangent to the curve at the point (2,−16) is parallel to the x-axis. Find i. the values of a and b ii. the equation of the curve

$$\mathrm{A}\:\mathrm{curve}\:\mathrm{passes}\:\mathrm{through}\:\mathrm{the}\:\mathrm{point}\:\left(\mathrm{1},−\mathrm{11}\right)\:\mathrm{and}\:\mathrm{its} \\ $$$$\mathrm{gradient}\:\mathrm{at}\:\mathrm{any}\:\mathrm{point}\:\mathrm{is}\:\boldsymbol{\mathrm{a}}\mathrm{x}^{\mathrm{2}} +\boldsymbol{\mathrm{b}},\:\mathrm{where}\:\boldsymbol{\mathrm{a}}\:\mathrm{and}\:\boldsymbol{\mathrm{b}}\:\mathrm{are} \\ $$$$\mathrm{constants}.\:\mathrm{The}\:\mathrm{tangent}\:\mathrm{to}\:\mathrm{the}\:\mathrm{curve}\:\mathrm{at}\:\mathrm{the}\:\mathrm{point} \\ $$$$\left(\mathrm{2},−\mathrm{16}\right)\:\mathrm{is}\:\mathrm{parallel}\:\mathrm{to}\:\mathrm{the}\:\boldsymbol{\mathrm{x}}-\mathrm{axis}.\:\mathrm{Find} \\ $$$$\mathrm{i}.\:\mathrm{the}\:\mathrm{values}\:\mathrm{of}\:\boldsymbol{\mathrm{a}}\:\mathrm{and}\:\boldsymbol{\mathrm{b}} \\ $$$$\mathrm{ii}.\:\mathrm{the}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{the}\:\mathrm{curve} \\ $$

Question Number 43922    Answers: 0   Comments: 0

Question Number 43921    Answers: 1   Comments: 0

prove that product of lengths of perpendiculars from any point of hyperbola to its asymptotes is constant

$${prove}\:{that}\:{product}\:{of}\:{lengths}\:{of}\:{perpendiculars} \\ $$$${from}\:{any}\:{point}\:{of}\:{hyperbola}\:{to}\:{its} \\ $$$${asymptotes}\:{is}\:{constant} \\ $$

Question Number 43919    Answers: 0   Comments: 0

The tangent at P to an ellipse meet diretrix at Q.prove that the line joining the corresponding focus to P and Q are perpendicular

$${The}\:{tangent}\:{at}\:{P}\:{to}\:{an}\:{ellipse}\:{meet}\: \\ $$$${diretrix}\:{at}\:{Q}.{prove}\:{that}\:{the}\:{line}\:{joining} \\ $$$${the}\:{corresponding}\:{focus}\:{to}\:{P}\:{and}\:{Q}\:{are}\:{perpendicular} \\ $$

Question Number 43918    Answers: 0   Comments: 3

1) find f(x) =∫_0 ^x ln(t)ln(1−t)dt with 0≤x≤1 2) find the value of ∫_0 ^1 ln(t)ln(1−t)dt .

$$\left.\mathrm{1}\right)\:{find}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{{x}} {ln}\left({t}\right){ln}\left(\mathrm{1}−{t}\right){dt}\:\:\:{with}\:\mathrm{0}\leqslant{x}\leqslant\mathrm{1} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left({t}\right){ln}\left(\mathrm{1}−{t}\right){dt}\:. \\ $$

Question Number 43914    Answers: 0   Comments: 1

find ∫ (x/(√(1+cosx)))dx .

$${find}\:\int\:\:\:\:\frac{{x}}{\sqrt{\mathrm{1}+{cosx}}}{dx}\:. \\ $$

Question Number 43913    Answers: 2   Comments: 4

find ∫ (dx/((√(1−cosx)) +(√(1+cosx))))

$${find}\:\int\:\:\:\:\:\frac{{dx}}{\sqrt{\mathrm{1}−{cosx}}\:+\sqrt{\mathrm{1}+{cosx}}} \\ $$$$ \\ $$

Question Number 43909    Answers: 1   Comments: 0

find f(a,t)=∫_0 ^(2π) (dx/(a +t sinx)) 2)calculate ∫_0 ^(2π) ((sinx)/((a+tsinx)^2 ))dx 3)calculate ∫_0 ^(2π) (dx/((a+tsinx)^2 )) .

$${find}\:{f}\left({a},{t}\right)=\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\frac{{dx}}{{a}\:+{t}\:{sinx}} \\ $$$$\left.\mathrm{2}\right){calculate}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\frac{{sinx}}{\left({a}+{tsinx}\right)^{\mathrm{2}} }{dx} \\ $$$$\left.\mathrm{3}\right){calculate}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\frac{{dx}}{\left({a}+{tsinx}\right)^{\mathrm{2}} }\:. \\ $$

Question Number 43907    Answers: 1   Comments: 2

find ∫ ((cos^2 x −sin^2 x)/(3 +tan^2 x))dx

$${find}\:\int\:\:\frac{{cos}^{\mathrm{2}} {x}\:−{sin}^{\mathrm{2}} {x}}{\mathrm{3}\:+{tan}^{\mathrm{2}} {x}}{dx} \\ $$

Question Number 43906    Answers: 0   Comments: 1

calculate ∫_0 ^∞ ((sin(x))/(sh(2x)))dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{sin}\left({x}\right)}{{sh}\left(\mathrm{2}{x}\right)}{dx} \\ $$

Question Number 43905    Answers: 1   Comments: 1

calculate ∫_1 ^(+∞) (dx/(x^3 (√(2+x^2 ))))

$${calculate}\:\int_{\mathrm{1}} ^{+\infty} \:\:\frac{{dx}}{{x}^{\mathrm{3}} \sqrt{\mathrm{2}+{x}^{\mathrm{2}} }} \\ $$

Question Number 43904    Answers: 1   Comments: 1

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

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

Question Number 43902    Answers: 2   Comments: 4

(√(a−b)) + (√(a+b)) = c (√(a−c)) + (√(a+c)) = b Solve for real b, and c ; in terms of real a.

$$\sqrt{{a}−{b}}\:+\:\sqrt{{a}+{b}}\:=\:{c} \\ $$$$\sqrt{{a}−{c}}\:+\:\sqrt{{a}+{c}}\:=\:{b} \\ $$$${Solve}\:{for}\:{real}\:{b},\:{and}\:{c}\:;\:{in}\:{terms}\: \\ $$$${of}\:{real}\:{a}. \\ $$

Question Number 43898    Answers: 0   Comments: 0

Question Number 43897    Answers: 0   Comments: 0

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