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Question Number 17481    Answers: 0   Comments: 0

Question Number 17480    Answers: 1   Comments: 0

why (d/dx)(∫_0 ^( y) e^t dt)=e^y (dy/dx)

$$\mathrm{why}\:\:\frac{\mathrm{d}}{\mathrm{dx}}\left(\int_{\mathrm{0}} ^{\:\:\mathrm{y}} \mathrm{e}^{\mathrm{t}} \mathrm{dt}\right)=\mathrm{e}^{\mathrm{y}} \frac{\mathrm{dy}}{\mathrm{dx}} \\ $$

Question Number 17475    Answers: 1   Comments: 0

Prove that ∫_e ^e^2 ((lnx)/((1+lnx)^2 ))dx=(e/6)(2e−3)

$$\mathrm{Prove}\:\mathrm{that}\:\int_{\mathrm{e}} ^{\mathrm{e}^{\mathrm{2}} } \frac{\mathrm{ln}{x}}{\left(\mathrm{1}+\mathrm{ln}{x}\right)^{\mathrm{2}} }\mathrm{d}{x}=\frac{\mathrm{e}}{\mathrm{6}}\left(\mathrm{2e}−\mathrm{3}\right) \\ $$

Question Number 17473    Answers: 1   Comments: 0

Prove that ∫_0 ^( 2a) (√(2ax−x^2 ))dx=((Πa^2 )/2)

$$\mathrm{Prove}\:\mathrm{that}\:\int_{\mathrm{0}} ^{\:\mathrm{2a}} \sqrt{\mathrm{2ax}−\mathrm{x}^{\mathrm{2}} }\mathrm{dx}=\frac{\Pi\mathrm{a}^{\mathrm{2}} }{\mathrm{2}} \\ $$

Question Number 17472    Answers: 1   Comments: 0

Prove that ∫_6 ^(11) (dx/(√((x−2)(x−3))))=2ln((3+2(√2))/(2+(√3)))

$$\mathrm{Prove}\:\mathrm{that}\:\int_{\mathrm{6}} ^{\mathrm{11}} \frac{\mathrm{dx}}{\sqrt{\left(\mathrm{x}−\mathrm{2}\right)\left(\mathrm{x}−\mathrm{3}\right)}}=\mathrm{2ln}\frac{\mathrm{3}+\mathrm{2}\sqrt{\mathrm{2}}}{\mathrm{2}+\sqrt{\mathrm{3}}} \\ $$

Question Number 17479    Answers: 0   Comments: 0

If f(x) is a periodic function with period time t ; prove that∫_a ^( a+t) f(x)dx is a indipendent.

$$\mathrm{If}\:{f}\left({x}\right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{periodic}\:\mathrm{function}\:\mathrm{with}\:\mathrm{period} \\ $$$$\mathrm{time}\:\mathrm{t}\:;\:\mathrm{prove}\:\mathrm{that}\int_{\mathrm{a}} ^{\:\mathrm{a}+\mathrm{t}} {f}\left({x}\right)\mathrm{d}{x}\:\mathrm{is}\:\mathrm{a}\:\mathrm{indipendent}. \\ $$

Question Number 17449    Answers: 1   Comments: 4

Between 2:00 and 2:15, what time is it exactly when the hour, minute, and second′s hand of a clock occupy the same angular position.

$$\mathrm{Between}\:\mathrm{2}:\mathrm{00}\:\mathrm{and}\:\mathrm{2}:\mathrm{15},\:\mathrm{what}\:\mathrm{time} \\ $$$$\mathrm{is}\:\mathrm{it}\:\mathrm{exactly}\:\mathrm{when}\:\mathrm{the}\:\mathrm{hour}, \\ $$$$\mathrm{minute},\:\mathrm{and}\:\mathrm{second}'\mathrm{s}\:\mathrm{hand}\:\mathrm{of}\:\mathrm{a} \\ $$$$\mathrm{clock}\:\mathrm{occupy}\:\mathrm{the}\:\mathrm{same}\:\mathrm{angular} \\ $$$$\mathrm{position}. \\ $$$$ \\ $$

Question Number 17492    Answers: 1   Comments: 0

The number of values of x lying in [−π, π] and satisfying 2 sin^2 θ = cos 2θ and sin 2θ + 2 cos 2θ − cos θ − 1 = 0 is

$$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{values}\:\mathrm{of}\:{x}\:\mathrm{lying}\:\mathrm{in} \\ $$$$\left[−\pi,\:\pi\right]\:\mathrm{and}\:\mathrm{satisfying}\:\mathrm{2}\:\mathrm{sin}^{\mathrm{2}} \:\theta\:=\:\mathrm{cos}\:\mathrm{2}\theta \\ $$$$\mathrm{and}\:\mathrm{sin}\:\mathrm{2}\theta\:+\:\mathrm{2}\:\mathrm{cos}\:\mathrm{2}\theta\:−\:\mathrm{cos}\:\theta\:−\:\mathrm{1}\:=\:\mathrm{0}\:\mathrm{is} \\ $$

Question Number 17446    Answers: 0   Comments: 2

Find the integer closest to 100(12 − (√(143))).

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{integer}\:\mathrm{closest}\:\mathrm{to} \\ $$$$\mathrm{100}\left(\mathrm{12}\:−\:\sqrt{\mathrm{143}}\right). \\ $$

Question Number 17444    Answers: 1   Comments: 0

Evaluate: ∫_0 ^(π/4) (dx/(cos^3 x (√(2 sin 2x))))

$$\mathrm{Evaluate}:\:\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{4}}} {\int}}\frac{{dx}}{\mathrm{cos}^{\mathrm{3}} \:{x}\:\sqrt{\mathrm{2}\:\mathrm{sin}\:\mathrm{2}{x}}} \\ $$

Question Number 17440    Answers: 1   Comments: 0

3x−4y=0,4y−5z=0,5z−3x=0 then x,y,z is AP,GP,HP,AGP??????

$$\mathrm{3}{x}−\mathrm{4}{y}=\mathrm{0},\mathrm{4}{y}−\mathrm{5}{z}=\mathrm{0},\mathrm{5}{z}−\mathrm{3}{x}=\mathrm{0} \\ $$$${then}\:{x},{y},{z}\:{is}\:{AP},{GP},{HP},{AGP}?????? \\ $$

Question Number 17645    Answers: 2   Comments: 1

Suppose that the point M lying in the interior of the parallelogram ABCD, two parallels to AB and AD are drawn, intersecting the sides of ABCD at the points P, Q, R, S (See Figure). Prove that M lies on the diagonal AC if and only if [MRDS] = [MPBQ].

$$\mathrm{Suppose}\:\mathrm{that}\:\mathrm{the}\:\mathrm{point}\:{M}\:\mathrm{lying}\:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{interior}\:\mathrm{of}\:\mathrm{the}\:\mathrm{parallelogram}\:{ABCD}, \\ $$$$\mathrm{two}\:\mathrm{parallels}\:\mathrm{to}\:{AB}\:\mathrm{and}\:{AD}\:\mathrm{are}\:\mathrm{drawn}, \\ $$$$\mathrm{intersecting}\:\mathrm{the}\:\mathrm{sides}\:\mathrm{of}\:{ABCD}\:\mathrm{at}\:\mathrm{the} \\ $$$$\mathrm{points}\:{P},\:{Q},\:{R},\:{S}\:\left(\mathrm{See}\:\mathrm{Figure}\right).\:\mathrm{Prove} \\ $$$$\mathrm{that}\:{M}\:\mathrm{lies}\:\mathrm{on}\:\mathrm{the}\:\mathrm{diagonal}\:{AC}\:\mathrm{if}\:\mathrm{and} \\ $$$$\mathrm{only}\:\mathrm{if}\:\left[{MRDS}\right]\:=\:\left[{MPBQ}\right]. \\ $$

Question Number 17438    Answers: 0   Comments: 0

find the mean value and root mean square of i=25sin100Πt ranging from 0 to 10

$$\mathrm{find}\:\mathrm{the}\:\mathrm{mean}\:\mathrm{value}\:\mathrm{and}\:\mathrm{root}\:\mathrm{mean}\:\mathrm{square}\:\mathrm{of}\: \\ $$$$\mathrm{i}=\mathrm{25sin100}\Pi\mathrm{t}\:\:\:\:\:\mathrm{ranging}\:\mathrm{from}\:\mathrm{0}\:\mathrm{to}\:\mathrm{10} \\ $$

Question Number 17435    Answers: 1   Comments: 0

Find the value of 4 sin (π/(24)) cos (π/(12)) cos(π/6).

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{4}\:\mathrm{sin}\:\frac{\pi}{\mathrm{24}}\:\:\mathrm{cos}\:\frac{\pi}{\mathrm{12}}\:\:\mathrm{cos}\frac{\pi}{\mathrm{6}}. \\ $$

Question Number 17421    Answers: 1   Comments: 0

The number of solutions of the equation 2^(∣x∣) = 1 + 2∣cos x∣ is

$$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{solutions}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\mathrm{2}^{\mid{x}\mid} \:=\:\mathrm{1}\:+\:\mathrm{2}\mid\mathrm{cos}\:{x}\mid\:\mathrm{is} \\ $$

Question Number 17420    Answers: 1   Comments: 0

tan^6 (π/9)−33tan^4 (π/9)+27tan^2 (π/9)=?

$$\mathrm{tan}^{\mathrm{6}} \frac{\pi}{\mathrm{9}}−\mathrm{33tan}^{\mathrm{4}} \frac{\pi}{\mathrm{9}}+\mathrm{27tan}^{\mathrm{2}} \frac{\pi}{\mathrm{9}}=? \\ $$

Question Number 17454    Answers: 1   Comments: 0

Find all integers n such that (n^2 − n − 1)^(n + 2) = 1

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{integers}\:{n}\:\mathrm{such}\:\mathrm{that} \\ $$$$\left({n}^{\mathrm{2}} \:−\:{n}\:−\:\mathrm{1}\right)^{{n}\:+\:\mathrm{2}} \:=\:\mathrm{1} \\ $$

Question Number 17401    Answers: 0   Comments: 4

Find the sum of 4-digit greatest number and the 5-digit smallest number, each number having three different digits.

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{4}-\mathrm{digit}\:\mathrm{greatest} \\ $$$$\mathrm{number}\:\mathrm{and}\:\mathrm{the}\:\mathrm{5}-\mathrm{digit}\:\mathrm{smallest} \\ $$$$\mathrm{number},\:\mathrm{each}\:\mathrm{number}\:\mathrm{having}\:\mathrm{three} \\ $$$$\mathrm{different}\:\mathrm{digits}. \\ $$

Question Number 17397    Answers: 1   Comments: 0

Question Number 17393    Answers: 1   Comments: 0

Find the modulus of z = 6 + 8i

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{modulus}\:\mathrm{of}\:\:\mathrm{z}\:=\:\mathrm{6}\:+\:\mathrm{8i} \\ $$

Question Number 17392    Answers: 2   Comments: 0

Find the cube root of z = − 1

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{cube}\:\mathrm{root}\:\mathrm{of}\:\:\mathrm{z}\:=\:−\:\mathrm{1} \\ $$

Question Number 17391    Answers: 1   Comments: 0

write z = (2 + 2(√3) i)^3 in polar form.

$$\mathrm{write}\:\:\:\mathrm{z}\:=\:\left(\mathrm{2}\:+\:\mathrm{2}\sqrt{\mathrm{3}}\:\mathrm{i}\right)^{\mathrm{3}} \:\:\mathrm{in}\:\mathrm{polar}\:\mathrm{form}. \\ $$

Question Number 17377    Answers: 1   Comments: 0

∫ ((cos(x))/(2 − cos(x))) dx

$$\int\:\:\frac{\mathrm{cos}\left(\mathrm{x}\right)}{\mathrm{2}\:−\:\mathrm{cos}\left(\mathrm{x}\right)}\:\mathrm{dx} \\ $$

Question Number 17374    Answers: 1   Comments: 1

Question Number 17373    Answers: 2   Comments: 0

Find the point in interior of a convex quadrilateral such that the sum of its distances to the 4 vertices is minimal. Find the point in interior of a convex quadrilateral such that the sum of its distances to the 4 sides is minimal.

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{point}\:\mathrm{in}\:\mathrm{interior}\:\mathrm{of}\:\mathrm{a}\:\mathrm{convex} \\ $$$$\mathrm{quadrilateral}\:\mathrm{such}\:\mathrm{that}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{its} \\ $$$$\mathrm{distances}\:\mathrm{to}\:\mathrm{the}\:\mathrm{4}\:\mathrm{vertices}\:\mathrm{is}\:\mathrm{minimal}. \\ $$$$ \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{point}\:\mathrm{in}\:\mathrm{interior}\:\mathrm{of}\:\mathrm{a}\:\mathrm{convex} \\ $$$$\mathrm{quadrilateral}\:\mathrm{such}\:\mathrm{that}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{its} \\ $$$$\mathrm{distances}\:\mathrm{to}\:\mathrm{the}\:\mathrm{4}\:\mathrm{sides}\:\mathrm{is}\:\mathrm{minimal}. \\ $$

Question Number 17386    Answers: 2   Comments: 0

Solve the equation: log _2 x log _3 x log _5 x=log _2 x log _3 x +log _3 x log _5 x +log _5 x log _2 x .

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{equation}: \\ $$$$\mathrm{log}\:_{\mathrm{2}} \mathrm{x}\:\mathrm{log}\:_{\mathrm{3}} \mathrm{x}\:\mathrm{log}\:_{\mathrm{5}} \mathrm{x}=\mathrm{log}\:_{\mathrm{2}} \mathrm{x}\:\mathrm{log}\:_{\mathrm{3}} \mathrm{x} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:+\mathrm{log}\:_{\mathrm{3}} \mathrm{x}\:\mathrm{log}\:_{\mathrm{5}} \mathrm{x} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:+\mathrm{log}\:_{\mathrm{5}} \mathrm{x}\:\mathrm{log}\:_{\mathrm{2}} \mathrm{x}\:. \\ $$

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