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Question Number 97554 Answers: 1 Comments: 5

Find all the triples of positive integers (x;y;z) so that ((x−y(√(2020)))/(y−z(√(2020)))) is a rational number and x^2 +y^2 +z^2 be a prime number.

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{the}\:\mathrm{triples}\:\mathrm{of}\:\mathrm{positive}\:\mathrm{integers}\:\left(\mathrm{x};\mathrm{y};\mathrm{z}\right) \\ $$$$\mathrm{so}\:\mathrm{that}\:\frac{\mathrm{x}−\mathrm{y}\sqrt{\mathrm{2020}}}{\mathrm{y}−\mathrm{z}\sqrt{\mathrm{2020}}}\:\mathrm{is}\:\mathrm{a}\:\mathrm{rational}\:\mathrm{number} \\ $$$$\mathrm{and}\:\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} +\mathrm{z}^{\mathrm{2}} \mathrm{be}\:\mathrm{a}\:\mathrm{prime}\:\mathrm{number}. \\ $$

Question Number 97552 Answers: 2 Comments: 0

∫_0 ^1 (√(x/(1−x))) ln((x/(1−x))) dx ?

$$\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\sqrt{\frac{{x}}{\mathrm{1}−{x}}}\:\mathrm{ln}\left(\frac{{x}}{\mathrm{1}−{x}}\right)\:{dx}\:? \\ $$

Question Number 97550 Answers: 0 Comments: 0

Question Number 97546 Answers: 0 Comments: 2

Question Number 97541 Answers: 1 Comments: 2

Determinate lim_(x→1/2) ((sin(2x−1))/(2x−1)) lim_(x→+∞ ) ((cosx)/(1+x^(2 ) ))

$${Determinate} \\ $$$$\underset{{x}\rightarrow\mathrm{1}/\mathrm{2}} {{lim}}\:\frac{{sin}\left(\mathrm{2}{x}−\mathrm{1}\right)}{\mathrm{2}{x}−\mathrm{1}} \\ $$$$\underset{{x}\rightarrow+\infty\:} {{lim}}\:\frac{{cosx}}{\mathrm{1}+{x}^{\mathrm{2}\:} } \\ $$

Question Number 97539 Answers: 2 Comments: 0

x^2 −3y^2 +2xy (dy/dx) = 0

$${x}^{\mathrm{2}} −\mathrm{3}{y}^{\mathrm{2}} +\mathrm{2}{xy}\:\frac{{dy}}{{dx}}\:=\:\mathrm{0} \\ $$

Question Number 97537 Answers: 1 Comments: 0

∫_0 ^1 ((−(√(1−x^2 )))/((yx^3 +x^2 −yx−1)))dx

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

Question Number 97531 Answers: 1 Comments: 0

5050((∫_0 ^1 (1−x^(50) )^(100) dx)/(∫_0 ^1 (1−x^(50) )^(101) dx))=

$$\mathrm{5050}\frac{\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{1}−{x}^{\mathrm{50}} \right)^{\mathrm{100}} {dx}}{\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{1}−{x}^{\mathrm{50}} \right)^{\mathrm{101}} {dx}}= \\ $$

Question Number 97527 Answers: 0 Comments: 0

∫((xdx)/(x!))=?

$$\int\frac{\mathrm{xdx}}{\mathrm{x}!}=? \\ $$

Question Number 97526 Answers: 1 Comments: 1

∫(dx/(1+sin x))=?

$$\int\frac{\mathrm{dx}}{\mathrm{1}+\mathrm{sin}\:\mathrm{x}}=? \\ $$

Question Number 97512 Answers: 0 Comments: 2

The value of k for which the quadratic equation (1−2k)x^2 −6kx−1=0 and kx^2 −x+1=0 have atleast one roots in common are ___

$$\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:\mathrm{k}\:\mathrm{for}\:\mathrm{which}\:\mathrm{the} \\ $$$$\mathrm{quadratic}\:\mathrm{equation}\:\left(\mathrm{1}−\mathrm{2k}\right)\mathrm{x}^{\mathrm{2}} −\mathrm{6kx}−\mathrm{1}=\mathrm{0} \\ $$$$\mathrm{and}\:\mathrm{kx}^{\mathrm{2}} −\mathrm{x}+\mathrm{1}=\mathrm{0}\:\mathrm{have}\:\mathrm{atleast} \\ $$$$\mathrm{one}\:\mathrm{roots}\:\mathrm{in}\:\mathrm{common}\:\mathrm{are}\:\_\_\_ \\ $$

Question Number 97506 Answers: 0 Comments: 0

please prove it cos x= J_0 (x) + 2Σ_(x−1) (−1)^x J_(2n) (x)

$${please}\:{prove}\:{it} \\ $$$$ \\ $$$$\mathrm{cos}\:{x}=\:{J}_{\mathrm{0}} \left({x}\right)\:+\:\mathrm{2}\sum_{{x}−\mathrm{1}} \left(−\mathrm{1}\right)^{{x}} {J}_{\mathrm{2}{n}} \left({x}\right) \\ $$

Question Number 97501 Answers: 0 Comments: 2

The natural number n for which the expression y = 5log^2 _3 (n) − log _3 (n^(12) )+9 , has the minimum value is ___

$$\mathrm{The}\:\mathrm{natural}\:\mathrm{number}\:\mathrm{n}\:\mathrm{for}\:\mathrm{which}\: \\ $$$$\mathrm{the}\:\mathrm{expression}\:\mathrm{y}\:=\:\mathrm{5log}^{\mathrm{2}} \:_{\mathrm{3}} \left(\mathrm{n}\right)\:− \\ $$$$\mathrm{log}\:_{\mathrm{3}} \left(\mathrm{n}^{\mathrm{12}} \right)+\mathrm{9}\:,\:\mathrm{has}\:\mathrm{the}\:\mathrm{minimum} \\ $$$$\mathrm{value}\:\mathrm{is}\:\_\_\_ \\ $$

Question Number 97505 Answers: 0 Comments: 0

Question Number 97497 Answers: 2 Comments: 0

Question Number 97496 Answers: 1 Comments: 0

if f(((2x+5)/(x−3)))=3x+5 find f(x) please solve it

$${if}\:\:\:\:\:\:{f}\left(\frac{\mathrm{2}{x}+\mathrm{5}}{{x}−\mathrm{3}}\right)=\mathrm{3}{x}+\mathrm{5}\:\:\:{find}\:\:\:{f}\left({x}\right) \\ $$$$ \\ $$$${please}\:{solve}\:{it} \\ $$

Question Number 97494 Answers: 1 Comments: 0

Question Number 97492 Answers: 2 Comments: 0

Question Number 97490 Answers: 1 Comments: 0

please prove it cosx= J_0 (x)+2Σ_(x−1) (−1)^x J_(2x) (x)

$${please}\:\:{prove}\:\:{it} \\ $$$$\mathrm{cos}{x}=\:{J}_{\mathrm{0}} \left({x}\right)+\mathrm{2}\sum_{{x}−\mathrm{1}} \left(−\mathrm{1}\right)^{{x}} {J}_{\mathrm{2}{x}} \left({x}\right) \\ $$

Question Number 97489 Answers: 2 Comments: 0

please prove it ∫_0 ^∞ e^(−ax^2 ) cos bx dx= (1/2)(√(π/a)).e^(−(b^2 /(4a)))

$${please}\:\:{prove}\:{it} \\ $$$$ \\ $$$$\int_{\mathrm{0}} ^{\infty} {e}^{−{ax}^{\mathrm{2}} } \mathrm{cos}\:{bx}\:\:{dx}=\:\frac{\mathrm{1}}{\mathrm{2}}\sqrt{\frac{\pi}{{a}}}.{e}^{−\frac{{b}^{\mathrm{2}} }{\mathrm{4}{a}}} \\ $$

Question Number 97485 Answers: 1 Comments: 3

Question Number 97483 Answers: 1 Comments: 4

In each week the growth of a plant is two−thirds the growth of the previous week. The plant grows 12 cm in the first week. (a) Calculate the growth of the plant in (b) the limiting height of the pant

$$\mathrm{In}\:\mathrm{each}\:\mathrm{week}\:\mathrm{the}\:\mathrm{growth}\:\mathrm{of}\:\mathrm{a}\:\mathrm{plant}\:\mathrm{is}\:\mathrm{two}−\mathrm{thirds} \\ $$$$\mathrm{the}\:\mathrm{growth}\:\mathrm{of}\:\mathrm{the}\:\mathrm{previous}\:\mathrm{week}. \\ $$$$\mathrm{The}\:\mathrm{plant}\:\mathrm{grows}\:\mathrm{12}\:\mathrm{cm}\:\mathrm{in}\:\mathrm{the}\:\mathrm{first}\:\mathrm{week}. \\ $$$$\left(\mathrm{a}\right)\:\mathrm{Calculate}\:\mathrm{the}\:\mathrm{growth}\:\mathrm{of}\:\mathrm{the}\:\mathrm{plant}\:\mathrm{in}\: \\ $$$$\left(\mathrm{b}\right)\:\mathrm{the}\:\mathrm{limiting}\:\mathrm{height}\:\mathrm{of}\:\mathrm{the}\:\mathrm{pant} \\ $$

Question Number 97479 Answers: 2 Comments: 2

Question Number 97478 Answers: 0 Comments: 0

Find the global parametrization of the curve { x^2 +y^2 +z^2 =1; x+y−z=0 }

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{global}\:\mathrm{parametrization} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{curve}\:\left\{\:\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} +\mathrm{z}^{\mathrm{2}} =\mathrm{1};\:\mathrm{x}+\mathrm{y}−\mathrm{z}=\mathrm{0}\:\right\}\: \\ $$

Question Number 97476 Answers: 0 Comments: 0

2F1((1/2),(1/2);(1/2);z)=(1−z)^(1/2) ∗∗1 by kummer transformation 2F1((1/2),(1/2);(1/2);z)=2F1((1/2),(1/2);1+(1/2)+(1/2)−(1/2);z) 2F1((1/2),(1/2);(1/2);z)=((sin^(−1) (√(1−z)))/(√(1−z)))∗∗2 why do i get different answer in ∗∗1 and 2∗∗

Question Number 97465 Answers: 2 Comments: 1

∫_0 ^∝ e^(−x^4 ) dx=(1/4) please prove it

$$\int_{\mathrm{0}} ^{\propto} {e}^{−{x}^{\mathrm{4}} } {dx}=\frac{\mathrm{1}}{\mathrm{4}} \\ $$$${please}\:{prove}\:{it} \\ $$

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