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Prove that if the lengths of a triangle form an arithmetic progression, then the centre of incircle and the centroid of triangle lie on a line parallel to the side of middle length of the triangle.

$${Prove}\:{that}\:{if}\:{the}\:{lengths}\:{of}\:{a}\: \\ $$$${triangle}\:{form}\:{an}\:{arithmetic} \\ $$$${progression},\:{then}\:{the}\:{centre}\:{of} \\ $$$${incircle}\:{and}\:{the}\:{centroid}\:{of} \\ $$$${triangle}\:{lie}\:{on}\:{a}\:{line}\:{parallel}\:{to} \\ $$$${the}\:{side}\:{of}\:{middle}\:{length}\:{of}\:{the} \\ $$$${triangle}. \\ $$

Question Number 62372 Answers: 1 Comments: 0

Solve for x , y 3x>2y ∧ 2x<3y where x,y∈N

$${Solve}\:{for}\:{x}\:,\:{y} \\ $$$$\mathrm{3}{x}>\mathrm{2}{y}\:\wedge\:\mathrm{2}{x}<\mathrm{3}{y}\: \\ $$$${where}\:{x},{y}\in\mathbb{N} \\ $$

Question Number 62363 Answers: 2 Comments: 0

Question Number 62347 Answers: 2 Comments: 0

Question Number 62343 Answers: 1 Comments: 1

calculate ∫_0 ^(π/4) {xΠ_(k=1) ^∞ cos((x/2^k ))}dx

$${calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \left\{{x}\prod_{{k}=\mathrm{1}} ^{\infty} \:{cos}\left(\frac{{x}}{\mathrm{2}^{{k}} }\right)\right\}{dx} \\ $$

Question Number 62342 Answers: 1 Comments: 4

let f(ξ) =∫ (x^2 /(√(1−ξx^2 )))dx with 0<ξ<1 1) determine a explicit form of f(ξ) 2) calculate lim_(ξ→1) f(ξ) 3) calculate ∫_0 ^(1/2) (x^2 /(√(1−sin^2 θ x^2 ))) dx with 0<θ<(π/2)

$${let}\:{f}\left(\xi\right)\:=\int\:\:\frac{{x}^{\mathrm{2}} }{\sqrt{\mathrm{1}−\xi{x}^{\mathrm{2}} }}{dx}\:\:\:{with}\:\:\mathrm{0}<\xi<\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{determine}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left(\xi\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{lim}_{\xi\rightarrow\mathrm{1}} \:\:\:{f}\left(\xi\right) \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\frac{\mathrm{1}}{\mathrm{2}}} \:\frac{{x}^{\mathrm{2}} }{\sqrt{\mathrm{1}−{sin}^{\mathrm{2}} \theta\:{x}^{\mathrm{2}} }}\:{dx}\:{with}\:\:\mathrm{0}<\theta<\frac{\pi}{\mathrm{2}} \\ $$

Question Number 62341 Answers: 2 Comments: 3

How many real root does the equation x^8 − x^7 + 2x^6 − 2x^5 + 3x^4 − 3x^3 + 4x^2 − 4x + (5/2) = 0 has

$$\mathrm{How}\:\mathrm{many}\:\mathrm{real}\:\mathrm{root}\:\mathrm{does}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\:\:\mathrm{x}^{\mathrm{8}} \:−\:\mathrm{x}^{\mathrm{7}} \:+\:\mathrm{2x}^{\mathrm{6}} \:−\:\mathrm{2x}^{\mathrm{5}} \:+\:\mathrm{3x}^{\mathrm{4}} \:−\:\mathrm{3x}^{\mathrm{3}} \:+\:\mathrm{4x}^{\mathrm{2}} \:−\:\mathrm{4x}\:+\:\frac{\mathrm{5}}{\mathrm{2}}\:\:=\:\:\mathrm{0}\:\:\:\:\:\:\:\mathrm{has} \\ $$

Question Number 62340 Answers: 0 Comments: 3

Question Number 62338 Answers: 0 Comments: 0

Question Number 62335 Answers: 0 Comments: 2

1) calculate f(x,y) =∫_0 ^∞ ((e^(−xt) cos(yt))/(√t)) dt and g(x,y) =∫_0 ^∞ ((e^(−xt) sin(yt))/(√t)) dt with x>0 and y>0 2) find the values of ∫_0 ^∞ ((e^(−2t) cos(t))/(√t)) dt and ∫_0 ^∞ ((e^(−t) cos(2t))/(√t)) dt

$$\left.\mathrm{1}\right)\:{calculate}\:{f}\left({x},{y}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\frac{{e}^{−{xt}} {cos}\left({yt}\right)}{\sqrt{{t}}}\:{dt}\:{and}\:{g}\left({x},{y}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{e}^{−{xt}} {sin}\left({yt}\right)}{\sqrt{{t}}}\:{dt} \\ $$$${with}\:{x}>\mathrm{0}\:\:{and}\:{y}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{values}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\frac{{e}^{−\mathrm{2}{t}} \:{cos}\left({t}\right)}{\sqrt{{t}}}\:{dt}\:{and}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{e}^{−{t}} {cos}\left(\mathrm{2}{t}\right)}{\sqrt{{t}}}\:{dt} \\ $$

Question Number 62334 Answers: 2 Comments: 0

if α^2 +β^2 = (α+β)^2 −2αβ evaluate(α−β)

$${if}\:\alpha^{\mathrm{2}} +\beta^{\mathrm{2}} =\:\left(\alpha+\beta\right)^{\mathrm{2}} −\mathrm{2}\alpha\beta\:{evaluate}\left(\alpha−\beta\right) \\ $$

Question Number 62332 Answers: 1 Comments: 0