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Question Number 76770    Answers: 0   Comments: 0

Question Number 76740    Answers: 0   Comments: 8

15 persons, among whom are A and B, sit down at random at a round table. The probability that there are 4 persons between A and B is

$$\mathrm{15}\:\mathrm{persons},\:\mathrm{among}\:\mathrm{whom}\:\mathrm{are}\:{A}\:\mathrm{and}\:{B}, \\ $$$$\mathrm{sit}\:\mathrm{down}\:\mathrm{at}\:\mathrm{random}\:\mathrm{at}\:\mathrm{a}\:\mathrm{round}\:\mathrm{table}. \\ $$$$\mathrm{The}\:\mathrm{probability}\:\mathrm{that}\:\mathrm{there}\:\mathrm{are}\:\mathrm{4}\:\mathrm{persons} \\ $$$$\mathrm{between}\:{A}\:\mathrm{and}\:{B}\:\mathrm{is} \\ $$

Question Number 76726    Answers: 3   Comments: 0

var(x) = 2 then var(2x −3)=? E(x) = 2 then E(2x −3) = ?

$$\:\mathrm{var}\left(\mathrm{x}\right)\:=\:\mathrm{2}\:\mathrm{then}\:\mathrm{var}\left(\mathrm{2x}\:−\mathrm{3}\right)=? \\ $$$$\mathrm{E}\left(\mathrm{x}\right)\:=\:\mathrm{2}\:\mathrm{then}\:\mathrm{E}\left(\mathrm{2x}\:−\mathrm{3}\right)\:=\:? \\ $$

Question Number 76725    Answers: 0   Comments: 1

To prove that x^2 >y^2 , it is sufficient to prove that A. x > y B. x^3 >y^3 C. ∣x∣ > ∣y∣ D. x > 3y

$$\mathrm{To}\:\mathrm{prove}\:\mathrm{that}\:\mathrm{x}^{\mathrm{2}} >\mathrm{y}^{\mathrm{2}} ,\:\mathrm{it}\:\mathrm{is}\:\mathrm{sufficient}\:\mathrm{to}\: \\ $$$$\mathrm{prove}\:\mathrm{that}\: \\ $$$$\mathrm{A}.\:\mathrm{x}\:>\:\mathrm{y} \\ $$$$\mathrm{B}.\:\mathrm{x}^{\mathrm{3}} >\mathrm{y}^{\mathrm{3}} \\ $$$$\mathrm{C}.\:\mid\mathrm{x}\mid\:>\:\mid\mathrm{y}\mid \\ $$$$\mathrm{D}.\:\mathrm{x}\:>\:\mathrm{3y} \\ $$

Question Number 76724    Answers: 1   Comments: 1

∫_0 ^3 ∣x^2 −1∣ dx ≡

$$\int_{\mathrm{0}} ^{\mathrm{3}} \mid\mathrm{x}^{\mathrm{2}} −\mathrm{1}\mid\:\mathrm{dx}\:\equiv\: \\ $$

Question Number 76723    Answers: 0   Comments: 0

the maclaurin expansion of ln (3 + 4x) is valid for A) −(3/4) ≤ x< (3/4) B) −(3/4)< x ≤ (3/4) C) −(1/4)< x ≤ (1/4) D) −(3/4)< x < (3/4)

$$\mathrm{the}\:\mathrm{maclaurin}\:\mathrm{expansion}\:\mathrm{of}\:\mathrm{ln}\:\left(\mathrm{3}\:+\:\mathrm{4}{x}\right)\:{is}\:{valid}\:{for} \\ $$$$\left.{A}\right)\:\:−\frac{\mathrm{3}}{\mathrm{4}}\:\leqslant\:\mathrm{x}<\:\frac{\mathrm{3}}{\mathrm{4}} \\ $$$$\left.\mathrm{B}\right)\:−\frac{\mathrm{3}}{\mathrm{4}}<\:\mathrm{x}\:\leqslant\:\frac{\mathrm{3}}{\mathrm{4}} \\ $$$$\left.\mathrm{C}\right)\:−\frac{\mathrm{1}}{\mathrm{4}}<\:\mathrm{x}\:\leqslant\:\frac{\mathrm{1}}{\mathrm{4}} \\ $$$$\left.\mathrm{D}\right)\:−\frac{\mathrm{3}}{\mathrm{4}}<\:\mathrm{x}\:<\:\frac{\mathrm{3}}{\mathrm{4}} \\ $$

Question Number 76721    Answers: 2   Comments: 0

what is the mean value of (1/(1+4x^(2 ) )) for 0≤x≤(1/2)

$$\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{mean}\:\mathrm{value}\:\mathrm{of}\:\frac{\mathrm{1}}{\mathrm{1}+\mathrm{4x}^{\mathrm{2}\:} }\:\:\mathrm{for}\:\:\mathrm{0}\leqslant\mathrm{x}\leqslant\frac{\mathrm{1}}{\mathrm{2}} \\ $$

Question Number 76718    Answers: 0   Comments: 0

For n ∈ N prove by mathematical induction that cos α+cos (α+β)+cos [α+(n−1)β]+...cos [α+(n−1)β]= ((cos [α+(((n−1)/2))β]sin ((nβ)/2))/(sin (n/2)))

$${For}\:{n}\:\in\:{N}\:{prove}\:{by}\:{mathematical} \\ $$$${induction}\:{that} \\ $$$$\mathrm{cos}\:\alpha+\mathrm{cos}\:\left(\alpha+\beta\right)+\mathrm{cos}\:\left[\alpha+\left({n}−\mathrm{1}\right)\beta\right]+...\mathrm{cos}\:\left[\alpha+\left({n}−\mathrm{1}\right)\beta\right]= \\ $$$$\frac{\mathrm{cos}\:\left[\alpha+\left(\frac{{n}−\mathrm{1}}{\mathrm{2}}\right)\beta\right]\mathrm{sin}\:\frac{{n}\beta}{\mathrm{2}}}{\mathrm{sin}\:\frac{{n}}{\mathrm{2}}} \\ $$

Question Number 76717    Answers: 1   Comments: 0

A triangle is formed by the three straight line y=m_1 x+(a/m_1 ) y=m_2 x+(a/m_2 ) y=m_3 x+(a/m_3 ) prove that its orthocenter always lies on the line x+a=0

$${A}\:{triangle}\:{is}\:{formed}\:{by} \\ $$$${the}\:{three}\:{straight}\:{line} \\ $$$${y}={m}_{\mathrm{1}} {x}+\frac{{a}}{{m}_{\mathrm{1}} } \\ $$$${y}={m}_{\mathrm{2}} {x}+\frac{{a}}{{m}_{\mathrm{2}} } \\ $$$${y}={m}_{\mathrm{3}} {x}+\frac{{a}}{{m}_{\mathrm{3}} } \\ $$$${prove}\:{that}\:{its}\:{orthocenter} \\ $$$${always}\:{lies}\:{on}\:{the}\:{line} \\ $$$${x}+{a}=\mathrm{0} \\ $$$$ \\ $$

Question Number 76716    Answers: 1   Comments: 4

prove that ∫_0 ^π ((xsin x)/(1+cos^2 x))=(π^2 /4)

$${prove}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\pi} \frac{{x}\mathrm{sin}\:{x}}{\mathrm{1}+\mathrm{cos}\:^{\mathrm{2}} {x}}=\frac{\pi^{\mathrm{2}} }{\mathrm{4}} \\ $$

Question Number 76715    Answers: 0   Comments: 3

If u=arcsin (x/y)+arctan (y/x) show that x(∂u/dx)+y(∂u/dy)=0

$${If}\:{u}={arc}\mathrm{sin}\:\frac{{x}}{{y}}+{arc}\mathrm{tan}\:\frac{{y}}{{x}} \\ $$$${show}\:{that}\: \\ $$$${x}\frac{\partial{u}}{{dx}}+{y}\frac{\partial{u}}{{dy}}=\mathrm{0} \\ $$

Question Number 76714    Answers: 1   Comments: 0

If y=(√(tan x+(√(tan x+(√(tan x+....∞)))))) prove that (dy/dx)=((sec^2 x)/(2y−1))

$${If}\:{y}=\sqrt{\mathrm{tan}\:{x}+\sqrt{\mathrm{tan}\:{x}+\sqrt{\mathrm{tan}\:{x}+....\infty}}}\: \\ $$$${prove}\:{that} \\ $$$$\frac{{dy}}{{dx}}=\frac{\mathrm{sec}\:^{\mathrm{2}} {x}}{\mathrm{2}{y}−\mathrm{1}} \\ $$

Question Number 76713    Answers: 0   Comments: 0

If cos y=xcos (a+y),show that (dy/dx)=((cos^2 (a+y))/(sin a))

$${If}\:\mathrm{cos}\:{y}={x}\mathrm{cos}\:\left({a}+{y}\right),{show} \\ $$$${that}\:\frac{{dy}}{{dx}}=\frac{\mathrm{cos}\:^{\mathrm{2}} \left({a}+{y}\right)}{\mathrm{sin}\:{a}} \\ $$

Question Number 76711    Answers: 1   Comments: 0

Question Number 76700    Answers: 1   Comments: 2

Question Number 76696    Answers: 2   Comments: 0

what is the value ln(0).?

$${what}\:{is}\:{the}\:{value}\:{ln}\left(\mathrm{0}\right).? \\ $$

Question Number 76694    Answers: 0   Comments: 0

Question Number 76688    Answers: 1   Comments: 0

calculate ∫ (1/(lnx))+ ln(lnx) dx .

$${calculate}\:\int\:\frac{\mathrm{1}}{{lnx}}+\:{ln}\left({lnx}\right)\:{dx}\:. \\ $$

Question Number 76680    Answers: 0   Comments: 4

prove that: ∫_0 ^1 (1−x^7 )^(1/3) dx=∫_0 ^1 (1−x^3 )^(1/7) dx

$${prove}\:{that}: \\ $$$$\:\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{1}−{x}^{\mathrm{7}} \right)^{\frac{\mathrm{1}}{\mathrm{3}}} {dx}=\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{1}−{x}^{\mathrm{3}} \right)^{\frac{\mathrm{1}}{\mathrm{7}}} {dx} \\ $$

Question Number 76669    Answers: 0   Comments: 2

Question Number 76667    Answers: 2   Comments: 1

Question Number 76664    Answers: 0   Comments: 1

∫_0 ^(+∞) ((t ln(t))/((t^2 +1)^2 )) dt

$$\int_{\mathrm{0}} ^{+\infty} \frac{{t}\:{ln}\left({t}\right)}{\left({t}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{2}} }\:{dt} \\ $$

Question Number 76663    Answers: 2   Comments: 0

Question Number 76661    Answers: 1   Comments: 0

calculate ∫ ((x^2 −2)/((x^4 +5x^2 +4) arctan (((x^2 +2)/x)))) dx

$${calculate}\:\int\:\frac{{x}^{\mathrm{2}} −\mathrm{2}}{\left({x}^{\mathrm{4}} +\mathrm{5}{x}^{\mathrm{2}} +\mathrm{4}\right)\:{arc}\mathrm{tan}\:\left(\frac{{x}^{\mathrm{2}} +\mathrm{2}}{{x}}\right)}\:{dx}\: \\ $$

Question Number 76657    Answers: 3   Comments: 0

prove that sin^(−1) (tanhx)=tan^(−1) (sinhx)

$${prove}\:{that}\:{sin}^{−\mathrm{1}} \left({tanhx}\right)={tan}^{−\mathrm{1}} \left({sinhx}\right) \\ $$

Question Number 76651    Answers: 1   Comments: 0

if a and b two numbers do not have to be chosen randomly and with returns from the set {1,2,3,4,5}. what probability that (a/b) is an integer ?

$${if}\:{a}\:{and}\:{b}\:{two}\:{numbers}\:{do}\:{not}\:{have}\: \\ $$$${to}\:{be}\:{chosen}\:{randomly}\:{and}\:{with}\: \\ $$$${returns}\:{from}\:{the}\:{set}\:\left\{\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4},\mathrm{5}\right\}.\:{what}\: \\ $$$${probability}\:{that}\:\frac{{a}}{{b}}\:{is}\:{an}\:{integer}\:? \\ $$

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