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Question Number 90581 Answers: 1 Comments: 0

given that α and β are roots of the equation aχ^2 +bχ+c=0. show that λμb^2 =ac(λ+μ)^(2 ) where (α/β)=(λ/μ)

$${given}\:{that}\:\alpha\:{and}\:\beta\:{are}\:{roots}\:{of}\:\:{the}\:{equation}\: \\ $$$${a}\chi^{\mathrm{2}} +{b}\chi+{c}=\mathrm{0}.\:{show}\:{that}\:\lambda\mu{b}^{\mathrm{2}} ={ac}\left(\lambda+\mu\right)^{\mathrm{2}\:} \\ $$$${where}\:\frac{\alpha}{\beta}=\frac{\lambda}{\mu} \\ $$

Question Number 90550 Answers: 0 Comments: 3

x^4 + (1/x^4 ) = 527 (x−1)(x−2)(x−3)(x−4) ?

$${x}^{\mathrm{4}} \:+\:\frac{\mathrm{1}}{{x}^{\mathrm{4}} }\:=\:\mathrm{527}\: \\ $$$$\left({x}−\mathrm{1}\right)\left({x}−\mathrm{2}\right)\left({x}−\mathrm{3}\right)\left({x}−\mathrm{4}\right)\:?\: \\ $$

Question Number 90544 Answers: 1 Comments: 0

∫ (dx/(√(2−cos x)))

$$\int\:\frac{{dx}}{\sqrt{\mathrm{2}−\mathrm{cos}\:{x}}} \\ $$

Question Number 90535 Answers: 1 Comments: 0

In a triangle ABC, a(b cos C−c cos B)=

$$\mathrm{In}\:\mathrm{a}\:\mathrm{triangle}\:{ABC},\:{a}\left({b}\:\mathrm{cos}\:{C}−{c}\:\mathrm{cos}\:{B}\right)= \\ $$

Question Number 90531 Answers: 0 Comments: 2

Please help me to find the value of ′n′ C_8 ^(n+3) =C_(20) ^(n+3)

$${Please}\:{help}\:{me}\:{to}\:{find}\:{the}\:{value} \\ $$$${of}\:\:\:'{n}' \\ $$$${C}_{\mathrm{8}} ^{{n}+\mathrm{3}} ={C}_{\mathrm{20}} ^{{n}+\mathrm{3}} \\ $$

Question Number 90520 Answers: 2 Comments: 1

Question Number 90514 Answers: 2 Comments: 6

Question Number 90512 Answers: 1 Comments: 2

∫_0 ^1 (1/x)dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}}{{x}}{dx} \\ $$

Question Number 90509 Answers: 2 Comments: 3

Question Number 90508 Answers: 0 Comments: 0

Σ_(n=1) ^∞ ((1/(n^m (1+n)^m ))) what is the general for this sum

$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(\frac{\mathrm{1}}{{n}^{{m}} \left(\mathrm{1}+{n}\right)^{{m}} }\right) \\ $$$${what}\:{is}\:{the}\:{general}\:{for}\:{this}\:{sum} \\ $$

Question Number 90507 Answers: 0 Comments: 2

if 2^(sin x) +2^(cos x) =2^(1+(1/(√2))) then find the value of x=?

$$ \\ $$$$\:\mathrm{if}\:\:\mathrm{2}^{\mathrm{sin}\:\mathrm{x}} +\mathrm{2}^{\mathrm{cos}\:\mathrm{x}} =\mathrm{2}^{\mathrm{1}+\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}} \:\:\mathrm{then}\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{x}=? \\ $$

Question Number 90503 Answers: 0 Comments: 2

Question Number 90489 Answers: 0 Comments: 0

Find f(x) if it equals Σ_(n=0) ^∞ (n^x /(n!))

$${Find}\:{f}\left({x}\right)\:{if}\:{it}\:{equals}\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{{n}^{{x}} }{{n}!} \\ $$

Question Number 90485 Answers: 0 Comments: 1

if x_0 =x_1 =1 and x_(n+1) =1996x_n +1997x_(n−1) for n≥2. Find the remainder of the division of x_(1996) by 3

$${if}\:{x}_{\mathrm{0}} ={x}_{\mathrm{1}} =\mathrm{1}\:{and}\:{x}_{{n}+\mathrm{1}} =\mathrm{1996}{x}_{{n}} +\mathrm{1997}{x}_{{n}−\mathrm{1}} \\ $$$${for}\:{n}\geqslant\mathrm{2}.\:{Find}\:{the}\:{remainder}\:{of} \\ $$$${the}\:{division}\:{of}\:{x}_{\mathrm{1996}} \:{by}\:\mathrm{3} \\ $$

Question Number 90483 Answers: 0 Comments: 0

prove that/ ((sin^3 a)/(sin b))+((cos^3 a)/(cos b))≥sec(a−b) for all a,b∈ (0,(π/2))

$${prove}\:{that}/\:\frac{{sin}^{\mathrm{3}} {a}}{{sin}\:{b}}+\frac{{cos}^{\mathrm{3}} {a}}{{cos}\:{b}}\geqslant{sec}\left({a}−{b}\right) \\ $$$${for}\:{all}\:{a},{b}\in\:\left(\mathrm{0},\frac{\pi}{\mathrm{2}}\right) \\ $$

Question Number 90480 Answers: 0 Comments: 0

∫e^x (((1+cos(x))(1−sin(x)))/((e^x cos(x)+1)^2 ))dx

$$\int{e}^{{x}} \frac{\left(\mathrm{1}+{cos}\left({x}\right)\right)\left(\mathrm{1}−{sin}\left({x}\right)\right)}{\left({e}^{{x}} \:{cos}\left({x}\right)+\mathrm{1}\right)^{\mathrm{2}} }{dx} \\ $$

Question Number 90475 Answers: 0 Comments: 3

Question Number 90473 Answers: 1 Comments: 2

Question Number 90472 Answers: 0 Comments: 3

Σ_(k=0) ^m ((2k+3)/2^(m−k) )

$$\underset{\mathrm{k}=\mathrm{0}} {\overset{\mathrm{m}} {\sum}}\frac{\mathrm{2k}+\mathrm{3}}{\mathrm{2}^{\mathrm{m}−\mathrm{k}} } \\ $$

Question Number 90471 Answers: 0 Comments: 0

is there a simple way to write Σ_(n=1) ^∞ ((2/(n(n+1))))^m for any m≥0

$${is}\:{there}\:{a}\:{simple}\:{way} \\ $$$${to}\:{write} \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(\frac{\mathrm{2}}{{n}\left({n}+\mathrm{1}\right)}\right)^{{m}} \:{for}\:{any}\:{m}\geqslant\mathrm{0} \\ $$

Question Number 90470 Answers: 0 Comments: 0

For any positive integer n, τ(n) is the number of its factors Prove, Σ_(i=1) ^n τ(i)=Σ_(i=1) ^n ⌊n/i⌋

$${For}\:{any}\:{positive}\:{integer}\:{n},\:\tau\left({n}\right)\:{is}\:{the}\:{number}\:{of}\:{its}\:{factors}\: \\ $$$${Prove}, \\ $$$$\sum_{{i}=\mathrm{1}} ^{{n}} \tau\left({i}\right)=\sum_{{i}=\mathrm{1}} ^{{n}} \lfloor{n}/{i}\rfloor \\ $$

Question Number 90468 Answers: 0 Comments: 1

Question Number 90499 Answers: 1 Comments: 6

Question Number 90463 Answers: 0 Comments: 1

Question Number 90458 Answers: 0 Comments: 1

∫ (√(x^2 +((13)/x))) dx ?

$$\int\:\sqrt{\mathrm{x}^{\mathrm{2}} +\frac{\mathrm{13}}{\mathrm{x}}}\:\mathrm{dx}\:? \\ $$

Question Number 90457 Answers: 0 Comments: 5

the range of y=(√x) is[0,+∞) if x≥0 or just [0,+∞) ?

$${the}\:{range}\:{of}\:{y}=\sqrt{{x}}\:\:{is}\left[\mathrm{0},+\infty\right)\:{if}\:{x}\geqslant\mathrm{0} \\ $$$${or}\:{just}\:\left[\mathrm{0},+\infty\right)\:? \\ $$

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