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AllQuestion and Answers: Page 1265

Question Number 80881 Answers: 1 Comments: 0

v=−(((2b+3cp)p)/(3+bp^2 )) ((3v^2 +2bpv+3cp+b)/(1+bp^2 +cp^3 )) > 0 b,c ∈ R , b<0 Any non-zero real value of p in terms of b,c obeying above condition?

$${v}=−\frac{\left(\mathrm{2}{b}+\mathrm{3}{cp}\right){p}}{\mathrm{3}+{bp}^{\mathrm{2}} } \\ $$$$\:\frac{\mathrm{3}{v}^{\mathrm{2}} +\mathrm{2}{bpv}+\mathrm{3}{cp}+{b}}{\mathrm{1}+{bp}^{\mathrm{2}} +{cp}^{\mathrm{3}} }\:>\:\mathrm{0}\:\: \\ $$$${b},{c}\:\in\:\mathbb{R}\:,\:{b}<\mathrm{0} \\ $$$${Any}\:{non}-{zero}\:{real}\:{value}\:{of}\:{p} \\ $$$${in}\:{terms}\:{of}\:{b},{c}\:\:{obeying}\:{above} \\ $$$${condition}? \\ $$

Question Number 80879 Answers: 0 Comments: 2

Question Number 80869 Answers: 1 Comments: 2

find the coordinate of the line prese nted with line 2x−3y+7=0.which is equadistance from points (−4,8) and (7,1)

$${find}\:{the}\:{coordinate}\:{of}\:{the}\:{line}\:{prese} \\ $$$${nted}\:{with}\:{line}\:\mathrm{2}{x}−\mathrm{3}{y}+\mathrm{7}=\mathrm{0}.{which}\:{is} \\ $$$${equadistance}\:{from}\:{points}\:\left(−\mathrm{4},\mathrm{8}\right)\:{and}\: \\ $$$$\left(\mathrm{7},\mathrm{1}\right) \\ $$

Question Number 80863 Answers: 1 Comments: 0

Let W the lambert function defined as W(xe^x )=x x≥0 Prove that ∫_0 ^1 (( W(−ulnu))/u)du=((ζ(2))/2)

$$\:{Let}\:{W}\:{the}\:{lambert}\:{function}\:{defined}\:{as}\:{W}\left({xe}^{{x}} \right)={x}\:\:\:{x}\geqslant\mathrm{0} \\ $$$${Prove}\:{that}\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\:{W}\left(−{ulnu}\right)}{{u}}{du}=\frac{\zeta\left(\mathrm{2}\right)}{\mathrm{2}}\:\: \\ $$

Question Number 80861 Answers: 1 Comments: 5

for x,y ∈R given f(x)+f(2x+y)+5xy= f(3x−y)+x^2 +1 find f(10)

$${for}\:{x},{y}\:\in\mathbb{R} \\ $$$${given}\:{f}\left({x}\right)+{f}\left(\mathrm{2}{x}+{y}\right)+\mathrm{5}{xy}= \\ $$$${f}\left(\mathrm{3}{x}−{y}\right)+{x}^{\mathrm{2}} +\mathrm{1} \\ $$$${find}\:{f}\left(\mathrm{10}\right) \\ $$

Question Number 80846 Answers: 1 Comments: 4

Question Number 80844 Answers: 1 Comments: 2

If (log_(10) (ax))(log_(10) (bx))=−1 find x in term a and b

$${If}\:\left(\mathrm{log}_{\mathrm{10}} \left({ax}\right)\right)\left(\mathrm{log}_{\mathrm{10}} \left({bx}\right)\right)=−\mathrm{1} \\ $$$${find}\:{x}\:{in}\:{term}\:{a}\:{and}\:{b}\: \\ $$

Question Number 80830 Answers: 1 Comments: 0

given A a square matrix non singular A≠ I. find A such that A^3 = I

$${given}\:{A}\:{a}\:{square}\:{matrix}\:{non} \\ $$$${singular}\:{A}\neq\:{I}. \\ $$$${find}\:{A}\:{such}\:{that}\:{A}^{\mathrm{3}} \:=\:{I} \\ $$

Question Number 80828 Answers: 0 Comments: 2

given lim_(x→−∞) [(√(2x+p)) −(√(2x+1))]×(√(2x−p ))= (1/4)p find p .

$${given}\: \\ $$$$\underset{{x}\rightarrow−\infty} {\mathrm{lim}}\:\left[\sqrt{\mathrm{2}{x}+{p}}\:−\sqrt{\mathrm{2}{x}+\mathrm{1}}\right]×\sqrt{\mathrm{2}{x}−{p}\:}=\:\frac{\mathrm{1}}{\mathrm{4}}{p} \\ $$$${find}\:{p}\:. \\ $$

Question Number 80823 Answers: 1 Comments: 0

show that lim_(x→∞) H_n =2F_1 (1,1;2,1) ln(4)−2ln(3)=2F_1 (1,1;2;((−1)/2))

$${show}\:{that}\: \\ $$$$\underset{{x}\rightarrow\infty} {{lim}}\:{H}_{{n}} =\mathrm{2}{F}_{\mathrm{1}} \left(\mathrm{1},\mathrm{1};\mathrm{2},\mathrm{1}\right) \\ $$$$ \\ $$$${ln}\left(\mathrm{4}\right)−\mathrm{2}{ln}\left(\mathrm{3}\right)=\mathrm{2}{F}_{\mathrm{1}} \left(\mathrm{1},\mathrm{1};\mathrm{2};\frac{−\mathrm{1}}{\mathrm{2}}\right) \\ $$

Question Number 80816 Answers: 1 Comments: 2

let 0<a<b prove that ln(1+(a/b))ln(1+(b/a))< (ln2)^2

$${let}\:\:\mathrm{0}<{a}<{b}\:\:{prove}\:{that} \\ $$$$\:{ln}\left(\mathrm{1}+\frac{{a}}{{b}}\right){ln}\left(\mathrm{1}+\frac{{b}}{{a}}\right)<\:\left({ln}\mathrm{2}\right)^{\mathrm{2}} \:\: \\ $$

Question Number 80815 Answers: 0 Comments: 0

find nsture of the serie Σ_(n=0) ^∞ (1+sin(n))^(1/n)

$${find}\:{nsture}\:{of}\:{the}\:{serie}\:\sum_{{n}=\mathrm{0}} ^{\infty} \left(\mathrm{1}+{sin}\left({n}\right)\right)^{\frac{\mathrm{1}}{{n}}} \\ $$

Question Number 80814 Answers: 0 Comments: 2

find nature of the serie Σ_(n=1) ^∞ (1−cos((π/n)))

$${find}\:{nature}\:{of}\:{the}\:{serie}\:\sum_{{n}=\mathrm{1}} ^{\infty} \left(\mathrm{1}−{cos}\left(\frac{\pi}{{n}}\right)\right) \\ $$

Question Number 80813 Answers: 1 Comments: 0

calculate Σ_(n=2) ^∞ ((ξ(n)−1)/n) with ξ(x)=Σ_(n=1) ^∞ (1/n^x ) (x>1)

$${calculate}\:\sum_{{n}=\mathrm{2}} ^{\infty} \:\frac{\xi\left({n}\right)−\mathrm{1}}{{n}}\:\:\:{with}\:\xi\left({x}\right)=\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{{n}^{{x}} }\:\:\:\left({x}>\mathrm{1}\right) \\ $$

Question Number 80809 Answers: 1 Comments: 5

Question Number 80804 Answers: 2 Comments: 2

Find [(√1)]+[(√2)]+[(√3)]+...+[(√(100))]=? with [x]=greatest integer function can we find a general formula for [(√1)]+[(√2)]+[(√3)]+...+[(√n)] in terms of n?

$${Find} \\ $$$$\left[\sqrt{\mathrm{1}}\right]+\left[\sqrt{\mathrm{2}}\right]+\left[\sqrt{\mathrm{3}}\right]+...+\left[\sqrt{\mathrm{100}}\right]=? \\ $$$${with}\:\left[{x}\right]={greatest}\:{integer}\:{function} \\ $$$$ \\ $$$${can}\:{we}\:{find}\:{a}\:{general}\:{formula}\:{for}\: \\ $$$$\left[\sqrt{\mathrm{1}}\right]+\left[\sqrt{\mathrm{2}}\right]+\left[\sqrt{\mathrm{3}}\right]+...+\left[\sqrt{{n}}\right] \\ $$$${in}\:{terms}\:{of}\:{n}? \\ $$

Question Number 80798 Answers: 0 Comments: 1

Question Number 80795 Answers: 0 Comments: 3

Question Number 80792 Answers: 1 Comments: 0

Π_(n=1) ^∞ [((2n)/(2n−1)).((2n)/(2n+1))] =?

$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\prod}}\left[\frac{\mathrm{2}{n}}{\mathrm{2}{n}−\mathrm{1}}.\frac{\mathrm{2}{n}}{\mathrm{2}{n}+\mathrm{1}}\right]\:=? \\ $$

Question Number 80788 Answers: 0 Comments: 0

Question Number 80786 Answers: 1 Comments: 5

Question Number 80780 Answers: 1 Comments: 1

2∙m^x + 3∙n^y = 18 min{ m^x ∙ n^y } = ?

$$\mathrm{2}\centerdot{m}^{{x}} \:+\:\mathrm{3}\centerdot{n}^{{y}} \:\:=\:\:\mathrm{18} \\ $$$${min}\left\{\:{m}^{{x}} \:\centerdot\:{n}^{{y}} \:\right\}\:=\:? \\ $$

Question Number 80777 Answers: 1 Comments: 0

If ^(n+2) C_8 :^(n−2) P_4 = 57 : 16, then the value of n is ......

$$\mathrm{If}\:\:^{{n}+\mathrm{2}} {C}_{\mathrm{8}} \::\:^{{n}−\mathrm{2}} {P}_{\mathrm{4}} =\:\mathrm{57}\::\:\mathrm{16},\:\mathrm{then}\:\mathrm{the}\:\mathrm{value} \\ $$$$\mathrm{of}\:{n}\:\mathrm{is}\:...... \\ $$

Question Number 80775 Answers: 0 Comments: 0

There are n straight lines in a plane, no two of which are parallel, and no three pass through the same point. Their points of intersection are joined. Then the number of fresh lines thus obtained is

$$\mathrm{There}\:\mathrm{are}\:{n}\:\mathrm{straight}\:\mathrm{lines}\:\mathrm{in}\:\mathrm{a}\:\mathrm{plane}, \\ $$$$\mathrm{no}\:\mathrm{two}\:\mathrm{of}\:\mathrm{which}\:\mathrm{are}\:\mathrm{parallel},\:\mathrm{and}\:\mathrm{no} \\ $$$$\mathrm{three}\:\mathrm{pass}\:\mathrm{through}\:\mathrm{the}\:\mathrm{same}\:\mathrm{point}. \\ $$$$\mathrm{Their}\:\mathrm{points}\:\mathrm{of}\:\mathrm{intersection}\:\mathrm{are}\:\mathrm{joined}. \\ $$$$\mathrm{Then}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{fresh}\:\mathrm{lines}\:\mathrm{thus}\: \\ $$$$\mathrm{obtained}\:\mathrm{is} \\ $$

Question Number 80770 Answers: 1 Comments: 1

∫x^2 +3x dx=..

$$\int\mathrm{x}^{\mathrm{2}} +\mathrm{3x}\:\mathrm{dx}=.. \\ $$

Question Number 80764 Answers: 1 Comments: 0

show that ∫_0 ^∞ x arctanh(e^(−αx) )dx=((7ζ(3))/(8α^2 ))

$${show}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\infty} {x}\:{arctanh}\left({e}^{−\alpha{x}} \right){dx}=\frac{\mathrm{7}\zeta\left(\mathrm{3}\right)}{\mathrm{8}\alpha^{\mathrm{2}} } \\ $$

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