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Question Number 63861 Answers: 0 Comments: 2

The number of non−zero terms in the expansion of (1+3(√2) x)^9 +(1−3(√2) x)^9 is

$$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{non}−\mathrm{zero}\:\mathrm{terms}\:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{expansion}\:\mathrm{of}\:\left(\mathrm{1}+\mathrm{3}\sqrt{\mathrm{2}}\:{x}\right)^{\mathrm{9}} +\left(\mathrm{1}−\mathrm{3}\sqrt{\mathrm{2}}\:{x}\right)^{\mathrm{9}} \:\mathrm{is} \\ $$

Question Number 63860 Answers: 1 Comments: 0

If n is even positive integer, then the condition that the greatest term in the expansion of (1+x)^n may have the greatest coefficient also is

$$\mathrm{If}\:{n}\:\mathrm{is}\:\mathrm{even}\:\mathrm{positive}\:\mathrm{integer},\:\mathrm{then}\:\mathrm{the} \\ $$$$\mathrm{condition}\:\mathrm{that}\:\mathrm{the}\:\mathrm{greatest}\:\mathrm{term}\:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{expansion}\:\mathrm{of}\:\left(\mathrm{1}+{x}\right)^{{n}} \:\mathrm{may}\:\mathrm{have}\:\mathrm{the} \\ $$$$\mathrm{greatest}\:\mathrm{coefficient}\:\mathrm{also}\:\mathrm{is} \\ $$

Question Number 63858 Answers: 1 Comments: 0

if a_1 , a_2 , a_3 , a_4 are the coefficient of any four four consecutive terms in the expansion of (1+x)^n then (a_1 /(a_2 +a_1 ))+(a_3 /(a_3 +a_4 )) is equal to...

Question Number 63857 Answers: 0 Comments: 2

Question Number 63852 Answers: 0 Comments: 0

prove that ∫_0 ^1 arctan(x) cot(((πx)/2)) dx = ((3 ln^2 (2))/(2π))+((lnπ ln2)/π)+∫_0 ^∞ ((ln(1+x^2 ))/(e^(2πx) +1)) dx

$${prove}\:{that} \\ $$$$ \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} {arctan}\left({x}\right)\:{cot}\left(\frac{\pi{x}}{\mathrm{2}}\right)\:{dx}\:=\:\frac{\mathrm{3}\:{ln}^{\mathrm{2}} \left(\mathrm{2}\right)}{\mathrm{2}\pi}+\frac{{ln}\pi\:{ln}\mathrm{2}}{\pi}+\int_{\mathrm{0}} ^{\infty} \frac{{ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}{{e}^{\mathrm{2}\pi{x}} +\mathrm{1}}\:{dx} \\ $$

Question Number 63845 Answers: 0 Comments: 1

Σ_(r=0) ^n ^n C_r ((1+r log_e 10)/((1+ log_e 10^n )^r ))=...

$$\underset{{r}=\mathrm{0}} {\overset{{n}} {\sum}}\:^{{n}} {C}_{{r}} \:\frac{\mathrm{1}+{r}\:\mathrm{log}_{{e}} \:\mathrm{10}}{\left(\mathrm{1}+\:\mathrm{log}_{{e}} \:\mathrm{10}^{{n}} \right)^{{r}} }=... \\ $$

Question Number 63844 Answers: 3 Comments: 3

∫(1+4x+x^2 )^m dx

$$\int\left(\mathrm{1}+\mathrm{4}{x}+{x}^{\mathrm{2}} \right)^{{m}} {dx} \\ $$

Question Number 63836 Answers: 0 Comments: 1

If n ∈ N, then the sum of the coefficients in the expansion of the binomial (5x−4y)^n is

$$\mathrm{If}\:{n}\:\in\:{N},\:\mathrm{then}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{coefficients} \\ $$$$\mathrm{in}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{of}\:\:\mathrm{the}\:\mathrm{binomial} \\ $$$$\left(\mathrm{5}{x}−\mathrm{4}{y}\right)^{{n}} \:\mathrm{is} \\ $$

Question Number 63835 Answers: 0 Comments: 1

If n ∈ N, then the sum of the coefficients in the expansion of the binomial (5x−4y)^n is

Question Number 63834 Answers: 0 Comments: 2

The 14th term from the end in the expansion of ((√x) − (√y))^(17) is

$$\mathrm{The}\:\mathrm{14th}\:\mathrm{term}\:\mathrm{from}\:\mathrm{the}\:\mathrm{end}\:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion} \\ $$$$\mathrm{of}\:\left(\sqrt{{x}}\:−\:\sqrt{{y}}\right)^{\mathrm{17}} \:\mathrm{is} \\ $$

Question Number 63833 Answers: 0 Comments: 0

If the binomial coefficients of 2nd, 3rd and 4th terms in the expansion of [(√2^(log_(10) (10−3^x )) ) + (2^((x−2) log_(10) 3) )^(1/5) ]^m are in AP and the 6th term is 21, then the value(s) of x is(are)

$$\mathrm{If}\:\mathrm{the}\:\mathrm{binomial}\:\mathrm{coefficients}\:\mathrm{of}\:\mathrm{2nd},\:\mathrm{3rd} \\ $$$$\mathrm{and}\:\mathrm{4th}\:\mathrm{terms}\:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{of} \\ $$$$\left[\sqrt{\mathrm{2}^{\mathrm{log}_{\mathrm{10}} \left(\mathrm{10}−\mathrm{3}^{{x}} \right)} }\:+\:\sqrt[{\mathrm{5}}]{\mathrm{2}^{\left({x}−\mathrm{2}\right)\:\mathrm{log}_{\mathrm{10}} \mathrm{3}} }\right]^{{m}} \:\mathrm{are}\:\mathrm{in} \\ $$$$\mathrm{AP}\:\mathrm{and}\:\mathrm{the}\:\mathrm{6th}\:\mathrm{term}\:\mathrm{is}\:\mathrm{21},\:\mathrm{then}\:\mathrm{the}\:\mathrm{value}\left(\mathrm{s}\right) \\ $$$$\mathrm{of}\:{x}\:\:\mathrm{is}\left(\mathrm{are}\right) \\ $$

Question Number 63832 Answers: 0 Comments: 5

The largest coefficient in the expansion of (1+x)^(24) is

$$\mathrm{The}\:\mathrm{largest}\:\mathrm{coefficient}\:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion} \\ $$$$\mathrm{of}\:\left(\mathrm{1}+{x}\right)^{\mathrm{24}} \:\mathrm{is} \\ $$

Question Number 63831 Answers: 1 Comments: 0

If C_r be the coefficient of x^r in (1+x)^n , then the value of Σ_(r=0) ^n (r+1)^2 C_r is

$$\mathrm{If}\:{C}_{{r}} \:\mathrm{be}\:\mathrm{the}\:\mathrm{coefficient}\:\mathrm{of}\:{x}^{{r}} \:\mathrm{in}\:\left(\mathrm{1}+{x}\right)^{{n}} , \\ $$$$\mathrm{then}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\underset{{r}=\mathrm{0}} {\overset{{n}} {\sum}}\left({r}+\mathrm{1}\right)^{\mathrm{2}} \:{C}_{{r}} \:\mathrm{is} \\ $$

Question Number 63824 Answers: 0 Comments: 1

solve y^′ (√(2x−1)) +y(x^2 +3) =xsin(2x)

$${solve}\:{y}^{'} \sqrt{\mathrm{2}{x}−\mathrm{1}}\:+{y}\left({x}^{\mathrm{2}} +\mathrm{3}\right)\:={xsin}\left(\mathrm{2}{x}\right) \\ $$

Question Number 63822 Answers: 0 Comments: 1

find ∫ (x^2 +1)(√((x+1)/(x−2)))dx

$${find}\:\int\:\:\left({x}^{\mathrm{2}} +\mathrm{1}\right)\sqrt{\frac{{x}+\mathrm{1}}{{x}−\mathrm{2}}}{dx} \\ $$

Question Number 63803 Answers: 1 Comments: 4

Question Number 63784 Answers: 0 Comments: 6

question 63639 again prove: ∀z∈C: ∣z+1∣+∣z^2 +z+1∣+∣z^3 +1∣≥1

$$\mathrm{question}\:\mathrm{63639}\:\mathrm{again} \\ $$$$\mathrm{prove}: \\ $$$$\forall{z}\in\mathbb{C}:\:\mid{z}+\mathrm{1}\mid+\mid{z}^{\mathrm{2}} +{z}+\mathrm{1}\mid+\mid{z}^{\mathrm{3}} +\mathrm{1}\mid\geqslant\mathrm{1} \\ $$

Question Number 63769 Answers: 1 Comments: 1

Question Number 63782 Answers: 1 Comments: 4

let f(a) =∫_(−∞) ^(+∞) (dx/((a^2 +x^2 )^3 )) with a>0 1) calculate f(a) 2)calculste also g(a) =∫_(−∞) ^(+∞) (dx/((a^2 +x^2 )^4 )) 3) find the values of integrals ∫_0 ^∞ (dx/((x^2 +1)^3 )) ∫_0 ^∞ (dx/((x^2 +2)^4 ))

$${let}\:{f}\left({a}\right)\:=\int_{−\infty} ^{+\infty} \:\:\:\frac{{dx}}{\left({a}^{\mathrm{2}} +{x}^{\mathrm{2}} \right)^{\mathrm{3}} }\:\:\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}\left({a}\right) \\ $$$$\left.\mathrm{2}\right){calculste}\:{also}\:{g}\left({a}\right)\:=\int_{−\infty} ^{+\infty} \:\:\:\frac{{dx}}{\left({a}^{\mathrm{2}} \:+{x}^{\mathrm{2}} \right)^{\mathrm{4}} } \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{values}\:{of}\:{integrals}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{\left({x}^{\mathrm{2}} \:+\mathrm{1}\right)^{\mathrm{3}} } \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\frac{{dx}}{\left({x}^{\mathrm{2}} +\mathrm{2}\right)^{\mathrm{4}} } \\ $$

Question Number 63812 Answers: 0 Comments: 2

A can terminate a work 9 hour earlier than B. A and B terminate that work after 20 hour together. A can terminate that work after .... hour.

$$ \\ $$$${A}\:\:{can}\:{terminate}\:{a}\:{work}\:\:\mathrm{9}\:{hour}\:{earlier} \\ $$$${than}\:{B}. \\ $$$${A}\:\:{and}\:\:{B}\:\:{terminate}\:{that}\:{work}\:\:{after}\:\mathrm{20}\:{hour}\:{together}. \\ $$$${A}\:{can}\:{terminate}\:{that}\:{work}\:{after}\:....\:{hour}. \\ $$$$ \\ $$$$ \\ $$

Question Number 63823 Answers: 0 Comments: 0

calculate ∫_(−∞) ^(+∞) ((3x^2 −1)/(x^4 −2x^2 +3))dx

$${calculate}\:\:\int_{−\infty} ^{+\infty} \:\:\frac{\mathrm{3}{x}^{\mathrm{2}} −\mathrm{1}}{{x}^{\mathrm{4}} −\mathrm{2}{x}^{\mathrm{2}} \:+\mathrm{3}}{dx} \\ $$

Question Number 63763 Answers: 0 Comments: 1