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Question Number 63907    Answers: 1   Comments: 1

The coefficient of x^5 in (1+2x+3x^2 +...)^(−3/2) is 21.

$$\mathrm{The}\:\mathrm{coefficient}\:\mathrm{of}\:{x}^{\mathrm{5}} \:\mathrm{in}\:\left(\mathrm{1}+\mathrm{2}{x}+\mathrm{3}{x}^{\mathrm{2}} +...\right)^{−\mathrm{3}/\mathrm{2}} \\ $$$$\mathrm{is}\:\mathrm{21}. \\ $$

Question Number 63904    Answers: 0   Comments: 1

Question Number 63903    Answers: 0   Comments: 0

Question Number 63900    Answers: 0   Comments: 1

Question Number 63895    Answers: 0   Comments: 0

Question Number 63894    Answers: 0   Comments: 1

sove the (de) x^2 y^′ −(2x+3)y =sin(x^2 ) with y(1)=2 and y^′ (1)=1 .

$${sove}\:{the}\:\left({de}\right)\:{x}^{\mathrm{2}} {y}^{'} \:−\left(\mathrm{2}{x}+\mathrm{3}\right){y}\:={sin}\left({x}^{\mathrm{2}} \right)\:\:{with}\:{y}\left(\mathrm{1}\right)=\mathrm{2}\:{and} \\ $$$${y}^{'} \left(\mathrm{1}\right)=\mathrm{1}\:. \\ $$

Question Number 63893    Answers: 0   Comments: 1

1) simplify W_n (z)=(1+z)(1+z^2 )....(1+z^2^n ) (z from C) 2) simplify P_n (θ) =(1+e^(iθ) )(1+e^(2iθ) ).....(1+e^(i2^n θ) ) and sove P_n (θ)=0

$$\left.\mathrm{1}\right)\:{simplify}\:{W}_{{n}} \left({z}\right)=\left(\mathrm{1}+{z}\right)\left(\mathrm{1}+{z}^{\mathrm{2}} \right)....\left(\mathrm{1}+{z}^{\mathrm{2}^{{n}} } \right)\:\left({z}\:{from}\:{C}\right) \\ $$$$\left.\mathrm{2}\right)\:{simplify}\:{P}_{{n}} \left(\theta\right)\:=\left(\mathrm{1}+{e}^{{i}\theta} \right)\left(\mathrm{1}+{e}^{\mathrm{2}{i}\theta} \right).....\left(\mathrm{1}+{e}^{{i}\mathrm{2}^{{n}} \theta} \right)\:{and}\:{sove} \\ $$$${P}_{{n}} \left(\theta\right)=\mathrm{0} \\ $$

Question Number 63892    Answers: 0   Comments: 3

calculate A=∫_0 ^∞ (x^(2017) /(1+x^(2019) )) dx and B =∫_0 ^∞ (x^(2019) /(1+x^(2021) )) dx calculate the fraction (A/B)

$${calculate}\:{A}=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{x}^{\mathrm{2017}} }{\mathrm{1}+{x}^{\mathrm{2019}} }\:{dx}\:\:{and}\:{B}\:=\int_{\mathrm{0}} ^{\infty} \:\frac{{x}^{\mathrm{2019}} }{\mathrm{1}+{x}^{\mathrm{2021}} }\:{dx} \\ $$$${calculate}\:{the}\:{fraction}\:\frac{{A}}{{B}} \\ $$

Question Number 63891    Answers: 0   Comments: 0

A bus is travelling along a straight road at 100Km/hr and the bus conductor walks at 6Km/hr on the floor of the bus and in the same direction as the bus. Find the speed of the conductor relative to the road, and relative to the bus. If the bus conductor now works at the same rate but in the opposite direction as the bus, find his new speed relative to the road. Answers in textbook: 106Km/hr, 64Km/hr, 94Km/hr

$$\mathrm{A}\:\mathrm{bus}\:\mathrm{is}\:\mathrm{travelling}\:\mathrm{along}\:\mathrm{a}\:\mathrm{straight}\:\mathrm{road}\:\mathrm{at}\:\mathrm{100Km}/\mathrm{hr}\:\mathrm{and} \\ $$$$\mathrm{the}\:\mathrm{bus}\:\mathrm{conductor}\:\mathrm{walks}\:\mathrm{at}\:\mathrm{6Km}/\mathrm{hr}\:\mathrm{on}\:\mathrm{the}\:\mathrm{floor}\:\mathrm{of}\:\mathrm{the}\:\mathrm{bus} \\ $$$$\mathrm{and}\:\mathrm{in}\:\mathrm{the}\:\mathrm{same}\:\mathrm{direction}\:\mathrm{as}\:\mathrm{the}\:\mathrm{bus}.\: \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{speed}\:\mathrm{of}\:\mathrm{the}\:\mathrm{conductor}\:\mathrm{relative}\:\mathrm{to}\:\mathrm{the}\:\mathrm{road},\:\mathrm{and} \\ $$$$\mathrm{relative}\:\mathrm{to}\:\mathrm{the}\:\mathrm{bus}.\:\mathrm{If}\:\mathrm{the}\:\mathrm{bus}\:\mathrm{conductor}\:\mathrm{now}\:\mathrm{works}\:\mathrm{at}\:\mathrm{the}\:\mathrm{same} \\ $$$$\mathrm{rate}\:\mathrm{but}\:\mathrm{in}\:\mathrm{the}\:\mathrm{opposite}\:\mathrm{direction}\:\mathrm{as}\:\mathrm{the}\:\mathrm{bus},\:\mathrm{find}\:\mathrm{his}\:\mathrm{new}\:\mathrm{speed} \\ $$$$\mathrm{relative}\:\mathrm{to}\:\mathrm{the}\:\mathrm{road}. \\ $$$$\mathrm{Answers}\:\mathrm{in}\:\mathrm{textbook}:\:\:\:\:\mathrm{106Km}/\mathrm{hr},\:\:\:\:\mathrm{64Km}/\mathrm{hr},\:\:\:\:\:\mathrm{94Km}/\mathrm{hr} \\ $$

Question Number 63888    Answers: 2   Comments: 0

y = log_2 [log_3 (log_5 x)] y = ?

$${y}\:=\:{log}_{\mathrm{2}} \left[{log}_{\mathrm{3}} \left({log}_{\mathrm{5}} {x}\right)\right] \\ $$$${y}\:=\:? \\ $$

Question Number 63881    Answers: 0   Comments: 1

∫e^x /Lnxdx

$$\int{e}^{{x}} /{Lnxdx} \\ $$

Question Number 63883    Answers: 0   Comments: 1

∫ln(x)ln(1−x)ln(1−2x)dx

$$\int{ln}\left({x}\right){ln}\left(\mathrm{1}−{x}\right){ln}\left(\mathrm{1}−\mathrm{2}{x}\right){dx} \\ $$

Question Number 63865    Answers: 0   Comments: 2

If ∫ ((4 e^x + 6 e^(−x) )/(9 e^x − 4 e^(−x) ))dx=Ax+B log(9e^(2x) −4)+C, then

$$\mathrm{If}\:\int\:\:\frac{\mathrm{4}\:{e}^{{x}} +\:\mathrm{6}\:{e}^{−{x}} }{\mathrm{9}\:{e}^{{x}} −\:\mathrm{4}\:{e}^{−{x}} }{dx}={Ax}+{B}\:\mathrm{log}\left(\mathrm{9}{e}^{\mathrm{2}{x}} −\mathrm{4}\right)+{C}, \\ $$$$\mathrm{then} \\ $$

Question Number 63862    Answers: 0   Comments: 2

If the 3rd term in the expansion of [x+x^(log_(10) x) ]^5 is 10^6 , then x may be

$$\mathrm{If}\:\mathrm{the}\:\mathrm{3rd}\:\mathrm{term}\:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{of} \\ $$$$\left[{x}+{x}^{\mathrm{log}_{\mathrm{10}} {x}} \right]^{\mathrm{5}} \mathrm{is}\:\mathrm{10}^{\mathrm{6}} ,\:\mathrm{then}\:{x}\:\mathrm{may}\:\mathrm{be} \\ $$

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...

$$\mathrm{if}\:\mathrm{a}_{\mathrm{1}} ,\:\mathrm{a}_{\mathrm{2}} ,\:\mathrm{a}_{\mathrm{3}} ,\:\mathrm{a}_{\mathrm{4}} \:\mathrm{are}\:\mathrm{the}\:\mathrm{coefficient} \\ $$$$\mathrm{of}\:\mathrm{any}\:\mathrm{four}\:\mathrm{four}\:\mathrm{consecutive} \\ $$$$\mathrm{terms}\:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{of}\:\left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{n}} \\ $$$$\mathrm{then}\:\frac{\mathrm{a}_{\mathrm{1}} }{\mathrm{a}_{\mathrm{2}} +\mathrm{a}_{\mathrm{1}} }+\frac{\mathrm{a}_{\mathrm{3}} }{\mathrm{a}_{\mathrm{3}} +\mathrm{a}_{\mathrm{4}} }\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to}... \\ $$

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

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

Question Number 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

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

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