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Question Number 63921 Answers: 0 Comments: 2

lim_(n→∞) (((n/(n+1)))^α +sin (1/n))^n where α∈Q is equal to

$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\left(\left(\frac{{n}}{{n}+\mathrm{1}}\right)^{\alpha} +\mathrm{sin}\:\frac{\mathrm{1}}{{n}}\right)^{{n}} \mathrm{where}\:\alpha\in\mathbb{Q} \\ $$$$\mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$

Question Number 63920 Answers: 0 Comments: 5

If α,β are root of quadratic equation ax^2 +bx+c then lim_(x→α) ((1−cos (ax^2 +bx+c))/((x−α)^2 ))=?

$$\mathrm{If}\:\alpha,\beta\:\mathrm{are}\:\mathrm{root}\:\mathrm{of}\:\mathrm{quadratic}\:\mathrm{equation} \\ $$$${ax}^{\mathrm{2}} +{bx}+{c}\:\mathrm{then} \\ $$$$\underset{{x}\rightarrow\alpha} {\mathrm{lim}}\frac{\mathrm{1}−\mathrm{cos}\:\left({ax}^{\mathrm{2}} +{bx}+{c}\right)}{\left({x}−\alpha\right)^{\mathrm{2}} }=? \\ $$

Question Number 63919 Answers: 0 Comments: 3

Π_(n=2) ^∞ (1−(1/n^2 ))=?

$$\underset{{n}=\mathrm{2}} {\overset{\infty} {\prod}}\left(\mathrm{1}−\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\right)=? \\ $$

Question Number 63918 Answers: 1 Comments: 0

lim_(x→π/2) (([1−tan x/2][1−sin x])/([1+tan x/2][π−2x]^3 ))=?

$$\underset{{x}\rightarrow\pi/\mathrm{2}} {\mathrm{lim}}\frac{\left[\mathrm{1}−\mathrm{tan}\:{x}/\mathrm{2}\right]\left[\mathrm{1}−\mathrm{sin}\:{x}\right]}{\left[\mathrm{1}+\mathrm{tan}\:{x}/\mathrm{2}\right]\left[\pi−\mathrm{2}{x}\right]^{\mathrm{3}} }=? \\ $$

Question Number 63917 Answers: 0 Comments: 2

lim_(x→∞) ((√(x+(√(x+(√x)))))−(√x))

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\left(\sqrt{{x}+\sqrt{{x}+\sqrt{{x}}}}−\sqrt{{x}}\right) \\ $$

Question Number 63915 Answers: 0 Comments: 0

If p is nearly equal to q and n > 1, such that (((n+1)p+(n−1)q)/((n−1)p+(n+1)q)) = ((p/q))^k , then the value of k is

$$\mathrm{If}\:{p}\:\mathrm{is}\:\mathrm{nearly}\:\mathrm{equal}\:\:\mathrm{to}\:{q}\:\:\mathrm{and}\:\:{n}\:>\:\mathrm{1},\:\mathrm{such} \\ $$$$\mathrm{that}\:\frac{\left({n}+\mathrm{1}\right){p}+\left({n}−\mathrm{1}\right){q}}{\left({n}−\mathrm{1}\right){p}+\left({n}+\mathrm{1}\right){q}}\:=\:\left(\frac{{p}}{{q}}\right)^{{k}} ,\:\mathrm{then}\:\mathrm{the} \\ $$$$\mathrm{value}\:\mathrm{of}\:{k}\:\mathrm{is} \\ $$

Question Number 63909 Answers: 1 Comments: 1

The integral part of (8 + 3(√7) )^(20) is even.

$$\mathrm{The}\:\mathrm{integral}\:\mathrm{part}\:\mathrm{of}\:\left(\mathrm{8}\:+\:\mathrm{3}\sqrt{\mathrm{7}}\:\right)^{\mathrm{20}} \:\mathrm{is}\:\mathrm{even}. \\ $$

Question Number 63908 Answers: 0 Comments: 1

If R=(5(√5) +11)^(2n+1) = [R]+ F, where [R] denotes the greatest integer in R, then RF= 2^(2n+1) .

$$\mathrm{If}\:{R}=\left(\mathrm{5}\sqrt{\mathrm{5}}\:+\mathrm{11}\right)^{\mathrm{2}{n}+\mathrm{1}} =\:\left[{R}\right]+\:{F},\:\mathrm{where} \\ $$$$\left[{R}\right]\:\mathrm{denotes}\:\mathrm{the}\:\mathrm{greatest}\:\mathrm{integer}\:\mathrm{in}\:{R},\: \\ $$$$\mathrm{then}\:{RF}=\:\mathrm{2}^{\mathrm{2}{n}+\mathrm{1}} . \\ $$

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

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...

Question Number 63857 Answers: 0 Comments: 2

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