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Question Number 55815 Answers: 1 Comments: 0

prove that (1+x)^n = 1+nx +((n(n−1))/(2!))x^2 +((n(n−1)(n−2))/(3!))x^3 +...n(n−n) using a suitable expansion method hence determine the expansion of (2.001)^(89)

$${prove}\:{that} \\ $$$$\left(\mathrm{1}+{x}\right)^{{n}} =\:\mathrm{1}+{nx}\:+\frac{{n}\left({n}−\mathrm{1}\right)}{\mathrm{2}!}{x}^{\mathrm{2}} +\frac{{n}\left({n}−\mathrm{1}\right)\left({n}−\mathrm{2}\right)}{\mathrm{3}!}{x}^{\mathrm{3}} +...{n}\left({n}−{n}\right) \\ $$$${using}\:{a}\:{suitable}\:{expansion}\:{method} \\ $$$${hence}\:{determine}\:{the}\:{expansion}\:{of} \\ $$$$\left(\mathrm{2}.\mathrm{001}\right)^{\mathrm{89}} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Question Number 55813 Answers: 0 Comments: 0

((0.8)/x)=((96)/(60))⇒x=0.5⇒0.8−0.5=0.3⇒0.4−0.3=0.1

$$\frac{\mathrm{0}.\mathrm{8}}{{x}}=\frac{\mathrm{96}}{\mathrm{60}}\Rightarrow{x}=\mathrm{0}.\mathrm{5}\Rightarrow\mathrm{0}.\mathrm{8}−\mathrm{0}.\mathrm{5}=\mathrm{0}.\mathrm{3}\Rightarrow\mathrm{0}.\mathrm{4}−\mathrm{0}.\mathrm{3}=\mathrm{0}.\mathrm{1} \\ $$

Question Number 55806 Answers: 1 Comments: 0

Solve for x: x^(log_3 2) = (√x) + 1

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{x}:\:\:\:\mathrm{x}^{\mathrm{log}_{\mathrm{3}} \mathrm{2}} \:\:=\:\:\sqrt{\mathrm{x}}\:\:+\:\:\mathrm{1} \\ $$

Question Number 55805 Answers: 0 Comments: 1

Question Number 55788 Answers: 1 Comments: 3

The sum of the last eight coefficients in the expansion of (1+x)^(16) is 2^(15) .

$$\mathrm{The}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{last}\:\mathrm{eight}\:\mathrm{coefficients}\:\mathrm{in} \\ $$$$\mathrm{the}\:\mathrm{expansion}\:\mathrm{of}\:\left(\mathrm{1}+{x}\right)^{\mathrm{16}} \:\mathrm{is}\:\mathrm{2}^{\mathrm{15}} \:. \\ $$

Question Number 55787 Answers: 2 Comments: 0

The remainder when 5^(99) is divided by 13 is

$$\mathrm{The}\:\mathrm{remainder}\:\mathrm{when}\:\mathrm{5}^{\mathrm{99}} \mathrm{is}\:\mathrm{divided}\:\mathrm{by}\:\mathrm{13}\:\mathrm{is} \\ $$

Question Number 55786 Answers: 1 Comments: 0

If the third term in the expansion of ((1/x) + x^(log_(10) x) )^5 is 1000, then the value of x is

$$\mathrm{If}\:\mathrm{the}\:\mathrm{third}\:\mathrm{term}\:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{of}\: \\ $$$$\left(\frac{\mathrm{1}}{{x}}\:+\:{x}^{\mathrm{log}_{\mathrm{10}} {x}} \right)^{\mathrm{5}} \:\mathrm{is}\:\mathrm{1000},\:\mathrm{then}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$${x}\:\mathrm{is} \\ $$

Question Number 55785 Answers: 2 Comments: 1

The coefficient of x^4 in the expansion of (1+x+x^2 +x^3 )^(11) is

$$\mathrm{The}\:\mathrm{coefficient}\:\mathrm{of}\:{x}^{\mathrm{4}} \:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{of} \\ $$$$\left(\mathrm{1}+{x}+{x}^{\mathrm{2}} +{x}^{\mathrm{3}} \right)^{\mathrm{11}} \:\mathrm{is} \\ $$

Question Number 55784 Answers: 1 Comments: 0

If n an odd natural number, then Σ_(r=0) ^n (((−1)^r )/(^n C_r )) equals

$$\mathrm{If}\:{n}\:\mathrm{an}\:\mathrm{odd}\:\mathrm{natural}\:\mathrm{number},\:\mathrm{then} \\ $$$$\underset{{r}=\mathrm{0}} {\overset{{n}} {\sum}}\:\frac{\left(−\mathrm{1}\right)^{{r}} }{\:^{{n}} {C}_{{r}} \:}\:\mathrm{equals} \\ $$

Question Number 55780 Answers: 1 Comments: 1

∫ (1/(√(x^2 +2x+1))) dx= A log ∣x+1∣+C for x>−1, then A=_____.

$$\int\:\frac{\mathrm{1}}{\sqrt{{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{1}}}\:{dx}=\:{A}\:\mathrm{log}\:\mid{x}+\mathrm{1}\mid+{C}\:\mathrm{for} \\ $$$${x}>−\mathrm{1},\:\mathrm{then}\:{A}=\_\_\_\_\_. \\ $$

Question Number 55779 Answers: 1 Comments: 2

If ∫ x log (1+(1/x))dx = f(x) ∙ log (x+1)+g(x) ∙ x^2 +Ax+C, then

$$\mathrm{If}\:\int\:{x}\:\mathrm{log}\:\left(\mathrm{1}+\frac{\mathrm{1}}{{x}}\right){dx}\: \\ $$$$\:\:\:\:\:\:=\:{f}\left({x}\right)\:\centerdot\:\mathrm{log}\:\left({x}+\mathrm{1}\right)+{g}\left({x}\right)\:\centerdot\:{x}^{\mathrm{2}} +{Ax}+{C}, \\ $$$$\mathrm{then} \\ $$

Question Number 55778 Answers: 1 Comments: 0

∫ ((2x−3)/((x^2 +x+1)^2 )) dx =

$$\int\:\:\frac{\mathrm{2}{x}−\mathrm{3}}{\left({x}^{\mathrm{2}} +{x}+\mathrm{1}\right)^{\mathrm{2}} }\:{dx}\:= \\ $$

Question Number 55777 Answers: 1 Comments: 0

∫_( 0) ^π cos^7 x sin 3x dx = 0

$$\underset{\:\mathrm{0}} {\overset{\pi} {\int}}\:\:\mathrm{cos}^{\mathrm{7}} {x}\:\mathrm{sin}\:\mathrm{3}{x}\:{dx}\:=\:\mathrm{0} \\ $$

Question Number 55776 Answers: 1 Comments: 0

lim_(n→∞) Σ_(r=1) ^n ((r^3 /(r^4 +n^4 ))) =

$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\:\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}\:\left(\frac{{r}^{\mathrm{3}} }{{r}^{\mathrm{4}} +{n}^{\mathrm{4}} }\right)\:= \\ $$

Question Number 55775 Answers: 1 Comments: 1

∫_( 0) ^1 ∣sin 2π x∣ dx =

$$\:\underset{\:\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\:\mid\mathrm{sin}\:\mathrm{2}\pi\:{x}\mid\:{dx}\:= \\ $$

Question Number 55774 Answers: 1 Comments: 0

∫_(−1/2) ^(1/2) cos x log (((1+x)/(1−x))) dx =

$$\:\underset{−\mathrm{1}/\mathrm{2}} {\overset{\mathrm{1}/\mathrm{2}} {\int}}\:\mathrm{cos}\:{x}\:\mathrm{log}\:\left(\frac{\mathrm{1}+{x}}{\mathrm{1}−{x}}\right)\:{dx}\:= \\ $$

Question Number 55773 Answers: 1 Comments: 0

∫_(−π) ^π sin x f(cos x) dx =

$$\:\underset{−\pi} {\overset{\pi} {\int}}\:\mathrm{sin}\:{x}\:{f}\left(\mathrm{cos}\:{x}\right)\:{dx}\:= \\ $$

Question Number 55772 Answers: 1 Comments: 1

∫_(−π/3) ^(π/3) ((x sin x)/(cos^2 x)) dx =

$$\:\underset{−\pi/\mathrm{3}} {\overset{\pi/\mathrm{3}} {\int}}\:\frac{{x}\:\mathrm{sin}\:{x}}{\mathrm{cos}^{\mathrm{2}} {x}}\:{dx}\:= \\ $$

Question Number 55771 Answers: 0 Comments: 0

Find all possible solutions (a, b, c) from Diophantine equation 2^a + 5^b = c^2

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{possible}\:\mathrm{solutions}\:\left({a},\:{b},\:{c}\right)\:\mathrm{from} \\ $$$$\mathrm{Diophantine}\:\mathrm{equation}\:\:\mathrm{2}^{{a}} \:+\:\mathrm{5}^{{b}} \:=\:{c}^{\mathrm{2}} \\ $$

Question Number 55770 Answers: 0 Comments: 0

Let A and B are matrices in R^(2017×2017) that satisfy A^(−1) = (A + B)^(−1) − B^(−1) and det(A^(−1) ) = 2017 Find det(B)

$$\mathrm{Let}\:{A}\:\mathrm{and}\:{B}\:\mathrm{are}\:\mathrm{matrices}\:\mathrm{in}\:\mathbb{R}^{\mathrm{2017}×\mathrm{2017}} \:\mathrm{that}\:\mathrm{satisfy} \\ $$$${A}^{−\mathrm{1}} \:=\:\left({A}\:+\:{B}\right)^{−\mathrm{1}} \:−\:{B}^{−\mathrm{1}} \\ $$$$\mathrm{and} \\ $$$$\mathrm{det}\left({A}^{−\mathrm{1}} \right)\:=\:\mathrm{2017} \\ $$$$\mathrm{Find}\:\:\:\mathrm{det}\left({B}\right) \\ $$

Question Number 55762 Answers: 0 Comments: 0

Question Number 55760 Answers: 0 Comments: 3

let f(x) =∫_0 ^∞ ((cos(xt))/((xt^2 +i)^2 ))dx with x from R and x≠0 1) find a explicit form of f(x) 2) extract A =Re(f(x)) and B =Im(f(x)) and find its values . 3) calculate ∫_0 ^∞ ((cos(2t))/((2t^2 +i)^2 ))dt 4) let U_n =∫_0 ^∞ ((cos(nt))/((nt^2 +i)^2 ))dt .calculate lim_(n→+∞) u_n and study the convergence of Σu_n

$${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{cos}\left({xt}\right)}{\left({xt}^{\mathrm{2}} +{i}\right)^{\mathrm{2}} }{dx}\:\:\:{with}\:{x}\:{from}\:{R}\:\:{and}\:{x}\neq\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{extract}\:\:{A}\:={Re}\left({f}\left({x}\right)\right)\:{and}\:\:{B}\:={Im}\left({f}\left({x}\right)\right)\:{and}\:{find}\:{its}\:{values}\:. \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{cos}\left(\mathrm{2}{t}\right)}{\left(\mathrm{2}{t}^{\mathrm{2}} \:+{i}\right)^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{4}\right)\:{let}\:{U}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\frac{{cos}\left({nt}\right)}{\left({nt}^{\mathrm{2}} \:+{i}\right)^{\mathrm{2}} }{dt}\:\:\:.{calculate}\:{lim}_{{n}\rightarrow+\infty} \:{u}_{{n}} \\ $$$${and}\:{study}\:{the}\:{convergence}\:{of}\:\Sigma{u}_{{n}} \\ $$$$ \\ $$

Question Number 55759 Answers: 1 Comments: 0

calculate I =∫_0 ^(2π) ((cost)/(3 +sin(2t)))dt and J =∫_0 ^(2π) ((sint)/(3 +cos(2t)))dt .

$${calculate}\:{I}\:=\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\frac{{cost}}{\mathrm{3}\:+{sin}\left(\mathrm{2}{t}\right)}{dt}\:{and}\:{J}\:=\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\frac{{sint}}{\mathrm{3}\:+{cos}\left(\mathrm{2}{t}\right)}{dt}\:. \\ $$

Question Number 55756 Answers: 1 Comments: 0

Find the product of all the real values of a that satisfies the equation 4∣a−4∣=∣a+4∣

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{product}\:\mathrm{of}\:\mathrm{all}\:\mathrm{the}\:\mathrm{real}\:\mathrm{values} \\ $$$$\mathrm{of}\:\:\:\mathrm{a}\:\:\mathrm{that}\:\mathrm{satisfies}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\mathrm{4}\mid\mathrm{a}−\mathrm{4}\mid=\mid\mathrm{a}+\mathrm{4}\mid \\ $$

Question Number 55748 Answers: 1 Comments: 1

Question Number 55737 Answers: 1 Comments: 2

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