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Question Number 52671 Answers: 0 Comments: 1

study the sequence u_0 =1 and u_(n+1) =(1/(1+u_n ^2 ))

$${study}\:{the}\:{sequence}\:{u}_{\mathrm{0}} =\mathrm{1}\:{and}\:{u}_{{n}+\mathrm{1}} \:\:=\frac{\mathrm{1}}{\mathrm{1}+{u}_{{n}} ^{\mathrm{2}} } \\ $$

Question Number 52670 Answers: 1 Comments: 1

study the convergence of Σ_(n=0) ^∞ sin(π(√(4n^2 +1)))

$${study}\:{the}\:{convergence}\:{of}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:{sin}\left(\pi\sqrt{\mathrm{4}{n}^{\mathrm{2}} +\mathrm{1}}\right) \\ $$

Question Number 52669 Answers: 0 Comments: 0

let S_(n ) (p)=Σ_(k=0) ^n k^p prove that S_n (p)=(1/(p+1)){ (n+1)^(p+1) −Σ_(k=0) ^(n−1) C_(p+1) ^k S_n (k)}

$${let}\:{S}_{{n}\:} \:\left({p}\right)=\sum_{{k}=\mathrm{0}} ^{{n}} \:{k}^{{p}} \\ $$$${prove}\:{that}\:{S}_{{n}} \left({p}\right)=\frac{\mathrm{1}}{{p}+\mathrm{1}}\left\{\:\left({n}+\mathrm{1}\right)^{{p}+\mathrm{1}} \:−\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:{C}_{{p}+\mathrm{1}} ^{{k}} \:{S}_{{n}} \left({k}\right)\right\} \\ $$

Question Number 52667 Answers: 1 Comments: 0

∫((x^4 +1)/(x^2 (√(x^4 −1)))) dx

$$\int\frac{{x}^{\mathrm{4}} +\mathrm{1}}{{x}^{\mathrm{2}} \sqrt{{x}^{\mathrm{4}} −\mathrm{1}}}\:{dx} \\ $$

Question Number 52663 Answers: 2 Comments: 1

Question Number 52655 Answers: 0 Comments: 0

Question Number 52653 Answers: 0 Comments: 0

2 X + 5^(2 ) = (√9) − X 2 X + X = 3 − 25 3 X = − 22 X = ((22)/3) 6 Z − 63 = 1 − ∣ − 20 ∣ 6 Z = 1 − 20 + 63 6 Z = 44 Z = ((44)/6) = ((22)/3) ∵ s(x)=s(z),∴X=Z

$$\mathrm{2}\:{X}\:\:+\:\:\mathrm{5}\:^{\mathrm{2}\:} \:\:=\:\:\:\:\sqrt{\mathrm{9}}\:\:\:\:−\:\:\:{X} \\ $$$$\mathrm{2}\:\:{X}\:\:\:+\:\:\:{X}\:\:\:=\:\:\:\mathrm{3}\:\:\:−\:\:\:\mathrm{25} \\ $$$$\mathrm{3}\:\:{X}\:\:\:=\:\:\:−\:\mathrm{22} \\ $$$${X}\:\:=\:\:\frac{\mathrm{22}}{\mathrm{3}} \\ $$$$ \\ $$$$\mathrm{6}\:{Z}\:\:−\:\mathrm{63}\:\:=\:\:\mathrm{1}\:\:−\:\:\mid\:\:\:−\:\:\mathrm{20}\:\:\mid \\ $$$$\mathrm{6}\:{Z}\:\:=\:\:\mathrm{1}\:\:−\:\:\mathrm{20}\:\:+\:\:\mathrm{63} \\ $$$$\mathrm{6}\:\:{Z}\:\:=\:\:\mathrm{44} \\ $$$${Z}\:\:=\:\:\frac{\mathrm{44}}{\mathrm{6}}\:\:=\:\:\frac{\mathrm{22}}{\mathrm{3}}\:\:\:\:\:\because\:{s}\left({x}\right)={s}\left({z}\right),\therefore{X}={Z} \\ $$

Question Number 52649 Answers: 0 Comments: 2

∫ ((4x^2 + 3)/((x^2 + x + 1)^2 )) dx

$$\int\:\frac{\mathrm{4x}^{\mathrm{2}} \:+\:\mathrm{3}}{\left(\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{x}\:+\:\mathrm{1}\right)^{\mathrm{2}} }\:\mathrm{dx} \\ $$

Question Number 52644 Answers: 0 Comments: 0

Question Number 52630 Answers: 0 Comments: 3

Question Number 52629 Answers: 2 Comments: 0

show that ((sinα+sin3α+sin5α)/(cosα+cos3α+cos5α))= tan3α

$${show}\:{that}\:\frac{{sin}\alpha+{sin}\mathrm{3}\alpha+{sin}\mathrm{5}\alpha}{{cos}\alpha+{cos}\mathrm{3}\alpha+{cos}\mathrm{5}\alpha}=\:{tan}\mathrm{3}\alpha \\ $$

Question Number 52627 Answers: 2 Comments: 1

Question Number 52619 Answers: 3 Comments: 4

1)∫_1 ^3 (dx/(x^2 +[x]^2 +1−2x[x])) 2)∫_(−1) ^1 [x[1+sinπx]+1]dx 3)∫_0 ^2 x^([x^2 +1]) dx 4)∫_0 ^1 e^(2x−[2x]) d(x−[x]) 5)∫_(−2) ^3 ∣x^2 −5x−6∣dx from question 1 to 4 [.]←greatest integer fuction ∣.∣←mod questions taken from B.Stat entrance exam...

$$\left.\mathrm{1}\right)\int_{\mathrm{1}} ^{\mathrm{3}} \frac{{dx}}{{x}^{\mathrm{2}} +\left[{x}\right]^{\mathrm{2}} +\mathrm{1}−\mathrm{2}{x}\left[{x}\right]} \\ $$$$\left.\mathrm{2}\right)\int_{−\mathrm{1}} ^{\mathrm{1}} \left[{x}\left[\mathrm{1}+{sin}\pi{x}\right]+\mathrm{1}\right]{dx} \\ $$$$\left.\mathrm{3}\right)\int_{\mathrm{0}} ^{\mathrm{2}} {x}^{\left[{x}^{\mathrm{2}} +\mathrm{1}\right]} {dx} \\ $$$$\left.\mathrm{4}\right)\int_{\mathrm{0}} ^{\mathrm{1}} {e}^{\mathrm{2}{x}−\left[\mathrm{2}{x}\right]} {d}\left({x}−\left[{x}\right]\right)\: \\ $$$$\left.\mathrm{5}\right)\int_{−\mathrm{2}} ^{\mathrm{3}} \mid{x}^{\mathrm{2}} −\mathrm{5}{x}−\mathrm{6}\mid{dx} \\ $$$${from}\:{question}\:\mathrm{1}\:{to}\:\mathrm{4}\:\left[.\right]\leftarrow{greatest}\:{integer}\:{fuction} \\ $$$$\mid.\mid\leftarrow{mod} \\ $$$${questions}\:{taken}\:{from}\:{B}.{Stat}\:{entrance}\:{exam}... \\ $$

Question Number 52611 Answers: 1 Comments: 5

Question Number 52610 Answers: 1 Comments: 4

please help me y=(√(9−x^2 )) Q find domain and range of given real function i found the domain=[−3,3] but for range i followed this way y^2 =9−x^2 x^2 =9−y^2 ⇒x=(√(9−y^2 ))⇒y=[−3,3] but the answer is [0,3] after checking the graph y≥0. yes its true we have to take positive root but how can i write it

$$\:{please}\:{help}\:{me} \\ $$$${y}=\sqrt{\mathrm{9}−{x}^{\mathrm{2}} } \\ $$$${Q}\:{find}\:{domain}\:{and}\:{range}\:{of}\:{given}\:{real}\:{function} \\ $$$${i}\:{found}\:{the}\:{domain}=\left[−\mathrm{3},\mathrm{3}\right] \\ $$$${but}\:{for}\:{range}\:{i}\:{followed}\:{this}\:{way} \\ $$$${y}^{\mathrm{2}} =\mathrm{9}−{x}^{\mathrm{2}} \\ $$$${x}^{\mathrm{2}} =\mathrm{9}−{y}^{\mathrm{2}} \Rightarrow{x}=\sqrt{\mathrm{9}−{y}^{\mathrm{2}} }\Rightarrow{y}=\left[−\mathrm{3},\mathrm{3}\right] \\ $$$${but}\:{the}\:{answer}\:{is}\:\left[\mathrm{0},\mathrm{3}\right] \\ $$$${after}\:{checking}\:{the}\:{graph}\:\:{y}\geqslant\mathrm{0}.\:{yes}\:{its}\:{true} \\ $$$${we}\:{have}\:{to}\:{take}\:{positive}\:{root}\:{but}\:{how}\:{can}\:{i} \\ $$$${write}\:{it} \\ $$

Question Number 52609 Answers: 1 Comments: 0

sin^2 A + sin^4 A=1then the value of tan^2 A−tan^4 A

$$ \\ $$$$ \\ $$$$\mathrm{sin}\:^{\mathrm{2}} \mathrm{A}\:+\:\mathrm{sin}\:^{\mathrm{4}} \mathrm{A}=\mathrm{1then}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{tan}\:^{\mathrm{2}} \mathrm{A}−\mathrm{tan}^{\mathrm{4}} \mathrm{A} \\ $$

Question Number 52600 Answers: 1 Comments: 1

If P_n denotes the product of the binomial coefficients in the expansion of (1+x)^n , then (P_(n+1) /P_n ) equals

$$\mathrm{If}\:{P}_{{n}} \:\mathrm{denotes}\:\mathrm{the}\:\mathrm{product}\:\mathrm{of}\:\mathrm{the}\:\mathrm{binomial}\: \\ $$$$\mathrm{coefficients}\:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{of}\:\left(\mathrm{1}+{x}\right)^{{n}} , \\ $$$$\mathrm{then}\:\frac{{P}_{{n}+\mathrm{1}} }{{P}_{{n}} }\:\mathrm{equals} \\ $$

Question Number 52593 Answers: 1 Comments: 1

Question Number 52592 Answers: 1 Comments: 0

if x+y+z=1 x^2 +y^2 +z^2 =2 x^3 +y^3 +z^3 =3 find x^4 +y^4 +z^4

$${if}\:{x}+{y}+{z}=\mathrm{1} \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} =\mathrm{2} \\ $$$${x}^{\mathrm{3}} +{y}^{\mathrm{3}} +{z}^{\mathrm{3}} =\mathrm{3} \\ $$$${find}\:{x}^{\mathrm{4}} +{y}^{\mathrm{4}} +{z}^{\mathrm{4}} \\ $$

Question Number 52590 Answers: 0 Comments: 1

Question Number 52573 Answers: 1 Comments: 0

Find the relation between p q and r if one of the root of the equation px^2 +qx+r=0 is double the other.

$${Find}\:{the}\:{relation}\:{between}\:{p}\:{q}\:{and}\:{r} \\ $$$${if}\:{one}\:{of}\:{the}\:{root}\:{of}\:{the}\:{equation} \\ $$$${px}^{\mathrm{2}} +{qx}+{r}=\mathrm{0}\:{is}\:{double}\:{the}\:{other}. \\ $$

Question Number 52572 Answers: 1 Comments: 0

In the equation ax^2 +bx+c=0.One root is the square of the other. Show that c(a−b)^3 =a(c−b)^3

$${In}\:{the}\:{equation}\:{ax}^{\mathrm{2}} +{bx}+{c}=\mathrm{0}.{One} \\ $$$${root}\:{is}\:{the}\:{square}\:{of}\:{the}\:{other}. \\ $$$${Show}\:{that}\:{c}\left({a}−{b}\right)^{\mathrm{3}} ={a}\left({c}−{b}\right)^{\mathrm{3}} \\ $$$$ \\ $$

Question Number 52558 Answers: 1 Comments: 6

Question Number 52550 Answers: 1 Comments: 1

∫_0 ^( ∞) (x/(e^x − 1)) dx

$$\:\:\int_{\mathrm{0}} ^{\:\infty} \:\:\:\frac{\mathrm{x}}{\mathrm{e}^{\mathrm{x}} \:−\:\mathrm{1}}\:\:\mathrm{dx}\: \\ $$

Question Number 52542 Answers: 1 Comments: 3

Let f(x) = (( 5x + 40 )/(x + 2)) (1) Calculate the derivative of f(4) without using the answer of question (2), (2) Calculate the derivative of f(x), (3) Calculate lim_(x → +∞) f(x) . All your answers should be correctly proved and detailed. Can you please help me for that exercise.

$$\mathrm{Let}\:{f}\left({x}\right)\:=\:\frac{\:\mathrm{5}{x}\:+\:\mathrm{40}\:}{{x}\:+\:\mathrm{2}} \\ $$$$\left(\mathrm{1}\right)\:\mathrm{Calculate}\:\mathrm{the}\:\mathrm{derivative}\:\mathrm{of}\:{f}\left(\mathrm{4}\right)\:\mathrm{without} \\ $$$$\mathrm{using}\:\mathrm{the}\:\mathrm{answer}\:\mathrm{of}\:\mathrm{question}\:\left(\mathrm{2}\right), \\ $$$$\left(\mathrm{2}\right)\:\mathrm{Calculate}\:\mathrm{the}\:\mathrm{derivative}\:\mathrm{of}\:{f}\left({x}\right), \\ $$$$\left(\mathrm{3}\right)\:\mathrm{Calculate}\:\:\underset{{x}\:\rightarrow\:+\infty} {\mathrm{lim}}\:{f}\left({x}\right)\:. \\ $$$$\mathrm{All}\:\mathrm{your}\:\mathrm{answers}\:\mathrm{should}\:\mathrm{be}\:\mathrm{correctly}\:\mathrm{proved}\:\mathrm{and}\:\mathrm{detailed}. \\ $$$$ \\ $$$${Can}\:{you}\:{please}\:{help}\:{me}\:{for}\:{that}\:{exercise}. \\ $$

Question Number 52539 Answers: 2 Comments: 0

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