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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

Question Number 52533 Answers: 0 Comments: 5

Question Number 52523 Answers: 0 Comments: 12

Question Number 52578 Answers: 0 Comments: 0

Question Number 52520 Answers: 0 Comments: 0

Please help John travels a distance of 24km from twon A on a bearing of 060° to town B. He then travels a distance of 18km to town C, which is 30km east of town A. (i) what is the bearing of town C from town B? (ii) calculate the bearing of town A from town C?

$$\mathrm{Please}\:\mathrm{he}{lp} \\ $$$$\:\mathrm{John}\:\mathrm{travels}\:\mathrm{a}\:\mathrm{distance}\:\mathrm{of}\:\mathrm{24km} \\ $$$$\mathrm{from}\:\mathrm{twon}\:\mathrm{A}\:\mathrm{on}\:\mathrm{a}\:\mathrm{bearing}\:\mathrm{of}\:\mathrm{060}°\:\mathrm{to} \\ $$$$\mathrm{town}\:\mathrm{B}.\:\mathrm{He}\:\mathrm{then}\:\mathrm{travels}\:\mathrm{a}\:\mathrm{distance} \\ $$$$\mathrm{of}\:\mathrm{18km}\:\mathrm{to}\:\mathrm{town}\:\mathrm{C},\:\mathrm{which}\:\mathrm{is}\:\mathrm{30km} \\ $$$$\mathrm{east}\:\mathrm{of}\:\mathrm{town}\:\mathrm{A}. \\ $$$$\left(\mathrm{i}\right)\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{bearing}\:\mathrm{of}\:\mathrm{town}\:\mathrm{C}\:\mathrm{from} \\ $$$$\:\:\:\:\:\:\:\mathrm{town}\:\mathrm{B}? \\ $$$$\left(\mathrm{ii}\right)\:\mathrm{calculate}\:\mathrm{the}\:\mathrm{bearing}\:\mathrm{of}\:\mathrm{town}\:\mathrm{A}\: \\ $$$$\:\:\:\:\:\:\:\:\mathrm{from}\:\mathrm{town}\:\mathrm{C}? \\ $$

Question Number 52516 Answers: 1 Comments: 0

find the value tanθ+2tan2θ+2^2 tan2^2 θ+2^3 tan2^3 θ+..+2^(n−1) tan2^(n−1) θ

$${find}\:{the}\:{value} \\ $$$${tan}\theta+\mathrm{2}{tan}\mathrm{2}\theta+\mathrm{2}^{\mathrm{2}} {tan}\mathrm{2}^{\mathrm{2}} \theta+\mathrm{2}^{\mathrm{3}} {tan}\mathrm{2}^{\mathrm{3}} \theta+..+\mathrm{2}^{{n}−\mathrm{1}} {tan}\mathrm{2}^{{n}−\mathrm{1}} \theta \\ $$

Question Number 52515 Answers: 2 Comments: 3

1)∫_0 ^∞ ((tan^(−1) (ax)−tan^(−1) (bx))/x)dx 2)∫_0 ^∞ ((e^(−x) sinx)/x)dx

$$\left.\mathrm{1}\right)\int_{\mathrm{0}} ^{\infty} \frac{{tan}^{−\mathrm{1}} \left({ax}\right)−{tan}^{−\mathrm{1}} \left({bx}\right)}{{x}}{dx} \\ $$$$\left.\mathrm{2}\right)\int_{\mathrm{0}} ^{\infty} \frac{{e}^{−{x}} {sinx}}{{x}}{dx} \\ $$

Question Number 52506 Answers: 1 Comments: 1

Question Number 52499 Answers: 0 Comments: 0

Let f is a 2^(nd) degree polynomial. (1/(2πi)) ∫_(∣z∣=2) ((z f ′(z))/(f(z))) dz = 0, and (1/(2πi)) ∫_(∣z∣=2) ((z^2 f ′(z))/(f(z))) dz = −2 If f(0) = 2017, find explicit form of f(z)

$$\mathrm{Let}\:{f}\:\mathrm{is}\:\mathrm{a}\:\mathrm{2}^{\mathrm{nd}} \:\mathrm{degree}\:\mathrm{polynomial}. \\ $$$$\frac{\mathrm{1}}{\mathrm{2}\pi{i}}\:\int_{\mid{z}\mid=\mathrm{2}} \frac{{z}\:{f}\:'\left({z}\right)}{{f}\left({z}\right)}\:{dz}\:=\:\mathrm{0},\:\:\mathrm{and}\:\frac{\mathrm{1}}{\mathrm{2}\pi{i}}\:\int_{\mid{z}\mid=\mathrm{2}} \frac{{z}^{\mathrm{2}} \:{f}\:'\left({z}\right)}{{f}\left({z}\right)}\:{dz}\:=\:−\mathrm{2} \\ $$$$\mathrm{If}\:{f}\left(\mathrm{0}\right)\:=\:\mathrm{2017},\:\mathrm{find}\:\mathrm{explicit}\:\mathrm{form}\:\mathrm{of}\:{f}\left({z}\right) \\ $$

Question Number 52498 Answers: 1 Comments: 1

Let z is complex number and satisfy z^(2017) = 1, z ≠ 1 Find (1 + z)(1 + z^2 )(1 + z^3 )…(1 + z^(2016) )

$$\mathrm{Let}\:{z}\:\mathrm{is}\:\mathrm{complex}\:\mathrm{number}\:\mathrm{and}\:\mathrm{satisfy}\:{z}^{\mathrm{2017}} \:=\:\mathrm{1},\:\:{z}\:\neq\:\mathrm{1} \\ $$$$\mathrm{Find}\:\left(\mathrm{1}\:+\:{z}\right)\left(\mathrm{1}\:+\:{z}^{\mathrm{2}} \right)\left(\mathrm{1}\:+\:{z}^{\mathrm{3}} \right)\ldots\left(\mathrm{1}\:+\:{z}^{\mathrm{2016}} \right) \\ $$

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