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Question Number 41622    Answers: 4   Comments: 9

let z_1 and z_2 the roots of x^2 −2x+2=0 1) calculate z_1 ^3 +z_2 ^3 then (1/z_1 ^3 ) +(1/z_2 ^3 ) 2) calculate z_1 ^4 +z_2 ^4 then (1/z_1 ^4 ) +(1/z_2 ^4 ) 3) let n from N simplify A_n = z_1 ^n +z_2 ^n and B_n = z_1 ^n −z_2 ^n 4) simplify S_n =Σ_(k=0) ^(n−1) (z_1 ^k +z_2 ^k )

$${let}\:{z}_{\mathrm{1}} \:{and}\:{z}_{\mathrm{2}} \:{the}\:{roots}\:{of}\:{x}^{\mathrm{2}} −\mathrm{2}{x}+\mathrm{2}=\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{z}_{\mathrm{1}} ^{\mathrm{3}} \:+{z}_{\mathrm{2}} ^{\mathrm{3}} \:\:\:{then}\:\:\frac{\mathrm{1}}{{z}_{\mathrm{1}} ^{\mathrm{3}} }\:+\frac{\mathrm{1}}{{z}_{\mathrm{2}} ^{\mathrm{3}} } \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{z}_{\mathrm{1}} ^{\mathrm{4}} \:+{z}_{\mathrm{2}} ^{\mathrm{4}} \:\:{then}\:\:\frac{\mathrm{1}}{{z}_{\mathrm{1}} ^{\mathrm{4}} }\:+\frac{\mathrm{1}}{{z}_{\mathrm{2}} ^{\mathrm{4}} } \\ $$$$\left.\mathrm{3}\right)\:{let}\:{n}\:{from}\:{N}\:\:{simplify} \\ $$$${A}_{{n}} =\:{z}_{\mathrm{1}} ^{{n}} \:+{z}_{\mathrm{2}} ^{{n}} \:\:\:\:\:\:\:{and}\:\:{B}_{{n}} =\:{z}_{\mathrm{1}} ^{{n}} \:−{z}_{\mathrm{2}} ^{{n}} \\ $$$$\left.\mathrm{4}\right)\:{simplify}\:\:{S}_{{n}} =\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:\:\left({z}_{\mathrm{1}} ^{{k}} \:\:\:+{z}_{\mathrm{2}} ^{{k}} \right) \\ $$

Question Number 41620    Answers: 2   Comments: 1

Question Number 41616    Answers: 0   Comments: 0

Question Number 41627    Answers: 1   Comments: 2

Question Number 41787    Answers: 1   Comments: 0

Question Number 41783    Answers: 1   Comments: 0

Question Number 41606    Answers: 2   Comments: 0

4x^4 +16x^3 +24x^2 −9x−1=0 using any method. find real value of x that satisfy the polynomial

$$\mathrm{4}\boldsymbol{\mathrm{x}}^{\mathrm{4}} +\mathrm{16}\boldsymbol{\mathrm{x}}^{\mathrm{3}} +\mathrm{24}\boldsymbol{\mathrm{x}}^{\mathrm{2}} −\mathrm{9}\boldsymbol{\mathrm{x}}−\mathrm{1}=\mathrm{0} \\ $$$$\boldsymbol{\mathrm{using}}\:\boldsymbol{\mathrm{any}}\:\boldsymbol{\mathrm{method}}.\:\boldsymbol{\mathrm{find}}\:\:\boldsymbol{\mathrm{real}}\:\boldsymbol{\mathrm{value}}\:\boldsymbol{\mathrm{of}}\:\:\boldsymbol{\mathrm{x}}\:\:\boldsymbol{\mathrm{that}}\:\boldsymbol{\mathrm{satisfy}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{polynomial}} \\ $$

Question Number 41586    Answers: 2   Comments: 1

f(x)=(√(−3+(√((x+1)/(x−1))))) ∫f(x)=? ∫f^(−1) (x)=?

$${f}\left({x}\right)=\sqrt{−\mathrm{3}+\sqrt{\frac{{x}+\mathrm{1}}{{x}−\mathrm{1}}}} \\ $$$$\int{f}\left({x}\right)=? \\ $$$$\int{f}^{−\mathrm{1}} \left({x}\right)=? \\ $$

Question Number 41579    Answers: 1   Comments: 1

If u_(10) = ∫_( 0) ^(π/2) x^(10) sin x dx, then the value of u_(10) +90 u_8 is

$$\mathrm{If}\:\:{u}_{\mathrm{10}} =\:\underset{\:\mathrm{0}} {\overset{\pi/\mathrm{2}} {\int}}{x}^{\mathrm{10}} \:\mathrm{sin}\:{x}\:{dx},\:\mathrm{then}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$${u}_{\mathrm{10}} +\mathrm{90}\:{u}_{\mathrm{8}} \:\:\mathrm{is} \\ $$

Question Number 41578    Answers: 0   Comments: 0

If u_(10) = ∫_( 0) ^(π/2) x^(10) sin x dx, then the value of u_(10) +90 u_8 is

$$\mathrm{If}\:\:{u}_{\mathrm{10}} =\:\underset{\:\mathrm{0}} {\overset{\pi/\mathrm{2}} {\int}}{x}^{\mathrm{10}} \:\mathrm{sin}\:{x}\:{dx},\:\mathrm{then}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$${u}_{\mathrm{10}} +\mathrm{90}\:{u}_{\mathrm{8}} \:\:\mathrm{is} \\ $$

Question Number 41561    Answers: 2   Comments: 3

∫ (dx/(3sin(x) + 4cos(x)))

$$\int\:\frac{\mathrm{dx}}{\mathrm{3sin}\left(\mathrm{x}\right)\:+\:\mathrm{4cos}\left(\mathrm{x}\right)} \\ $$

Question Number 41577    Answers: 0   Comments: 0

If u_(10) = ∫_( 0) ^(π/2) x^(10) sin x dx, then the value of u_(10) +90 u_8 is

$$\mathrm{If}\:\:{u}_{\mathrm{10}} =\:\underset{\:\mathrm{0}} {\overset{\pi/\mathrm{2}} {\int}}{x}^{\mathrm{10}} \:\mathrm{sin}\:{x}\:{dx},\:\mathrm{then}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$${u}_{\mathrm{10}} +\mathrm{90}\:{u}_{\mathrm{8}} \:\:\mathrm{is} \\ $$

Question Number 41555    Answers: 5   Comments: 0

Question Number 41536    Answers: 3   Comments: 0

if y=(√(((1+sinx)/(1−sinx)) )) show that (dy/dx)=(1/(1−sinx))

$$\boldsymbol{\mathrm{if}}\:\boldsymbol{{y}}=\sqrt{\frac{\mathrm{1}+\boldsymbol{\mathrm{sin}{x}}}{\mathrm{1}−\boldsymbol{\mathrm{sin}{x}}}\:}\:\boldsymbol{\mathrm{show}}\:\boldsymbol{\mathrm{that}} \\ $$$$\frac{\boldsymbol{{dy}}}{\boldsymbol{{dx}}}=\frac{\mathrm{1}}{\mathrm{1}−\boldsymbol{\mathrm{sin}{x}}} \\ $$

Question Number 41534    Answers: 1   Comments: 0

Three chidren are playing the game of claping hands,the first child claping hands in every after 1sec,the second child clap hands in every after 10sec and the third child claps in every after 5sec. for how long do all three children will clap their hands together at the same time?

$$\boldsymbol{\mathrm{Three}}\:\boldsymbol{\mathrm{chidren}}\:\boldsymbol{\mathrm{are}}\:\boldsymbol{\mathrm{playing}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{game}} \\ $$$$\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{claping}}\:\boldsymbol{\mathrm{hands}},\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{first}}\:\boldsymbol{\mathrm{child}}\:\boldsymbol{\mathrm{claping}}\:\boldsymbol{\mathrm{hands}} \\ $$$$\boldsymbol{\mathrm{in}}\:\boldsymbol{\mathrm{every}}\:\boldsymbol{\mathrm{after}}\:\mathrm{1}\boldsymbol{\mathrm{sec}},\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{second}}\: \\ $$$$\boldsymbol{\mathrm{child}}\:\boldsymbol{\mathrm{clap}}\:\boldsymbol{\mathrm{hands}}\:\boldsymbol{\mathrm{in}}\:\boldsymbol{\mathrm{every}}\:\boldsymbol{\mathrm{after}}\:\mathrm{10}\boldsymbol{\mathrm{sec}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{the}} \\ $$$$\boldsymbol{\mathrm{third}}\:\boldsymbol{\mathrm{child}}\:\boldsymbol{\mathrm{claps}}\:\boldsymbol{\mathrm{in}}\:\boldsymbol{\mathrm{every}}\:\boldsymbol{\mathrm{after}}\:\mathrm{5}\boldsymbol{\mathrm{sec}}.\:\boldsymbol{\mathrm{for}}\:\boldsymbol{\mathrm{how}}\:\boldsymbol{\mathrm{long}}\:\boldsymbol{\mathrm{do}}\:\boldsymbol{\mathrm{all}} \\ $$$$\boldsymbol{\mathrm{three}}\:\boldsymbol{\mathrm{children}}\:\boldsymbol{\mathrm{will}}\:\boldsymbol{\mathrm{clap}}\:\boldsymbol{\mathrm{their}}\:\boldsymbol{\mathrm{hands}}\:\boldsymbol{\mathrm{together}}\:\boldsymbol{\mathrm{at}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{same}}\:\boldsymbol{\mathrm{time}}? \\ $$

Question Number 41530    Answers: 1   Comments: 0

Question Number 41522    Answers: 2   Comments: 2

Find the cube root of 26 − 15(√3)

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{cube}\:\mathrm{root}\:\mathrm{of}\:\:\:\mathrm{26}\:−\:\mathrm{15}\sqrt{\mathrm{3}} \\ $$

Question Number 41521    Answers: 0   Comments: 0

let A(−1,1) and B(0,3) find image of the line (AB) by 1) translation t_u^→ with u^→ (1−2i) 2) rotation R(w,(π/3)) with w(1+i)

$${let}\:{A}\left(−\mathrm{1},\mathrm{1}\right)\:{and}\:{B}\left(\mathrm{0},\mathrm{3}\right)\:\:\:{find}\:\:\:{image}\:{of}\:{the}\:{line}\:\left({AB}\right)\:{by} \\ $$$$\left.\mathrm{1}\right)\:{translation}\:{t}_{\overset{\rightarrow} {{u}}} \:\:\:\:{with}\:\overset{\rightarrow} {{u}}\left(\mathrm{1}−\mathrm{2}{i}\right) \\ $$$$\left.\mathrm{2}\right)\:{rotation}\:{R}\left({w},\frac{\pi}{\mathrm{3}}\right)\:\:{with}\:{w}\left(\mathrm{1}+{i}\right) \\ $$

Question Number 41520    Answers: 3   Comments: 0

let Z = cos(((2π)/7)) +isin(((2π)/7)) and A= Z+Z^2 +Z^4 B=Z^3 +Z^5 +Z^6 1) prove that A^− =B 2) prove that A+B =−1 and A.B =2 3) find A and B.

$${let}\:{Z}\:=\:{cos}\left(\frac{\mathrm{2}\pi}{\mathrm{7}}\right)\:+{isin}\left(\frac{\mathrm{2}\pi}{\mathrm{7}}\right)\:{and}\:\:{A}=\:{Z}+{Z}^{\mathrm{2}} \:+{Z}^{\mathrm{4}} \\ $$$${B}={Z}^{\mathrm{3}} \:+{Z}^{\mathrm{5}} \:+{Z}^{\mathrm{6}} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\overset{−} {{A}}={B} \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:{A}+{B}\:=−\mathrm{1}\:{and}\:{A}.{B}\:=\mathrm{2} \\ $$$$\left.\mathrm{3}\right)\:{find}\:\:{A}\:{and}\:{B}. \\ $$$$ \\ $$

Question Number 41519    Answers: 2   Comments: 0

let z=(√(2−(√3))) −i(√(2+(√3))) calculate ∣z^n ∣ and arg(z^n )

$${let}\:{z}=\sqrt{\mathrm{2}−\sqrt{\mathrm{3}}}\:\:−{i}\sqrt{\mathrm{2}+\sqrt{\mathrm{3}}} \\ $$$${calculate}\:\mid{z}^{{n}} \mid\:\:{and}\:{arg}\left({z}^{{n}} \right) \\ $$

Question Number 41518    Answers: 3   Comments: 0

calculate A_n = ∫_0 ^1 (1−t^2 )^n dt with n integr natural

$${calculate}\:\:{A}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{1}−{t}^{\mathrm{2}} \right)^{{n}} {dt}\:\:\:{with}\:{n}\:{integr}\:{natural} \\ $$

Question Number 41517    Answers: 1   Comments: 1

let S_n = 1 +(1/((^3 (√2)))) + (1/((^3 (√3)))) + ....+(1/((^3 (√n)))) calculate lim _(n→+∞) S_n

$${let}\:\:{S}_{{n}} =\:\mathrm{1}\:+\frac{\mathrm{1}}{\left(^{\mathrm{3}} \sqrt{\mathrm{2}}\right)}\:+\:\frac{\mathrm{1}}{\left(^{\mathrm{3}} \sqrt{\mathrm{3}}\right)}\:+\:....+\frac{\mathrm{1}}{\left(^{\mathrm{3}} \sqrt{{n}}\right)} \\ $$$${calculate}\:{lim}\:_{{n}\rightarrow+\infty} \:{S}_{{n}} \\ $$

Question Number 41516    Answers: 1   Comments: 1

calculate ∫_0 ^1 ((ln(1+x))/((1+x)^4 )) dx

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{ln}\left(\mathrm{1}+{x}\right)}{\left(\mathrm{1}+{x}\right)^{\mathrm{4}} }\:{dx} \\ $$

Question Number 41515    Answers: 1   Comments: 3

let f_n (x) =((sin(2(n+1)x))/(sinx)) if x∈]0,(π/2)] and f_n (0)=2(n+1) let u_n = ∫_0 ^(π/2) f_n (x)dx 1) prove that ∀n fromN u_(n+1) −u_n =2(((−1)^(n+1) )/(2n+3)) 2)find lim_(n→+∞) u_n

$$\left.{l}\left.{et}\:\:{f}_{{n}} \left({x}\right)\:=\frac{{sin}\left(\mathrm{2}\left({n}+\mathrm{1}\right){x}\right)}{{sinx}}\:{if}\:\:{x}\in\right]\mathrm{0},\frac{\pi}{\mathrm{2}}\right]\:{and}\:{f}_{{n}} \left(\mathrm{0}\right)=\mathrm{2}\left({n}+\mathrm{1}\right)\:\:{let} \\ $$$${u}_{{n}} =\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:{f}_{{n}} \left({x}\right){dx} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\:\forall{n}\:{fromN}\:\:{u}_{{n}+\mathrm{1}} −{u}_{{n}} =\mathrm{2}\frac{\left(−\mathrm{1}\right)^{{n}+\mathrm{1}} }{\mathrm{2}{n}+\mathrm{3}} \\ $$$$\left.\mathrm{2}\right){find}\:{lim}_{{n}\rightarrow+\infty} \:{u}_{{n}} \\ $$

Question Number 41514    Answers: 1   Comments: 0

find ∫ cos(lnx)dx

$${find}\:\:\:\int\:\:{cos}\left({lnx}\right){dx}\: \\ $$

Question Number 41513    Answers: 2   Comments: 0

let A_n = ∫_0 ^∞ e^(−nx^2 ) cos(x^2 ) dx and B_n =∫_0 ^∞ e^(−nx^2 ) sin(x^2 )dx (n∈ N^★ ) 1) calculate A_n and B_n 2) find lim_(n→+∞) (A_n /B_n )

$${let}\:\:{A}_{{n}} =\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{nx}^{\mathrm{2}} } {cos}\left({x}^{\mathrm{2}} \right)\:{dx}\:\:{and}\:\:{B}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{nx}^{\mathrm{2}} } {sin}\left({x}^{\mathrm{2}} \right){dx}\:\:\:\:\left({n}\in\:{N}^{\bigstar} \right) \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:\:{A}_{{n}} \:{and}\:\:{B}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{n}\rightarrow+\infty} \:\:\frac{{A}_{{n}} }{{B}_{{n}} } \\ $$

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