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Question Number 213504 Answers: 1 Comments: 0

Question Number 213503 Answers: 1 Comments: 0

Question Number 213499 Answers: 0 Comments: 0

∫_0 ^( 2π) ((z∙sin(z))/(1+cos^2 (z)))dz =? (contour integral) pls help.....

$$\int_{\mathrm{0}} ^{\:\mathrm{2}\pi} \frac{{z}\centerdot\mathrm{sin}\left({z}\right)}{\mathrm{1}+\mathrm{cos}^{\mathrm{2}} \left({z}\right)}\mathrm{d}{z}\:=?\:\left(\mathrm{contour}\:\mathrm{integral}\right) \\ $$$$\mathrm{pls}\:\mathrm{help}..... \\ $$

Question Number 213492 Answers: 0 Comments: 1

Question Number 213484 Answers: 0 Comments: 0

∫_0 ^( 2π) ((z∙sin(z))/(1+cos^2 (z)))dz (Contour integral) ∮_( ∣z∣=2) (1/(z^2 +1)) dz ∮_( ∣z∣=2) ((sin(z))/(z^2 +1)) dz

$$\int_{\mathrm{0}} ^{\:\mathrm{2}\pi} \:\frac{{z}\centerdot\mathrm{sin}\left({z}\right)}{\mathrm{1}+\mathrm{cos}^{\mathrm{2}} \left({z}\right)}\mathrm{d}{z}\:\:\left(\mathrm{Contour}\:\mathrm{integral}\right)\: \\ $$$$\oint_{\:\mid{z}\mid=\mathrm{2}} \:\frac{\mathrm{1}}{{z}^{\mathrm{2}} +\mathrm{1}}\:\mathrm{d}{z} \\ $$$$\oint_{\:\mid{z}\mid=\mathrm{2}} \:\:\frac{\mathrm{sin}\left({z}\right)}{{z}^{\mathrm{2}} +\mathrm{1}}\:\mathrm{d}{z} \\ $$

Question Number 213486 Answers: 1 Comments: 0

x^5 +5x−(6/x)=0 x?

$$\:\:\boldsymbol{\mathrm{x}}^{\mathrm{5}} +\mathrm{5}\boldsymbol{\mathrm{x}}−\frac{\mathrm{6}}{\boldsymbol{\mathrm{x}}}=\mathrm{0}\:\:\:\:\:\:\boldsymbol{\mathrm{x}}? \\ $$

Question Number 213482 Answers: 1 Comments: 0

Question Number 213485 Answers: 2 Comments: 0

prove that (1/2^2 )+(1/3^2 )+...+(1/(2021^2 ))<((25)/(36))

$${prove}\:{that} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{2}} }+\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{2}} }+...+\frac{\mathrm{1}}{\mathrm{2021}^{\mathrm{2}} }<\frac{\mathrm{25}}{\mathrm{36}} \\ $$

Question Number 213468 Answers: 1 Comments: 0

show that the sequence {a_n } defined recurssively by a_1 = (3/2) a_(n ) = (√(3a_(n−1 ) −2 )) for n≥2 converges and find its limit.

$$\:\mathrm{show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{sequence}\:\left\{\mathrm{a}_{\mathrm{n}} \right\}\:\mathrm{defined}\: \\ $$$$\mathrm{recurssively}\:\mathrm{by}\:\mathrm{a}_{\mathrm{1}} =\:\frac{\mathrm{3}}{\mathrm{2}} \\ $$$$\mathrm{a}_{\mathrm{n}\:} =\:\sqrt{\mathrm{3a}_{\mathrm{n}−\mathrm{1}\:} \:−\mathrm{2}\:}\:\:\:\:\mathrm{for}\:\mathrm{n}\geqslant\mathrm{2}\:\:\mathrm{converges}\: \\ $$$$\mathrm{and}\:\mathrm{find}\:\mathrm{its}\:\mathrm{limit}. \\ $$

Question Number 213463 Answers: 1 Comments: 0

For p,q and r prime numbers satisfying { ((p(q+1)(r+1)=1064)),((r(p+1)(q+1)=1554)) :} find the value p(q+1)r

$$\:\:\mathrm{For}\:\mathrm{p},\mathrm{q}\:\mathrm{and}\:\mathrm{r}\:\mathrm{prime}\:\mathrm{numbers}\: \\ $$$$\:\:\mathrm{satisfying}\:\begin{cases}{\mathrm{p}\left(\mathrm{q}+\mathrm{1}\right)\left(\mathrm{r}+\mathrm{1}\right)=\mathrm{1064}}\\{\mathrm{r}\left(\mathrm{p}+\mathrm{1}\right)\left(\mathrm{q}+\mathrm{1}\right)=\mathrm{1554}}\end{cases} \\ $$$$\:\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{p}\left(\mathrm{q}+\mathrm{1}\right)\mathrm{r}\: \\ $$

Question Number 213467 Answers: 1 Comments: 3

$$\:\:\:\underbrace{\boldsymbol{{B}}} \\ $$

Question Number 213459 Answers: 0 Comments: 0

Find tupple natural numbers (a,b,c) such that { ((max{((a+b)/2)+((∣a−b∣)/2) , ((b+c)/2)+((∣b−c∣)/2) ,((c+a)/2)+((∣c−a∣)/2)}=a)),((min{((a+b)/2)−((∣a−b∣)/2) , ((b+c)/2)−((∣b−c∣)/2) , ((c+a)/2)−((∣c−a∣)/2)}=b)) :} where a+b+c = 10

$$\:\:\mathrm{Find}\:\mathrm{tupple}\:\mathrm{natural}\:\mathrm{numbers}\:\left(\mathrm{a},\mathrm{b},\mathrm{c}\right) \\ $$$$\:\mathrm{such}\:\mathrm{that}\: \\ $$$$\:\:\:\:\begin{cases}{\mathrm{max}\left\{\frac{\mathrm{a}+\mathrm{b}}{\mathrm{2}}+\frac{\mid\mathrm{a}−\mathrm{b}\mid}{\mathrm{2}}\:,\:\frac{\mathrm{b}+\mathrm{c}}{\mathrm{2}}+\frac{\mid\mathrm{b}−\mathrm{c}\mid}{\mathrm{2}}\:,\frac{\mathrm{c}+\mathrm{a}}{\mathrm{2}}+\frac{\mid\mathrm{c}−\mathrm{a}\mid}{\mathrm{2}}\right\}=\mathrm{a}}\\{\mathrm{min}\left\{\frac{\mathrm{a}+\mathrm{b}}{\mathrm{2}}−\frac{\mid\mathrm{a}−\mathrm{b}\mid}{\mathrm{2}}\:,\:\frac{\mathrm{b}+\mathrm{c}}{\mathrm{2}}−\frac{\mid\mathrm{b}−\mathrm{c}\mid}{\mathrm{2}}\:,\:\frac{\mathrm{c}+\mathrm{a}}{\mathrm{2}}−\frac{\mid\mathrm{c}−\mathrm{a}\mid}{\mathrm{2}}\right\}=\mathrm{b}}\end{cases} \\ $$$$\:\:\mathrm{where}\:\mathrm{a}+\mathrm{b}+\mathrm{c}\:=\:\mathrm{10} \\ $$

Question Number 213451 Answers: 1 Comments: 0

∫ (1/(z^6 −1)) dz=??

$$\int\:\:\frac{\mathrm{1}}{{z}^{\mathrm{6}} −\mathrm{1}}\:\mathrm{d}{z}=?? \\ $$

Question Number 213439 Answers: 1 Comments: 5

Question Number 213436 Answers: 2 Comments: 0

Question Number 213435 Answers: 0 Comments: 0

A ∈ M_(2×2) , A^3 = A^2 + A ⇒ det ( A −2I )=?

$$ \\ $$$$\:\:\:\:\:\:\:{A}\:\in\:\mathrm{M}_{\mathrm{2}×\mathrm{2}} \:\:,\:\:\:{A}^{\mathrm{3}} \:=\:{A}^{\mathrm{2}} \:+\:{A} \\ $$$$\:\:\:\:\:\:\:\:\Rightarrow\:\:{det}\:\left(\:{A}\:−\mathrm{2}{I}\:\right)=? \\ $$$$\:\:\:\:\:\: \\ $$

Question Number 213430 Answers: 1 Comments: 0

x,y,z ∈ R { ((x + [y] + {z} = 9,4)),(([x] + {y} + z = 11,3)),(({x} + y + [z] = 10,5)) :} find: x = ?

$$\mathrm{x},\mathrm{y},\mathrm{z}\:\in\:\mathbb{R} \\ $$$$\begin{cases}{\mathrm{x}\:+\:\left[\mathrm{y}\right]\:+\:\left\{\mathrm{z}\right\}\:=\:\mathrm{9},\mathrm{4}}\\{\left[\mathrm{x}\right]\:+\:\left\{\mathrm{y}\right\}\:+\:\mathrm{z}\:=\:\mathrm{11},\mathrm{3}}\\{\left\{\mathrm{x}\right\}\:+\:\mathrm{y}\:+\:\left[\mathrm{z}\right]\:=\:\mathrm{10},\mathrm{5}}\end{cases}\:\:\:\:\:\mathrm{find}:\:\boldsymbol{\mathrm{x}}\:=\:? \\ $$

Question Number 213428 Answers: 0 Comments: 0

Question Number 213427 Answers: 0 Comments: 0

Question Number 213423 Answers: 2 Comments: 0

⌊ (1/2)x−1⌋ + ⌊ (2/2)x−2⌋+⌊(3/2)x−3⌋+...+⌊((100)/2)x−100⌋ ≤10100 for x non negative integers. find the possible value of x

$$\:\:\lfloor\:\frac{\mathrm{1}}{\mathrm{2}}\mathrm{x}−\mathrm{1}\rfloor\:+\:\lfloor\:\frac{\mathrm{2}}{\mathrm{2}}\mathrm{x}−\mathrm{2}\rfloor+\lfloor\frac{\mathrm{3}}{\mathrm{2}}\mathrm{x}−\mathrm{3}\rfloor+...+\lfloor\frac{\mathrm{100}}{\mathrm{2}}\mathrm{x}−\mathrm{100}\rfloor\:\leqslant\mathrm{10100} \\ $$$$\:\:\mathrm{for}\:\mathrm{x}\:\mathrm{non}\:\mathrm{negative}\:\mathrm{integers}. \\ $$$$\:\:\:\mathrm{find}\:\mathrm{the}\:\mathrm{possible}\:\mathrm{value}\:\mathrm{of}\:\mathrm{x} \\ $$

Question Number 213413 Answers: 2 Comments: 0

Find: A= [(√1)] + [(√2)] + [(√3) ]+...+ [(√(323))] = ?

$$\mathrm{Find}: \\ $$$$\mathrm{A}=\:\left[\sqrt{\mathrm{1}}\right]\:+\:\left[\sqrt{\mathrm{2}}\right]\:+\:\left[\sqrt{\mathrm{3}}\:\right]+...+\:\left[\sqrt{\mathrm{323}}\right]\:=\:? \\ $$

Question Number 213417 Answers: 2 Comments: 0

a , b , c , d ∈ N a + b + c + d = 63 Find: maksimum(ab + bc + cd) = ?

$$\mathrm{a}\:,\:\mathrm{b}\:,\:\mathrm{c}\:,\:\mathrm{d}\:\in\:\mathbb{N} \\ $$$$\mathrm{a}\:+\:\mathrm{b}\:+\:\mathrm{c}\:+\:\mathrm{d}\:=\:\mathrm{63} \\ $$$$\mathrm{Find}:\:\:\:\mathrm{maksimum}\left(\mathrm{ab}\:+\:\mathrm{bc}\:+\:\mathrm{cd}\right)\:=\:? \\ $$

Question Number 213404 Answers: 0 Comments: 3

pls teach me above question ↓↓ (prove real analysis pls) and sorry Mr gaster i cant believe you answer....

$$\mathrm{pls}\:\mathrm{teach}\:\mathrm{me}\:\mathrm{above}\:\mathrm{question} \\ $$$$\downarrow\downarrow\:\left(\mathrm{prove}\:\mathrm{real}\:\mathrm{analysis}\:\mathrm{pls}\right) \\ $$$$\mathrm{and}\:\mathrm{sorry}\:\mathrm{Mr}\:\mathrm{gaster} \\ $$$$\mathrm{i}\:\mathrm{cant}\:\mathrm{believe}\:\mathrm{you}\:\mathrm{answer}.... \\ $$

Question Number 213398 Answers: 1 Comments: 0

One simple Equation pls prove this property Σ_(j=1) ^N a_j ∙Σ_(k=1) ^M b_k =Σ_(j=1) ^N ∙Σ_(k=1) ^M a_j b_k .. and Σ_(j=0) ^N f(a+((b−a)/N)j)∙((b−a)/N)∙Σ_(k=0) ^M g(a+((b−a)/M)k)∙((b−a)/M) =Σ_(j=0) ^N Σ_(k=0) ^M f(a+((b−a)/N)j)g(a+((b−a)/M)k)(((b−a)^2 )/(MN)) But..... that Sum not euqal to ∫_a ^( b) f(z)g(z)dz... why integral form dosen′t work like Summation Σ_(j=1) ^N f(j) ∙Σ_(k=1) ^M g(k)=Σ_(j=1) ^N Σ_(k=1) ^M f(j)g(k) is True But..... ∫_a ^b f(u)du∙ ∫_a ^b g(v)dv isn′t equal to ∫_a ^b f(w)g(w)dw

$$\mathrm{One}\:\mathrm{simple}\:\mathrm{Equation} \\ $$$$\mathrm{pls}\:\mathrm{prove}\:\mathrm{this}\:\mathrm{property} \\ $$$$\underset{{j}=\mathrm{1}} {\overset{{N}} {\sum}}\:{a}_{{j}} \centerdot\underset{{k}=\mathrm{1}} {\overset{{M}} {\sum}}{b}_{{k}} =\underset{{j}=\mathrm{1}} {\overset{{N}} {\sum}}\centerdot\underset{{k}=\mathrm{1}} {\overset{{M}} {\sum}}\:{a}_{{j}} {b}_{{k}} .. \\ $$$$\:\:\mathrm{and} \\ $$$$\underset{{j}=\mathrm{0}} {\overset{{N}} {\sum}}\:{f}\left({a}+\frac{{b}−{a}}{{N}}{j}\right)\centerdot\frac{{b}−{a}}{{N}}\centerdot\underset{{k}=\mathrm{0}} {\overset{{M}} {\sum}}\:\mathrm{g}\left({a}+\frac{{b}−{a}}{{M}}{k}\right)\centerdot\frac{{b}−{a}}{{M}} \\ $$$$=\underset{{j}=\mathrm{0}} {\overset{{N}} {\sum}}\:\underset{{k}=\mathrm{0}} {\overset{{M}} {\sum}}\:{f}\left({a}+\frac{{b}−{a}}{{N}}{j}\right)\mathrm{g}\left({a}+\frac{{b}−{a}}{{M}}{k}\right)\frac{\left({b}−{a}\right)^{\mathrm{2}} }{{MN}} \\ $$$$\mathrm{But}..... \\ $$$$\mathrm{that}\:\mathrm{Sum}\:\mathrm{not}\:\mathrm{euqal}\:\mathrm{to}\:\int_{{a}} ^{\:{b}} \:{f}\left({z}\right)\mathrm{g}\left({z}\right)\mathrm{d}{z}... \\ $$$$\mathrm{why}\:\mathrm{integral}\:\mathrm{form}\:\mathrm{dosen}'\mathrm{t}\:\mathrm{work} \\ $$$$\:\mathrm{like}\:\mathrm{Summation} \\ $$$$\underset{{j}=\mathrm{1}} {\overset{{N}} {\sum}}\:{f}\left({j}\right)\:\centerdot\underset{{k}=\mathrm{1}} {\overset{{M}} {\sum}}\:\mathrm{g}\left({k}\right)=\underset{{j}=\mathrm{1}} {\overset{{N}} {\sum}}\underset{{k}=\mathrm{1}} {\overset{{M}} {\sum}}\:{f}\left({j}\right)\mathrm{g}\left({k}\right) \\ $$$$\:\mathrm{is}\:\mathrm{True}\:\mathrm{But}..... \\ $$$$\int_{{a}} ^{{b}} \:{f}\left({u}\right)\mathrm{d}{u}\centerdot\:\int_{{a}} ^{{b}} \:\mathrm{g}\left({v}\right)\mathrm{d}{v}\:\:\mathrm{isn}'\mathrm{t}\:\mathrm{equal}\:\mathrm{to}\: \\ $$$$\int_{{a}} ^{{b}} \:{f}\left({w}\right)\mathrm{g}\left({w}\right)\mathrm{d}{w} \\ $$

Question Number 213397 Answers: 1 Comments: 0

Question Number 213391 Answers: 1 Comments: 0

Solve the system of equations where a,b,c≥0 a−2bc=b−2ac=c−2ab a+b+c=2

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{system}\:\mathrm{of}\:\mathrm{equations}\:\mathrm{where}\:{a},{b},{c}\geqslant\mathrm{0} \\ $$$${a}−\mathrm{2}{bc}={b}−\mathrm{2}{ac}={c}−\mathrm{2}{ab} \\ $$$${a}+{b}+{c}=\mathrm{2}\: \\ $$

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