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Question Number 150376 Answers: 1 Comments: 2

Question Number 150374 Answers: 1 Comments: 0

Σ_(k=0) ^∞ ((2^k + 3^k )/5^k ) = ?

$$\underset{\boldsymbol{\mathrm{k}}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{\mathrm{2}^{\boldsymbol{\mathrm{k}}} \:+\:\mathrm{3}^{\boldsymbol{\mathrm{k}}} }{\mathrm{5}^{\boldsymbol{\mathrm{k}}} }\:\:=\:? \\ $$

Question Number 150372 Answers: 0 Comments: 1

Question Number 150450 Answers: 0 Comments: 0

show the?connection between the beta distribution(n,p) and hypergeometric distribution(N,k,n)in a limiting case

$$\mathrm{show}\:\mathrm{the}?\mathrm{connection}\:\mathrm{between}\:\mathrm{the} \\ $$$$\mathrm{beta}\:\mathrm{distribution}\left(\mathrm{n},\mathrm{p}\right)\:\mathrm{and}\:\mathrm{hypergeometric} \\ $$$$\mathrm{distribution}\left(\mathrm{N},\mathrm{k},\mathrm{n}\right)\mathrm{in}\:\mathrm{a}\:\mathrm{limiting}\:\mathrm{case} \\ $$

Question Number 150366 Answers: 0 Comments: 1

Question Number 150405 Answers: 2 Comments: 0

Question Number 150404 Answers: 4 Comments: 0

Ω =∫_( 0) ^( ∞) ((log(x))/((1 + x^2 )^2 )) dx = ?

$$\Omega\:=\underset{\:\mathrm{0}} {\overset{\:\infty} {\int}}\:\frac{\mathrm{log}\left(\mathrm{x}\right)}{\left(\mathrm{1}\:+\:\mathrm{x}^{\mathrm{2}} \right)^{\mathrm{2}} }\:\mathrm{dx}\:=\:? \\ $$

Question Number 150402 Answers: 1 Comments: 1

For m≥1 𝛀 =∫_( 0) ^( ∞) ((x ln^m (x))/(e^x − 1)) = 2 ln^m 𝛇(3)

$$\mathrm{For}\:\:\mathrm{m}\geqslant\mathrm{1} \\ $$$$\boldsymbol{\Omega}\:=\underset{\:\mathrm{0}} {\overset{\:\infty} {\int}}\:\frac{\mathrm{x}\:\mathrm{ln}^{\boldsymbol{\mathrm{m}}} \:\left(\mathrm{x}\right)}{\mathrm{e}^{\boldsymbol{\mathrm{x}}} \:−\:\mathrm{1}}\:=\:\mathrm{2}\:\mathrm{ln}^{\boldsymbol{\mathrm{m}}} \:\boldsymbol{\zeta}\left(\mathrm{3}\right) \\ $$

Question Number 150400 Answers: 0 Comments: 0

Question Number 150363 Answers: 2 Comments: 5

Question Number 150469 Answers: 1 Comments: 0

If f(x) = (4^x /(4^x + 2)) find f((1/(17))) + f(((16)/(17))) =^(?)

$$\mathrm{If}\:\:\:\mathrm{f}\left(\mathrm{x}\right)\:=\:\frac{\mathrm{4}^{\boldsymbol{\mathrm{x}}} }{\mathrm{4}^{\boldsymbol{\mathrm{x}}} \:+\:\mathrm{2}}\:\:\:\mathrm{find}\:\:\:\mathrm{f}\left(\frac{\mathrm{1}}{\mathrm{17}}\right)\:+\:\mathrm{f}\left(\frac{\mathrm{16}}{\mathrm{17}}\right)\:\overset{?} {=} \\ $$

Question Number 150351 Answers: 1 Comments: 2

Question Number 150350 Answers: 0 Comments: 0

Question Number 150346 Answers: 0 Comments: 2

If x;y and z are positive integers, then determine the smallest positive integer N=x+y+z+xy+yz+zx, which is bigger than 2022.

$$\mathrm{If}\:\:\boldsymbol{\mathrm{x}};\boldsymbol{\mathrm{y}}\:\mathrm{and}\:\boldsymbol{\mathrm{z}}\:\mathrm{are}\:\mathrm{positive}\:\mathrm{integers},\:\mathrm{then} \\ $$$$\mathrm{determine}\:\mathrm{the}\:\mathrm{smallest}\:\mathrm{positive}\:\mathrm{integer} \\ $$$$\boldsymbol{\mathrm{N}}=\mathrm{x}+\mathrm{y}+\mathrm{z}+\mathrm{xy}+\mathrm{yz}+\mathrm{zx},\:\mathrm{which}\:\mathrm{is} \\ $$$$\mathrm{bigger}\:\mathrm{than}\:\mathrm{2022}. \\ $$

Question Number 150332 Answers: 0 Comments: 0

Question Number 150331 Answers: 0 Comments: 0

ultimately Q.149894 boils down to finding s_(max) , s_(min) ∀ {h−scos (θ−(π/6))}^2 +{k−ssin (θ+(π/6))}^2 =r^2

$${ultimately}\:\:{Q}.\mathrm{149894}\:\:{boils}\: \\ $$$${down}\:{to}\:{finding}\:{s}_{{max}} ,\:{s}_{{min}} \:\forall \\ $$$$\:\:\:\:\left\{{h}−{s}\mathrm{cos}\:\left(\theta−\frac{\pi}{\mathrm{6}}\right)\right\}^{\mathrm{2}} \\ $$$$+\left\{{k}−{s}\mathrm{sin}\:\left(\theta+\frac{\pi}{\mathrm{6}}\right)\right\}^{\mathrm{2}} ={r}^{\mathrm{2}} \\ $$$$ \\ $$

Question Number 150328 Answers: 0 Comments: 0

Find all the solutions to the equation: x^3 − y^3 = xy + 61

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{the}\:\mathrm{solutions}\:\mathrm{to}\:\mathrm{the}\:\mathrm{equation}: \\ $$$$\boldsymbol{\mathrm{x}}^{\mathrm{3}} \:−\:\boldsymbol{\mathrm{y}}^{\mathrm{3}} \:=\:\boldsymbol{\mathrm{xy}}\:+\:\mathrm{61} \\ $$

Question Number 150327 Answers: 2 Comments: 0

1) 33x ≡ 48 (mod 654) 2) 5^(1000 000) ≡ x (mod 41) Find x=?

$$\left.\mathrm{1}\right)\:\mathrm{33}\boldsymbol{\mathrm{x}}\:\:\equiv\:\:\mathrm{48}\:\left(\mathrm{mod}\:\mathrm{654}\right) \\ $$$$\left.\mathrm{2}\right)\:\mathrm{5}^{\mathrm{1000}\:\mathrm{000}} \:\:\equiv\:\:\boldsymbol{\mathrm{x}}\:\left(\mathrm{mod}\:\mathrm{41}\right) \\ $$$$\mathrm{Find}\:\:\boldsymbol{\mathrm{x}}=? \\ $$

Question Number 150326 Answers: 0 Comments: 2

Find ax^5 +by^5 if the real numbers a,b,x and y satisf the equations: ax + by = 3 ax^2 + by^2 = 7 ax^3 + by^3 = 16 ax^4 + by^4 = 42

$$\mathrm{Find}\:\:\boldsymbol{\mathrm{ax}}^{\mathrm{5}} +\boldsymbol{\mathrm{by}}^{\mathrm{5}} \:\:\mathrm{if}\:\mathrm{the}\:\mathrm{real}\:\mathrm{numbers} \\ $$$$\boldsymbol{\mathrm{a}},\boldsymbol{\mathrm{b}},\boldsymbol{\mathrm{x}}\:\:\mathrm{and}\:\:\boldsymbol{\mathrm{y}}\:\:\mathrm{satisf}\:\mathrm{the}\:\mathrm{equations}: \\ $$$$\mathrm{ax}\:+\:\mathrm{by}\:=\:\mathrm{3} \\ $$$$\mathrm{ax}^{\mathrm{2}} \:+\:\mathrm{by}^{\mathrm{2}} \:=\:\mathrm{7} \\ $$$$\mathrm{ax}^{\mathrm{3}} \:+\:\mathrm{by}^{\mathrm{3}} \:=\:\mathrm{16} \\ $$$$\mathrm{ax}^{\mathrm{4}} \:+\:\mathrm{by}^{\mathrm{4}} \:=\:\mathrm{42} \\ $$

Question Number 150325 Answers: 2 Comments: 0

Solve..... 𝛗= ∫_0 ^( (π/2)) cos^( 2) (x ). ln (cot( x ))dx=? .....m.n.....

$$\:\:\:\: \\ $$$$\:\:\:\:\:\:\mathrm{Solve}..... \\ $$$$\:\:\:\:\:\:\:\:\: \\ $$$$\:\boldsymbol{\phi}=\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} {cos}^{\:\mathrm{2}} \left({x}\:\right).\:{ln}\:\left({cot}\left(\:{x}\:\right)\right){dx}=?\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:.....{m}.{n}..... \\ $$

Question Number 150323 Answers: 0 Comments: 0

a pmf of a random variable Xis given as f(x)=(e^(−10) . 10^x )/x! X=0 , 1 ,2 ... find P(x<16)

$$\mathrm{a}\:\mathrm{pmf}\:\mathrm{of}\:\mathrm{a}\:\mathrm{random}\:\mathrm{variable}\:\mathrm{Xis}\:\mathrm{given}\:\mathrm{as} \\ $$$$\mathrm{f}\left(\mathrm{x}\right)=\left(\mathrm{e}^{−\mathrm{10}} .\:\mathrm{10}^{\mathrm{x}} \right)/\mathrm{x}!\:\:\mathrm{X}=\mathrm{0}\:,\:\mathrm{1}\:,\mathrm{2}\:...\:\mathrm{find}\: \\ $$$$\mathrm{P}\left(\mathrm{x}<\mathrm{16}\right) \\ $$

Question Number 150313 Answers: 2 Comments: 0

Question Number 150312 Answers: 0 Comments: 0

Find n lowest integers that divide (1^2 + 2^2 + 3^2 + …+ n^2 ) .

$${Find}\:\:{n}\:\:{lowest}\:\:{integers}\:\:{that}\:\:{divide}\:\:\left(\mathrm{1}^{\mathrm{2}} \:+\:\mathrm{2}^{\mathrm{2}} \:+\:\mathrm{3}^{\mathrm{2}} \:+\:\ldots+\:{n}^{\mathrm{2}} \right)\:. \\ $$

Question Number 150308 Answers: 1 Comments: 1

Question Number 150305 Answers: 1 Comments: 1

Question Number 150303 Answers: 1 Comments: 0

If x;y;z∈[0;∞) then: 2^x +2^y +2^z +2^(x+y+z) ≥ 4^(√(xy)) +4^(√(yz)) +4^(√(zx)) +1

$$\mathrm{If}\:\:\:\mathrm{x};\mathrm{y};\mathrm{z}\in\left[\mathrm{0};\infty\right)\:\:\mathrm{then}: \\ $$$$\mathrm{2}^{\boldsymbol{\mathrm{x}}} +\mathrm{2}^{\boldsymbol{\mathrm{y}}} +\mathrm{2}^{\boldsymbol{\mathrm{z}}} +\mathrm{2}^{\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{y}}+\boldsymbol{\mathrm{z}}} \:\geqslant\:\mathrm{4}^{\sqrt{\boldsymbol{\mathrm{xy}}}} +\mathrm{4}^{\sqrt{\boldsymbol{\mathrm{yz}}}} +\mathrm{4}^{\sqrt{\boldsymbol{\mathrm{zx}}}} +\mathrm{1} \\ $$

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