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Question Number 226562 Answers: 0 Comments: 0

If x + y i = ((a + i)/(a − i)) , prove that ay − 1 = x. (x+yi)(a−i)=a+i ax − xi + ayi − yi^2 = a + i (ax + y) + (ay − x)i = a + i ay − x = 1 x = ay −1

$$\mathrm{If}\:\mathrm{x}\:+\:\mathrm{y}\:{i}\:=\:\frac{\mathrm{a}\:+\:{i}}{\mathrm{a}\:−\:{i}}\:,\:\mathrm{prove}\:\mathrm{that}\:\mathrm{ay}\:−\:\mathrm{1}\:=\:\mathrm{x}. \\ $$$$\:\left(\mathrm{x}+\mathrm{y}{i}\right)\left(\mathrm{a}−{i}\right)=\mathrm{a}+{i} \\ $$$$\:\:\:\mathrm{ax}\:−\:\mathrm{x}{i}\:+\:\mathrm{ay}{i}\:−\:\mathrm{y}{i}^{\mathrm{2}} \:=\:\mathrm{a}\:+\:{i} \\ $$$$\:\:\left(\mathrm{ax}\:+\:\mathrm{y}\right)\:+\:\left(\mathrm{ay}\:−\:\mathrm{x}\right){i}\:=\:\mathrm{a}\:+\:{i} \\ $$$$\:\:\mathrm{ay}\:−\:\mathrm{x}\:=\:\mathrm{1} \\ $$$$\:\:\mathrm{x}\:=\:\mathrm{ay}\:−\mathrm{1} \\ $$

Question Number 226561 Answers: 4 Comments: 0

Question Number 226558 Answers: 1 Comments: 0

Question Number 226554 Answers: 3 Comments: 0

Question Number 226550 Answers: 3 Comments: 0

a+b+c = x lim_(x→0) ((a^3 +b^3 +c^3 )/(abc)) =?

$$\:\:\:\: {a}+{b}+{c}\:=\:{x}\: \\ $$$$\:\:\:\: \underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}} }{{abc}}\:=? \\ $$

Question Number 226542 Answers: 0 Comments: 1

Question Number 226572 Answers: 0 Comments: 0

Question Number 226569 Answers: 0 Comments: 0

a^4 + b^4 + c^4 = 2d^2 Prove that the equation has an infinite number of natural solutions

$$\mathrm{a}^{\mathrm{4}} \:+\:\mathrm{b}^{\mathrm{4}} \:+\:\mathrm{c}^{\mathrm{4}} \:=\:\mathrm{2d}^{\mathrm{2}} \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{has}\:\mathrm{an}\:\mathrm{infinite} \\ $$$$\mathrm{number}\:\mathrm{of}\:\mathrm{natural}\:\mathrm{solutions} \\ $$

Question Number 226538 Answers: 0 Comments: 0

Question Number 226536 Answers: 2 Comments: 0

If (x+(2a^2 +5))(x−(2a^2 +7)) ≤ 0 x∈[−(a^2 +8a−10) ; (a^2 +9a−11)] Find: a = ?

$$\mathrm{If}\:\:\:\left(\mathrm{x}+\left(\mathrm{2a}^{\mathrm{2}} +\mathrm{5}\right)\right)\left(\mathrm{x}−\left(\mathrm{2a}^{\mathrm{2}} +\mathrm{7}\right)\right)\:\leqslant\:\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\mathrm{x}\in\left[−\left(\mathrm{a}^{\mathrm{2}} +\mathrm{8a}−\mathrm{10}\right)\:;\:\left(\mathrm{a}^{\mathrm{2}} +\mathrm{9a}−\mathrm{11}\right)\right] \\ $$$$\mathrm{Find}:\:\boldsymbol{\mathrm{a}}\:=\:? \\ $$

Question Number 226533 Answers: 1 Comments: 0

Question Number 226534 Answers: 0 Comments: 2

Question Number 226526 Answers: 1 Comments: 0

Question Number 226525 Answers: 2 Comments: 0

Question Number 226524 Answers: 1 Comments: 0

Question Number 226515 Answers: 2 Comments: 2

Question Number 226514 Answers: 1 Comments: 0

Question Number 226513 Answers: 2 Comments: 0

Find gcd(a^2 +ab+b^2 ,ab) if gcd(a,b)=1

$${Find}\:{gcd}\left({a}^{\mathrm{2}} +{ab}+{b}^{\mathrm{2}} ,{ab}\right)\:{if}\:{gcd}\left({a},{b}\right)=\mathrm{1} \\ $$

Question Number 226507 Answers: 1 Comments: 0

Question Number 226509 Answers: 1 Comments: 0

Find: Σ_(n=1) ^∞ (1/(n∙(2n + 1)^2 )) = ?

$$\mathrm{Find}:\:\:\:\underset{\boldsymbol{\mathrm{n}}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{\mathrm{n}\centerdot\left(\mathrm{2n}\:+\:\mathrm{1}\right)^{\mathrm{2}} }\:=\:? \\ $$

Question Number 226486 Answers: 2 Comments: 3

Question Number 226485 Answers: 1 Comments: 0

Question Number 226471 Answers: 2 Comments: 2

If, x^2 +2y^2 ∞xy then prove that, 2x^2 +y^2 ∞xy

$$\:{If},\:{x}^{\mathrm{2}} +\mathrm{2}{y}^{\mathrm{2}} \infty{xy}\: \\ $$$$\:\:{then}\:{prove}\:{that},\:\mathrm{2}{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \infty{xy} \\ $$

Question Number 226469 Answers: 2 Comments: 1

Question Number 226464 Answers: 0 Comments: 2

Question Number 226455 Answers: 1 Comments: 0

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