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AlgebraQuestion and Answers: Page 79

Question Number 190882    Answers: 1   Comments: 2

Question Number 190820    Answers: 1   Comments: 0

Question Number 190802    Answers: 1   Comments: 0

log_x x=x^(5x−10) x=?

$${log}_{{x}} {x}={x}^{\mathrm{5}{x}−\mathrm{10}} \:\:\:\:\:{x}=? \\ $$

Question Number 190784    Answers: 0   Comments: 0

If , 0 ⇢ M′ ⇢^f M⇢^g M′′⇢0 is a short exact sequence and M′ , M′′ are two finitely generated R −modules then prove M is finitely generated. Hint: f , g are two R − homomorphism.

$$ \\ $$$$\:\:\mathrm{If}\:,\:\mathrm{0}\:\dashrightarrow\:\mathrm{M}'\:\overset{{f}} {\dashrightarrow}\mathrm{M}\overset{{g}} {\dashrightarrow}\mathrm{M}''\dashrightarrow\mathrm{0}\:\mathrm{is} \\ $$$$\:\:\:\:\mathrm{a}\:\mathrm{short}\:\mathrm{exact}\:\mathrm{sequence}\:\:\mathrm{and}\:\:\mathrm{M}'\:,\:\mathrm{M}''\:\mathrm{are} \\ $$$$\:\:\:\mathrm{two}\:\:\mathrm{finitely}\:\mathrm{generated}\:\:\mathrm{R}\:−\mathrm{modules} \\ $$$$\:\:\:\mathrm{then}\:\:\mathrm{prove}\:\:\:\mathrm{M}\:\:\mathrm{is}\:\:\mathrm{finitely}\:\mathrm{generated}.\: \\ $$$$\:\:\:\mathrm{Hint}:\:\:{f}\:\:,\:\:{g}\:\:\:{are}\:\:{two}\:\:\:{R}\:−\:{homomorphism}. \\ $$

Question Number 190774    Answers: 0   Comments: 0

Question Number 190765    Answers: 1   Comments: 0

Question Number 190746    Answers: 2   Comments: 0

Question Number 190705    Answers: 1   Comments: 0

Question Number 190704    Answers: 1   Comments: 0

f(x+y)=f(x)+f(y)+x∙y f(4)=10 finde f(2022)=?

$${f}\left({x}+{y}\right)={f}\left({x}\right)+{f}\left({y}\right)+{x}\centerdot{y} \\ $$$${f}\left(\mathrm{4}\right)=\mathrm{10}\:\:{finde}\:{f}\left(\mathrm{2022}\right)=? \\ $$

Question Number 190677    Answers: 0   Comments: 0

If x ∈ R a_1 ,a_2 ,a_3 , b_1 ,b_2 ,b_3 > 0 Then prove that: a_1 ^(sin^2 x) b_1 ^(cos^2 x) + a_2 ^(sin^2 x) b_2 ^(cos^2 x) + a_3 ^(sin^2 x) b_3 ^(cos^2 x) ≤ ≤ (a_1 + a_2 + a_3 )^(sin^2 x) (b_1 + b_2 + b_3 )^(cos^2 x)

$$\mathrm{If}\:\:\mathrm{x}\:\in\:\mathbb{R} \\ $$$$\:\:\:\:\:\mathrm{a}_{\mathrm{1}} ,\mathrm{a}_{\mathrm{2}} ,\mathrm{a}_{\mathrm{3}} \:,\:\mathrm{b}_{\mathrm{1}} ,\mathrm{b}_{\mathrm{2}} ,\mathrm{b}_{\mathrm{3}} \:>\:\mathrm{0} \\ $$$$\mathrm{Then}\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\mathrm{a}_{\mathrm{1}} ^{\boldsymbol{\mathrm{sin}}^{\mathrm{2}} \boldsymbol{\mathrm{x}}} \:\mathrm{b}_{\mathrm{1}} ^{\boldsymbol{\mathrm{cos}}^{\mathrm{2}} \boldsymbol{\mathrm{x}}} \:+\:\mathrm{a}_{\mathrm{2}} ^{\boldsymbol{\mathrm{sin}}^{\mathrm{2}} \boldsymbol{\mathrm{x}}} \:\mathrm{b}_{\mathrm{2}} ^{\boldsymbol{\mathrm{cos}}^{\mathrm{2}} \boldsymbol{\mathrm{x}}} \:+\:\mathrm{a}_{\mathrm{3}} ^{\boldsymbol{\mathrm{sin}}^{\mathrm{2}} \boldsymbol{\mathrm{x}}} \:\mathrm{b}_{\mathrm{3}} ^{\boldsymbol{\mathrm{cos}}^{\mathrm{2}} \boldsymbol{\mathrm{x}}} \:\leqslant \\ $$$$\leqslant\:\left(\mathrm{a}_{\mathrm{1}} +\:\mathrm{a}_{\mathrm{2}} +\:\mathrm{a}_{\mathrm{3}} \right)^{\boldsymbol{\mathrm{sin}}^{\mathrm{2}} \boldsymbol{\mathrm{x}}} \:\left(\mathrm{b}_{\mathrm{1}} +\:\mathrm{b}_{\mathrm{2}} +\:\mathrm{b}_{\mathrm{3}} \right)^{\boldsymbol{\mathrm{cos}}^{\mathrm{2}} \boldsymbol{\mathrm{x}}} \\ $$

Question Number 190659    Answers: 0   Comments: 0

Prove that: ((cos (π/9)))^(1/3) − ((cos ((2π)/9)))^(1/3) − ((cos ((4π)/9)))^(1/3) = ((3 − (3/2) (9)^(1/3) ))^(1/3)

$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\sqrt[{\mathrm{3}}]{\mathrm{cos}\:\frac{\pi}{\mathrm{9}}}\:−\:\sqrt[{\mathrm{3}}]{\mathrm{cos}\:\frac{\mathrm{2}\pi}{\mathrm{9}}}\:−\:\sqrt[{\mathrm{3}}]{\mathrm{cos}\:\frac{\mathrm{4}\pi}{\mathrm{9}}} \\ $$$$=\:\sqrt[{\mathrm{3}}]{\mathrm{3}\:−\:\frac{\mathrm{3}}{\mathrm{2}}\:\sqrt[{\mathrm{3}}]{\mathrm{9}}}\: \\ $$

Question Number 190625    Answers: 1   Comments: 0

solve in R : ⌊ (1/x) ⌋ + ⌊ x ⌋ = 2

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\mathrm{solve}\:\mathrm{in}\:\:\:\mathbb{R}\:\:\:: \\ $$$$\:\:\:\:\:\:\:\:\lfloor\:\frac{\mathrm{1}}{{x}}\:\rfloor\:\:+\:\lfloor\:{x}\:\rfloor\:=\:\mathrm{2}\:\:\:\:\: \\ $$$$ \\ $$

Question Number 190611    Answers: 0   Comments: 2

Montrer que: 1•c^2 =a^2 +b^2 2• rayon r=(c/(1+(√2)))−((a+b)/(2+(√2)))

$$\mathrm{Montrer}\:\mathrm{que}: \\ $$$$\mathrm{1}\bullet\boldsymbol{\mathrm{c}}^{\mathrm{2}} =\boldsymbol{\mathrm{a}}^{\mathrm{2}} +\boldsymbol{\mathrm{b}}^{\mathrm{2}} \\ $$$$\mathrm{2}\bullet\:\mathrm{rayon}\:\:\:\:\:\boldsymbol{\mathrm{r}}=\frac{\boldsymbol{\mathrm{c}}}{\mathrm{1}+\sqrt{\mathrm{2}}}−\frac{\boldsymbol{\mathrm{a}}+\boldsymbol{\mathrm{b}}}{\mathrm{2}+\sqrt{\mathrm{2}}} \\ $$$$ \\ $$

Question Number 190607    Answers: 0   Comments: 0

Question Number 190536    Answers: 1   Comments: 0

If p,q and r are the roots of equation x^3 −3x^2 +1 = 0 then find the value of ((3p−2))^(1/3) +((3q−2))^(1/3) +((3r−2))^(1/3)

$$\:\mathrm{If}\:\mathrm{p},\mathrm{q}\:\mathrm{and}\:\mathrm{r}\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{equation} \\ $$$$\:\mathrm{x}^{\mathrm{3}} −\mathrm{3x}^{\mathrm{2}} +\mathrm{1}\:=\:\mathrm{0}\:\mathrm{then}\:\mathrm{find}\:\mathrm{the}\:\mathrm{value} \\ $$$$\:\mathrm{of}\:\sqrt[{\mathrm{3}}]{\mathrm{3p}−\mathrm{2}}\:+\sqrt[{\mathrm{3}}]{\mathrm{3q}−\mathrm{2}}+\sqrt[{\mathrm{3}}]{\mathrm{3r}−\mathrm{2}}\: \\ $$

Question Number 190523    Answers: 1   Comments: 0

Question Number 190522    Answers: 1   Comments: 0

Question Number 190520    Answers: 2   Comments: 0

if a,b and c root of the x^3 −16x^2 −57x+1=0 thi find thd volue of a^(1/5) +b^(1/5) +c^(1/5) =?

$${if}\:{a},{b}\:{and}\:{c}\:{root}\:{of}\:{the} \\ $$$${x}^{\mathrm{3}} −\mathrm{16}{x}^{\mathrm{2}} −\mathrm{57}{x}+\mathrm{1}=\mathrm{0} \\ $$$${thi}\:{find}\:{thd}\:{volue}\:{of} \\ $$$${a}^{\frac{\mathrm{1}}{\mathrm{5}}} +{b}^{\frac{\mathrm{1}}{\mathrm{5}}} +{c}^{\frac{\mathrm{1}}{\mathrm{5}}} =? \\ $$

Question Number 190448    Answers: 1   Comments: 0

Question Number 190435    Answers: 2   Comments: 0

how is solution lim_(x→0) ((x^(10) ∙sin^4 x∙cos^8 x∙(x+1)^3 )/(x^4 +3x^3 +3x^2 +x))=?

$$ \\ $$$$\mathrm{how}\:\mathrm{is}\:\mathrm{solution} \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{x}^{\mathrm{10}} \centerdot\mathrm{sin}^{\mathrm{4}} \mathrm{x}\centerdot\mathrm{cos}^{\mathrm{8}} \mathrm{x}\centerdot\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{3}} }{\mathrm{x}^{\mathrm{4}} +\mathrm{3x}^{\mathrm{3}} +\mathrm{3x}^{\mathrm{2}} +\mathrm{x}}=? \\ $$$$ \\ $$

Question Number 190407    Answers: 2   Comments: 0

Question Number 190392    Answers: 1   Comments: 1

If a + b = 3 Find: ((a^2 + b^2 − 2a − 2b)/(a^2 − b^2 − 4a + 4))

$$\mathrm{If}\:\:\:\mathrm{a}\:+\:\mathrm{b}\:=\:\mathrm{3} \\ $$$$\mathrm{Find}:\:\:\:\frac{\mathrm{a}^{\mathrm{2}} \:+\:\mathrm{b}^{\mathrm{2}} \:−\:\mathrm{2a}\:−\:\mathrm{2b}}{\mathrm{a}^{\mathrm{2}} \:−\:\mathrm{b}^{\mathrm{2}} \:−\:\mathrm{4a}\:+\:\mathrm{4}} \\ $$

Question Number 190357    Answers: 0   Comments: 1

Question Number 190325    Answers: 1   Comments: 0

Question Number 190324    Answers: 1   Comments: 1

Question Number 190321    Answers: 0   Comments: 1

What is the length x of triangle equilateral A^′ B′C′,such that Area(triangle ABC)=Area(triangle A′B^′ C′)

$${What}\:{is}\:{the}\:{length}\:\:\boldsymbol{{x}}\:\:{of}\:{triangle} \\ $$$${equilateral}\:{A}^{'} {B}'{C}',{such}\:{that} \\ $$$${Area}\left({triangle}\:{ABC}\right)={Area}\left({triangle}\:{A}'{B}^{'} {C}'\right) \\ $$$$ \\ $$

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