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AlgebraQuestion and Answers: Page 46

Question Number 204999    Answers: 2   Comments: 0

Solve for x∈C x^3 +(4−3i)x^2 −(51+49i)x−442+170i=0

$$\mathrm{Solve}\:\mathrm{for}\:{x}\in\mathbb{C} \\ $$$${x}^{\mathrm{3}} +\left(\mathrm{4}−\mathrm{3i}\right){x}^{\mathrm{2}} −\left(\mathrm{51}+\mathrm{49i}\right){x}−\mathrm{442}+\mathrm{170i}=\mathrm{0} \\ $$

Question Number 204926    Answers: 0   Comments: 2

Prove that in any △ABC (m_a + m_b + m_c )^2 ≥ 9(√3) F

$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{in}\:\mathrm{any}\:\bigtriangleup\mathrm{ABC} \\ $$$$\left(\mathrm{m}_{\boldsymbol{\mathrm{a}}} \:+\:\mathrm{m}_{\boldsymbol{\mathrm{b}}} \:+\:\mathrm{m}_{\boldsymbol{\mathrm{c}}} \right)^{\mathrm{2}} \:\geqslant\:\mathrm{9}\sqrt{\mathrm{3}}\:\mathrm{F} \\ $$

Question Number 204920    Answers: 1   Comments: 0

16^(y+x^2 ) + 16^(y^2 +x) = 1 x+y =?

$$\:\:\:\:\:\mathrm{16}^{\mathrm{y}+\mathrm{x}^{\mathrm{2}} } \:+\:\mathrm{16}^{\mathrm{y}^{\mathrm{2}} +\mathrm{x}} \:=\:\mathrm{1}\: \\ $$$$\:\:\:\:\mathrm{x}+\mathrm{y}\:=? \\ $$

Question Number 204916    Answers: 0   Comments: 8

How many axes of symmetry does an open angle have?

$$ \\ $$How many axes of symmetry does an open angle have?

Question Number 204869    Answers: 1   Comments: 0

How many distinct positive integer valued solution exist the equation (x^2 − 7x + 11)^((x^2 −13x + 42)) = 1 (a) 2 (b) 4 (c) 6 (d) 8

$$\mathrm{How}\:\mathrm{many}\:\mathrm{distinct}\:\mathrm{positive}\:\mathrm{integer}\:\mathrm{valued}\:\mathrm{solution}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\mathrm{exist}\:\mathrm{the}\:\mathrm{equation}\:\left({x}^{\mathrm{2}} \:−\:\mathrm{7}{x}\:+\:\mathrm{11}\right)^{\left({x}^{\mathrm{2}} \:−\mathrm{13}{x}\:+\:\mathrm{42}\right)} \:=\:\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\left(\mathrm{a}\right)\:\mathrm{2}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{b}\right)\:\mathrm{4}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{c}\right)\:\mathrm{6}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{d}\right)\:\mathrm{8} \\ $$

Question Number 204853    Answers: 2   Comments: 0

f(x)=(√(1−log_((2x+5)) ((x+1)^2 ))) Find the domain of this function

$${f}\left({x}\right)=\sqrt{\mathrm{1}−\mathrm{log}_{\left(\mathrm{2}{x}+\mathrm{5}\right)} \left(\left({x}+\mathrm{1}\right)^{\mathrm{2}} \right)} \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{domain}\:\mathrm{of}\:\mathrm{this}\:\mathrm{function} \\ $$

Question Number 204815    Answers: 2   Comments: 0

Given (3p^2 −p+q^3 )^(12) , find the coefficient of p^(10) q^6

$$\:\:\:\:\:\mathrm{Given}\:\left(\mathrm{3p}^{\mathrm{2}} −\mathrm{p}+\mathrm{q}^{\mathrm{3}} \right)^{\mathrm{12}} \:,\:\mathrm{find}\:\mathrm{the}\: \\ $$$$\:\:\:\:\mathrm{coefficient}\:\mathrm{of}\:\mathrm{p}^{\mathrm{10}} \mathrm{q}^{\mathrm{6}} \\ $$

Question Number 204805    Answers: 0   Comments: 5

Why lim_(n→∞) (x^n u_n )=0 ,When u_n is bounded

$$\mathrm{Why}\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\left({x}^{{n}} {u}_{{n}} \right)=\mathrm{0}\:,\mathrm{When}\:{u}_{{n}} \:\mathrm{is}\:\mathrm{bounded}\: \\ $$

Question Number 204756    Answers: 1   Comments: 23

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Question Number 204754    Answers: 1   Comments: 0

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Question Number 204739    Answers: 2   Comments: 0

Question Number 204742    Answers: 1   Comments: 4

Solve for real x

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{real}\:{x} \\ $$

Question Number 204733    Answers: 2   Comments: 0

Find: ((59^2 + 48^2 + 41^2 − 30^2 )/(68^2 + 52^2 + 32^2 − 48^2 )) = ?

$$\mathrm{Find}:\:\:\:\frac{\mathrm{59}^{\mathrm{2}} \:+\:\mathrm{48}^{\mathrm{2}} \:+\:\mathrm{41}^{\mathrm{2}} \:−\:\mathrm{30}^{\mathrm{2}} }{\mathrm{68}^{\mathrm{2}} \:+\:\mathrm{52}^{\mathrm{2}} \:+\:\mathrm{32}^{\mathrm{2}} \:−\:\mathrm{48}^{\mathrm{2}} }\:=\:? \\ $$

Question Number 204712    Answers: 5   Comments: 1

Find all real solution (√(3x^2 +x−1)) +(√(x^2 −2x−3)) = (√(3x^2 +3x+5)) + (√(x^2 +3))

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{real}\:\mathrm{solution}\: \\ $$$$\:\:\:\:\sqrt{\mathrm{3x}^{\mathrm{2}} +\mathrm{x}−\mathrm{1}}\:+\sqrt{\mathrm{x}^{\mathrm{2}} −\mathrm{2x}−\mathrm{3}}\:=\: \\ $$$$\:\:\sqrt{\mathrm{3x}^{\mathrm{2}} +\mathrm{3x}+\mathrm{5}}\:+\:\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{3}}\: \\ $$

Question Number 204702    Answers: 1   Comments: 0

prove that : cl(Q×Q )=^? R^2 note: (X ,d ) is a metric space , A ⊆ X : x∈ A^( −) =cl(A) ⇔ ∀ r >0 , B_r (x) ∩ A ≠ φ

$$ \\ $$$$\:\:\:\mathrm{prove}\:\mathrm{that}\:: \\ $$$$\:\:\:\:\:\:\:\mathrm{cl}\left(\mathbb{Q}×\mathbb{Q}\:\right)\overset{?} {=}\:\mathbb{R}^{\mathrm{2}} \\ $$$$\:\:\:\:\:\:{note}:\:\:\:\left({X}\:,{d}\:\right)\:{is}\:{a}\:{metric}\:{space} \\ $$$$\:\:\:\:\:\:\:\:\:\:,\:\:\:{A}\:\subseteq\:{X}\::\:\:\:\:\:{x}\in\:\overset{\:\:−} {{A}}=\mathrm{cl}\left({A}\right)\:\Leftrightarrow\:\forall\:{r}\:>\mathrm{0}\:,\:{B}_{{r}} \:\left({x}\right)\:\cap\:{A}\:\neq\:\phi \\ $$

Question Number 204701    Answers: 3   Comments: 0

Question Number 204666    Answers: 0   Comments: 0

this is a closed curve: f(θ)=(e^(iθ) )^((e^(iθ) )) =e^(−θsin θ) e^(iθcos θ) ; −π<θ≤π f: { ((x(θ)=e^(−θsin θ) cos (θcos θ))),((y(θ)=e^(−θsin θ) sin (θcos θ))) :} find the area

$$\mathrm{this}\:\mathrm{is}\:\mathrm{a}\:\mathrm{closed}\:\mathrm{curve}: \\ $$$${f}\left(\theta\right)=\left(\mathrm{e}^{\mathrm{i}\theta} \right)^{\left(\mathrm{e}^{\mathrm{i}\theta} \right)} =\mathrm{e}^{−\theta\mathrm{sin}\:\theta} \mathrm{e}^{\mathrm{i}\theta\mathrm{cos}\:\theta} ;\:−\pi<\theta\leqslant\pi \\ $$$${f}:\:\begin{cases}{{x}\left(\theta\right)=\mathrm{e}^{−\theta\mathrm{sin}\:\theta} \mathrm{cos}\:\left(\theta\mathrm{cos}\:\theta\right)}\\{{y}\left(\theta\right)=\mathrm{e}^{−\theta\mathrm{sin}\:\theta} \mathrm{sin}\:\left(\theta\mathrm{cos}\:\theta\right)}\end{cases} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{area} \\ $$

Question Number 204664    Answers: 2   Comments: 0

8+(√(8^2 +(√(8^4 +(√(8^8 +(√(8^(16) +(√(...)))))))))) = ?

$$\:\:\mathrm{8}+\sqrt{\mathrm{8}^{\mathrm{2}} +\sqrt{\mathrm{8}^{\mathrm{4}} +\sqrt{\mathrm{8}^{\mathrm{8}} +\sqrt{\mathrm{8}^{\mathrm{16}} +\sqrt{...}}}}}\:=\:?\: \\ $$

Question Number 204658    Answers: 2   Comments: 0

If a = (9)^(1/3) − (3)^(1/3) + 1 Find (((4 − a)/a))^6 = ?

$$\mathrm{If}\:\:\:\mathrm{a}\:=\:\sqrt[{\mathrm{3}}]{\mathrm{9}}\:−\:\sqrt[{\mathrm{3}}]{\mathrm{3}}\:+\:\mathrm{1} \\ $$$$\mathrm{Find}\:\:\:\left(\frac{\mathrm{4}\:−\:\mathrm{a}}{\mathrm{a}}\right)^{\mathrm{6}} =\:? \\ $$

Question Number 204647    Answers: 4   Comments: 0

Question Number 204632    Answers: 3   Comments: 0

Question Number 204621    Answers: 3   Comments: 0

a , b , c ∈ R^+ If (√a) + (√b) + (√c) = 1 Prove that: a + b + c ≥ (1/3)

$$\mathrm{a}\:,\:\mathrm{b}\:,\:\mathrm{c}\:\in\:\mathbb{R}^{+} \\ $$$$\mathrm{If}\:\:\:\sqrt{\mathrm{a}}\:+\:\sqrt{\mathrm{b}}\:+\:\sqrt{\mathrm{c}}\:=\:\mathrm{1} \\ $$$$\mathrm{Prove}\:\mathrm{that}:\:\:\:\mathrm{a}\:+\:\mathrm{b}\:+\:\mathrm{c}\:\geqslant\:\frac{\mathrm{1}}{\mathrm{3}} \\ $$

Question Number 204610    Answers: 1   Comments: 0

If , f(x) = { (( 2^(2x) − log_3 ( x+3 ) ; x ≥5)),(( f (1+ x ) −4 ; x < 5)) :} ⇒ f (0 )= ?

$$ \\ $$$$\:\:\:{If}\:,\:\:\:{f}\left({x}\right)\:=\:\begin{cases}{\:\mathrm{2}^{\mathrm{2}{x}} −\:{log}_{\mathrm{3}} \:\left(\:{x}+\mathrm{3}\:\right)\:\:\:\:;\:\:\:{x}\:\geqslant\mathrm{5}}\\{\:{f}\:\left(\mathrm{1}+\:{x}\:\right)\:\:−\mathrm{4}\:\:\:\:\:\:\:\:\:\:\:\:\:\:;\:\:{x}\:<\:\mathrm{5}}\end{cases}\:\:\:\:\: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\Rightarrow\:\:{f}\:\left(\mathrm{0}\:\right)=\:? \\ $$$$ \\ $$

Question Number 204590    Answers: 1   Comments: 0

Question Number 204583    Answers: 1   Comments: 0

For z = a − bi If (∣z∣ − z)∙(∣z∣ + z^(−) ) = 4bi Find ∣z∣ = ?

$$\mathrm{For}\:\:\:\mathrm{z}\:=\:\mathrm{a}\:−\:\mathrm{bi} \\ $$$$\mathrm{If}\:\:\:\left(\mid\mathrm{z}\mid\:−\:\mathrm{z}\right)\centerdot\left(\mid\mathrm{z}\mid\:+\:\overline {\mathrm{z}}\right)\:=\:\mathrm{4bi} \\ $$$$\mathrm{Find}\:\:\:\mid\mathrm{z}\mid\:=\:? \\ $$

Question Number 204545    Answers: 2   Comments: 0

If a = (1/2^2 ) + (1/3^2 ) + ... + (1/(100^2 )) b = 0,99 Prove that: a < b

$$\mathrm{If} \\ $$$$\mathrm{a}\:=\:\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{2}} }\:\:+\:\:\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{2}} }\:\:+\:\:...\:\:+\:\:\frac{\mathrm{1}}{\mathrm{100}^{\mathrm{2}} } \\ $$$$\mathrm{b}\:=\:\mathrm{0},\mathrm{99} \\ $$$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\mathrm{a}\:<\:\mathrm{b} \\ $$

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