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AlgebraQuestion and Answers: Page 46

Question Number 205116 Answers: 2 Comments: 0

let x^2 −3x+p = 0 has two positive roots ′a′ and ′b′ then inf((4/a)+(1/b)) is

$$\:\:\mathrm{let}\:\mathrm{x}^{\mathrm{2}} −\mathrm{3x}+\mathrm{p}\:=\:\mathrm{0}\:\mathrm{has}\:\mathrm{two}\:\mathrm{positive}\:\mathrm{roots} \\ $$$$\:'\mathrm{a}'\:\mathrm{and}\:'\mathrm{b}'\:\mathrm{then}\:\:\mathrm{inf}\left(\frac{\mathrm{4}}{\mathrm{a}}+\frac{\mathrm{1}}{\mathrm{b}}\right)\:\mathrm{is}\: \\ $$

Question Number 205107 Answers: 0 Comments: 2

y = log_2 (sin(x)+cos(x)) ⇒ R_y = ?(Range )

$$ \\ $$$$\:\:\:\:{y}\:=\:{log}_{\mathrm{2}} \left({sin}\left({x}\right)+{cos}\left({x}\right)\right) \\ $$$$\:\:\:\Rightarrow\:\:{R}_{{y}} \:=\:?\left({Range}\:\right) \\ $$$$ \\ $$

Question Number 205101 Answers: 1 Comments: 0

given that there are real constant a,b, c, d such the identity λx^2 +2xy+y^2 = (ax+by)^2 +(cx+dy)^2 holds for all x,y ∈ R this implies (a) λ=−5 (b) λ≥1 (c)0<λ<1 (d) there is no such λ∈R

$$\:\:\mathrm{given}\:\mathrm{that}\:\mathrm{there}\:\mathrm{are}\:\mathrm{real}\:\mathrm{constant}\:\mathrm{a},\mathrm{b},\:\mathrm{c},\:\mathrm{d} \\ $$$$\:\:\mathrm{such}\:\mathrm{the}\:\mathrm{identity} \\ $$$$\:\lambda\mathrm{x}^{\mathrm{2}} +\mathrm{2xy}+\mathrm{y}^{\mathrm{2}} =\:\left(\mathrm{ax}+\mathrm{by}\right)^{\mathrm{2}} +\left(\mathrm{cx}+\mathrm{dy}\right)^{\mathrm{2}} \:\mathrm{holds} \\ $$$$\:\mathrm{for}\:\mathrm{all}\:\mathrm{x},\mathrm{y}\:\in\:\mathbb{R}\:\mathrm{this}\:\mathrm{implies} \\ $$$$\left({a}\right)\:\lambda=−\mathrm{5}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left({b}\right)\:\lambda\geqslant\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\:\:\left({c}\right)\mathrm{0}<\lambda<\mathrm{1} \\ $$$$\:\left({d}\right)\:\mathrm{there}\:\mathrm{is}\:\mathrm{no}\:\mathrm{such}\:\lambda\in\mathbb{R} \\ $$

Question Number 205091 Answers: 0 Comments: 0

f:z ⇒ z f:z ⇒ z_n f:z_n ⇒ z_n How many homomorphism can be define

$${f}:{z}\:\Rightarrow\:{z} \\ $$$${f}:{z}\:\Rightarrow\:{z}_{{n}} \\ $$$${f}:{z}_{{n}} \Rightarrow\:{z}_{{n}} \\ $$$${How}\:{many}\:{homomorphism}\:{can}\:{be}\:{define} \\ $$

Question Number 205083 Answers: 1 Comments: 0

Question Number 205073 Answers: 6 Comments: 0

if a, b, c are the roots of f(x)=x^3 −2024x^2 +2024x+2024 find (1/(1−a^2 ))+(1/(1−b^2 ))+(1/(1−c^2 ))=?

$${if}\:{a},\:{b},\:{c}\:{are}\:{the}\:{roots}\:{of} \\ $$$${f}\left({x}\right)={x}^{\mathrm{3}} −\mathrm{2024}{x}^{\mathrm{2}} +\mathrm{2024}{x}+\mathrm{2024} \\ $$$${find}\:\frac{\mathrm{1}}{\mathrm{1}−{a}^{\mathrm{2}} }+\frac{\mathrm{1}}{\mathrm{1}−{b}^{\mathrm{2}} }+\frac{\mathrm{1}}{\mathrm{1}−{c}^{\mathrm{2}} }=? \\ $$

Question Number 205092 Answers: 2 Comments: 0

Question Number 205051 Answers: 2 Comments: 0

Find all values of k such that the expression x^3 + kx^2 −7x+6 can be resolved into three linear real factors.

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{values}\:\mathrm{of}\:\:\mathrm{k}\:\mathrm{such}\:\mathrm{that}\:\mathrm{the} \\ $$$$\mathrm{expr}{e}\mathrm{ssion}\:\mathrm{x}^{\mathrm{3}} +\:\mathrm{kx}^{\mathrm{2}} −\mathrm{7x}+\mathrm{6}\:\mathrm{can}\:\mathrm{be} \\ $$$$\mathrm{re}{s}\mathrm{olved}\:\mathrm{into}\:\mathrm{three}\:\mathrm{linear}\:\mathrm{real}\:\mathrm{factors}. \\ $$

Question Number 205053 Answers: 0 Comments: 1

Question Number 205021 Answers: 2 Comments: 0

x^2 + 5x +6 = 0 & x^2 + kx + 1 = 0 have a common root then k = ?

$${x}^{\mathrm{2}} \:+\:\mathrm{5}{x}\:+\mathrm{6}\:=\:\mathrm{0}\:\&\:{x}^{\mathrm{2}} \:+\:{kx}\:+\:\mathrm{1}\:=\:\mathrm{0}\:{have}\:{a}\: \\ $$$${common}\:{root}\:\mathrm{then}\:\:{k}\:=\:? \\ $$

Question Number 205018 Answers: 1 Comments: 2

For what value of ′k′ can be expression x^3 + kx^2 −7x +6 be resolved into three linear factors? (a) 0 (b) 1 (c) 2 (d) 3

$$\mathrm{For}\:\mathrm{what}\:\mathrm{value}\:\mathrm{of}\:\:'\mathrm{k}'\:\mathrm{can}\:\mathrm{be}\:\mathrm{expression}\:{x}^{\mathrm{3}} \:+\:{kx}^{\mathrm{2}} \:−\mathrm{7}{x}\:+\mathrm{6}\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\mathrm{be}\:\mathrm{resolved}\:\mathrm{into}\:\mathrm{three}\:\mathrm{linear}\:\mathrm{factors}? \\ $$$$\left(\mathrm{a}\right)\:\mathrm{0}\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{b}\right)\:\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{c}\right)\:\mathrm{2}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{d}\right)\:\mathrm{3} \\ $$

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

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

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