Let - - be real and z be a complex number- If z 2- z -0 has two distinct roots on the line Re z -1- then it is necessary thatA- -1- -B- -0-1C- -1-0D- -1

The correct option is A β ϵ (1, ∞) Let the roots of the given equation be 1+ip and 1 ip, p ϵ R, p ≠ 0 ⇒β = product of roots =(1+ip)(1 ip)=1+p^2 gt;1, ∀ ...

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Byju's Answer

Standard XI

Mathematics

Complex Numbers

Let α, β be r...

Question

Let $α, β$ be real and z be a complex number. If $z^{2} + α z + β = 0$ has two distinct roots on the line Re(z) = 1, then it is necessary that

$β ϵ (1, \infty)$

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$β ϵ (0, 1)$

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$β ϵ (- 1, 0)$

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$| β | = 1$

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Solution

The correct option is A
$β ϵ (1, \infty)$

Let the roots of the given equation be $1 + i p a n d 1 - i p, p ϵ R, p \neq 0 \Rightarrow β = p r o d u c t o f r o o t s = (1 + i p) (1 - i p) = 1 + p^{2} > 1, \forall p ϵ R \Rightarrow β ϵ (1, \infty)$

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Let α,β be real and z be a complex number. If z2+αz+β=0 has two distinct roots on the line Re(z) = 1, then it is necessary that

The correct option is A β ϵ (1,∞) Let the roots of the given equation be 1+ip and 1−ip, p ϵ R, p≠0⇒β=product of roots=(1+ip)(1−ip)=1+p2>1, ∀ p ϵ R⇒β ϵ (1,∞)

Let $α, β$ be real and z be a complex number. If $z^{2} + α z + β = 0$ has two distinct roots on the line Re(z) = 1, then it is necessary that

The correct option is A
$β ϵ (1, \infty)$

Let the roots of the given equation be $1 + i p a n d 1 - i p, p ϵ R, p \neq 0 \Rightarrow β = p r o d u c t o f r o o t s = (1 + i p) (1 - i p) = 1 + p^{2} > 1, \forall p ϵ R \Rightarrow β ϵ (1, \infty)$