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Let α and β be real numbers and z be a complex number. If z2 + αz + β = 0 has two distinct non-real roots with real roots Re(z) = 1, then it is necessary that
  • a)
    β ∈ (−1,0)
  • b)
    β ∈ (1, ∞)
  • c)
    |β| = 1
  • d)
    β ∈ (0, 1)
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
Let α and β be real numbers and zbe a complex number. If z2...
Let z = x + iy
(x + iy)2+ α(x + iy) + β = 0
⇒ x2 − y2 + 2ixy + αx + iαy + β = 0
Equating real and imaginary parts separately, we get
x2 − y2 + αx + β = 0 , (2x + α)y = 0
Now, 2x + α = 0 (∵ y = 0)
⇒ α = −2 (∵ x = Re z = 1)
Now, 1 − y2 − 2 + β = 0
⇒ β = 1 + y2 > 1 (∵ y ∈ R, y ≠ 0)
⇒ β ∈ (1, ∞)
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Let α and β be real numbers and zbe a complex number. If z2 +αz + β = 0 has two distinct non-real roots with real roots Re(z) = 1, then it is necessary thata)β ∈ (−1,0)b)β ∈ (1, ∞)c)|β| = 1d)β ∈ (0, 1)Correct answer is option 'C'. Can you explain this answer?
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Let α and β be real numbers and zbe a complex number. If z2 +αz + β = 0 has two distinct non-real roots with real roots Re(z) = 1, then it is necessary thata)β ∈ (−1,0)b)β ∈ (1, ∞)c)|β| = 1d)β ∈ (0, 1)Correct answer is option 'C'. Can you explain this answer? for Defence 2024 is part of Defence preparation. The Question and answers have been prepared according to the Defence exam syllabus. Information about Let α and β be real numbers and zbe a complex number. If z2 +αz + β = 0 has two distinct non-real roots with real roots Re(z) = 1, then it is necessary thata)β ∈ (−1,0)b)β ∈ (1, ∞)c)|β| = 1d)β ∈ (0, 1)Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Defence 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let α and β be real numbers and zbe a complex number. If z2 +αz + β = 0 has two distinct non-real roots with real roots Re(z) = 1, then it is necessary thata)β ∈ (−1,0)b)β ∈ (1, ∞)c)|β| = 1d)β ∈ (0, 1)Correct answer is option 'C'. Can you explain this answer?.
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