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STD 11 - 4.1 complex nubers — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsSTD 11 - 4.1 complex nubers1 Mark
MCQ
Let $z$ be a complex number, then the equation ${z^4} + z + 2 = 0$ cannot have a root, such that
  • ✓
    $|z|\, < 1$
  • B
    $|z|\, = 1$
  • C
    $|z|\, > 1$
  • D
    None of these
✓

Answer

Correct option: A.
$|z|\, < 1$
a
(a)Suppose there exists a complex number $z$ which satisfies the given equation and is such that $|z|\, < 1$.
Then ${z^4} + z + 2 = 0$ ==> $ - 2 = {z^4} + z$==> $| - 2|\, = \,|{z^4} + z|$
==> $2 \le \,|{z^4}| + |z|$==> $2 < 2,$ because$|z|\, < 1$
But $2 < 2$ is not possible. Hence given equation cannot have a root $z$ such that $|z| < 1$

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