If n is a positive integer greater than unity and z is a complex … - Vidyadip

MCQ

If $n$ is a positive integer greater than unity and $z$ is a complex number satisfying the equation ${z^n} = {(z + 1)^n}$, then

✓
${\mathop{\rm Re}\nolimits} (z) < 0$
B
${\mathop{\rm Re}\nolimits} (z) > 0$
C
${\mathop{\rm Re}\nolimits} (z) = 0$
D
None of these

✓

Answer

Correct option: A.

${\mathop{\rm Re}\nolimits} (z) < 0$

a
(a)We have ${z^n} = {(1 + z)^n}\,\,\, \Rightarrow {\left( {\frac{z}{{z + 1}}} \right)^n} = 1$
==> $\frac{z}{{z + 1}} = {1^{1/n}}$

==> $\frac{z}{{z + 1}}$is a $n$th root of unity
==> $\left| {\frac{z}{{z + 1}}} \right| = 1$==>$\frac{{|z|}}{{|z + 1|}} = 1$

==> $|z|\, = \,|z + 1|$
==> $x + \frac{1}{2} = 0$==> $x = \frac{{ - 1}}{2}$

==> ${\mathop{\rm Re}\nolimits} (z) < 0$.

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