If z 0 is a complex number, then - Vidyadip

MCQ

If $z \ne 0$ is a complex number, then

✓
${\mathop{\rm Re}\nolimits} (z) = 0 \Rightarrow {\mathop{\rm Im}\nolimits} ({z^2}) = 0$
B
${\mathop{\rm Re}\nolimits} ({z^2}) = 0 \Rightarrow {\mathop{\rm Im}\nolimits} ({z^2}) = 0$
C
${\mathop{\rm Re}\nolimits} (z) = 0 \Rightarrow {\mathop{\rm Re}\nolimits} ({z^2}) = 0$
D
None of these

✓

Answer

Correct option: A.

${\mathop{\rm Re}\nolimits} (z) = 0 \Rightarrow {\mathop{\rm Im}\nolimits} ({z^2}) = 0$

a
(a) If $z \ne 0$. Let $z = x + iy$ ==> ${z^2} = {x^2} - {y^2} + i(2xy)$
$Re(z)= 0$ ==> $x = 0$. Therefore ${\mathop{\rm Im}\nolimits} ({z^2}) = 2xy = 0$
Thus ${\mathop{\rm Re}\nolimits} (z) = 0 \Rightarrow {\mathop{\rm Im}\nolimits} ({z^2}) = 0$.

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