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

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 11 - 4.1 complex nubers4 Marks
Question
If $(1 - i)^n = 2^n,$ then $n = $
✓

Answer

b
(b) If $(1 - i)^n = 2^n$ ......$(i)$
We know that if two complex numbers are equal, their moduli must also be equal, therefore from $(i)$, we have
$|(1 - i)^n|\, = \,|2^n|$

$ \Rightarrow $ $|1 - i|^n = \,|2|^n$,$(\because \,\,2^n > 0)$
==> $\left[ \sqrt {{1^2} + {{( - 1)}^2}}  \right]^n = 2^n$

==> $(\sqrt 2 )^n = 2^n$
==> $2^{n/2} = 2^n$

==> $\frac{n}{2} = n$

==>$n = 0$
Trick : By inspection, ${(1 - i)^0} = {2^0}\,\,\,\, \Rightarrow 1 = 1$.

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