Complex Numbers And Quadratic Equations question 388
Question: If $ {{(1-i)}^{n}}=2^{n}, $ then $ n= $ [RPET 1990]
Options:
A) 1
B) 0
C) $ -1 $
D) None of these
Show Answer
Answer:
Correct Answer: B
Solution:
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) $
Þ $ {{[ \sqrt{1^{2}+{{(-1)}^{2}}} ]}^{n}}=2^{n} $
Þ $ {{(\sqrt{2})}^{n}}=2^{n} $
Þ $ {2^{n/2}}=2^{n} $
Þ $ \frac{n}{2}=n $
Þ $ n=0 $ Trick: By inspection, $ {{(1-i)}^{0}}=2^{0}\Rightarrow 1=1 $
BETA
