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JAMIA MILLIA ISLAMIA Jamia Millia Islamia Solved Paper-2006

  • question_answer
        The smallest positive integer n for which\[{{\left( \frac{1+i}{1-i} \right)}^{n}}=1\]is

    A)  \[n=8\]                               

    B)  \[n=12\]

    C)  \[n=16\]                             

    D)  none of these

    Correct Answer: D

    Solution :

                    Now, \[\frac{1+i}{1-i}=\frac{(1+i)(1+i)}{(1-i)(1+i)}=\frac{1+{{i}^{2}}+2i}{2}=i\] Now, \[{{\left( \frac{1+i}{1-i} \right)}^{n}}=1\] \[\Rightarrow \]               \[{{(i)}^{n}}=1\]                                                ???(i) \[\Rightarrow \]Least value of n for which Eq. (i) is true is 4 \[\because \]                     \[{{(i)}^{4}}=1\]


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