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If Z = ( 1 + I 1 − I ) Then Z4 Equals - Mathematics

MCQ

If \[z = \left( \frac{1 + i}{1 - i} \right)\] then z4 equals

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Solution

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\[\text {Let } z = \frac{1 + i}{1 - i}\]

Rationalising the denominator:

\[z=\frac{1 + i}{1 - i}\times\frac{1 + i}{1 + i}\]

\[\Rightarrow z = \frac{1 + i^2 + 2i}{1 - i^2}\]

\[\Rightarrow z = \frac{2i}{2}\]

\[ \Rightarrow z = i\]

\[\Rightarrow z^4 = i^4 \]

\[\text { Since} i^2 = - 1,\text {  we have }: \]

\[ \Rightarrow z^4 = i^2 \times i^2 \]

\[ \Rightarrow z^4 = 1\]

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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 13 Complex Numbers
Q 19 | Page 65
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