Complex Numbers 1 Question 9
9. The smallest positive integer $n$ for which $\frac{1+i^{n}}{1-i}=1$, is
(a) 8
(b) 16
(c) 12
(d) None of these
$(1980,2 M)$
Objective Question II
(One or more than one correct option)
Show Answer
Answer:
Correct Answer: 9. (d)
Solution:
- Since, $\frac{1+i}{1-i}^{n}=1 \Rightarrow \frac{1+i}{1-i} \times \frac{1+i}{1+i}^{n}=1$
$$ \begin{aligned} \Rightarrow & & \frac{2 i^{n}}{2} & =1 \\ \Rightarrow & & i^{n} & =1 \end{aligned} $$
The smallest positive integer $n$ for which $i^{n}=1$ is 4 .
$$ \therefore \quad n=4 $$
