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Question: The smallest positive integer \(- 3\)for which \(x_{1}.x_{2}......\infty\) is....

The smallest positive integer −3- 3for which x1.x2......∞x_{1}.x_{2}......\infty is.

A

1

B

2

C

3

D

4

Answer

2

Explanation

Solution

We have =cos⁡(11π+π2)+isin⁡(11π+π2)=0+i(−1)=−i= \cos\left( 11\pi + \frac{\pi}{2} \right) + i\sin\left( 11\pi + \frac{\pi}{2} \right) = 0 + i( - 1) = - i

⇒(1+i1−i)2n=1\Rightarrow \left( \frac { 1 + i } { 1 - i } \right) ^ { 2 n } = 1 i2=−1(i2)2=(−1)2=1⇒i4n=1ni^{2} = - 1(i^{2})^{2} = ( - 1)^{2} = 1 \Rightarrow i^{4n} = 1^{n}

i4n−1=−i∵i^{4n - 1} = - i\because ⇒2n=4\Rightarrow 2 n = 4 (1+i1−i)4n+1=i4n+1=ii4n=i(∵i4n=1)\left( \frac{1 + i}{1 - i} \right)^{4n + 1} = i^{4n + 1} = ii^{4n} = i(\because i^{4n} = 1).