Write the least positive integeral vlaue of n for which 1-i-1-in is real-

For n = 1 (1+i/1 i )^1=1+i/1 i =1+i/1 i×1+i/1+i =(1+i)^2/1^2+1^2 =1^2+i^2+2i/2 =1 1+2i/2=2i/2 =i For n = 2 (1+i/1 i )^2=(i)^2= 1, which is real Hence least ...

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Standard XII

Mathematics

Properties of Iota

Write the lea...

Question

Write the least positive integeral vlaue of n for which ${(\frac{1 + i}{1 - i})}^{n}$ is real.

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Solution

For n = 1

${(\frac{1 + i}{1 - i})}^{1} = \frac{1 + i}{1 - i} = \frac{1 + i}{1 - i} \times \frac{1 + i}{1 + i} = \frac{(1 + i)^{2}}{1^{2} + 1^{2}} = \frac{1^{2} + i^{2} + 2 i}{2} = \frac{1 - 1 + 2 i}{2} = \frac{2 i}{2} = i For n = 2 {(\frac{1 + i}{1 - i})}^{2} = (i)^{2} = - 1, which is real$

Hence least positive integral value of n is 2.

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Write the least positive integeral vlaue of n for which (1+i1−i)n is real.

For n = 1 (1+i1−i)1=1+i1−i=1+i1−i×1+i1+i=(1+i)212+12=12+i2+2i2=1−1+2i2=2i2=iFor n = 2(1+i1−i)2=(i)2=−1, which is real Hence least positive integral value of n is 2.

Write the least positive integeral vlaue of n for which ${(\frac{1 + i}{1 - i})}^{n}$ is real.