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NCERT Class XI Mathematics - Complex Numbers and Quadratic Equations - Solutions

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Question : 47 of 52
Marks: +1, -0
Find the modulus of 1+i1−i−1−i1+i\frac{1+i}{1-i} - \frac{1-i}{1+i}
Solution:  
We have, 1+i1−i−1−i1+i\frac{1+i}{1-i} - \frac{1-i}{1+i} = (1+i)2−(1−i)2(1−i)(1+i)\frac{(1+i)^2 - (1-i)^2}{(1-i)(1+i)}
= (1−1+2i)−(1−1−2i)1+1\frac{(1-1+2i)-(1-1-2i)}{1+1} = 2i+2i2\frac{2i+2i}{2} = 4i2\frac{4i}{2} = 2i
∴ ∣1+i1−i−1−i1+i∣\left| \frac{1+i}{1-i} - \frac{1-i}{1+i} \right| = 22\sqrt{2^2} = 2
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