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  • If  then the expansion of <img alt="(x+i y)^{n}-(x-i y)^{n}" src="http://entrancecor

If i=\sqrt{-1} then the expansion of (x+i y)^{n}-(x-i y)^{n} is

Option: 1

Purely real


Option: 2

Purely imaginary


Option: 3

purely real if n is even


Option: 4

Real and imaginary


Answers (1)

best_answer

As

\mathrm{(x+y)^{n}-(x-y)^{n}=2\left[^{n} C_{1}\; x^{n-1} \;y^{1}+^{n} C_{3} \;x^{n-3}\; y^{3}+^{n} C_{5} \;x^{n-5}\; y^{5}+\ldots \ldots\right]}

Hence,

(x+iy)^{n}-(x-iy)^{n}=2\left[^{n} C_{1} x^{n-1} (iy)^{1}+^{n} C_{3}\; x^{n-3} (iy)^{3}\cdots\cdots]\right.

it is clear that the index of i is odd and odd power of i remains imaginary

hence the expansion becomes purely imaginary

 

option B is correct

Posted by

Rishi

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