If n is a positive integer, then ( 3 + 1 )^ 2n - ( 3 - 1 )^ 2n is - Vidyadip

MCQ

If $n $ is a positive integer, then ${\left( {\sqrt 3 + 1} \right)^{2n}} - {\left( {\sqrt 3 - 1} \right)^{2n}}$ is

✓
an irrational number
B
an odd positive integer
C
an even positive integer
D
a rational number other than positive integers

✓

Answer

Correct option: A.

an irrational number

a
$1+i=\sqrt{2}\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right)$

$1-i=\sqrt{2}\left(\cos \left(-\frac{\pi}{4}\right)+i \sin \left(-\frac{\pi}{4}\right)\right)$

$(1+i)^{n}=(\sqrt{2})^{n}\left(\cos \frac{n \pi}{4}+i \sin \frac{n \pi}{4}\right)$

$(1-i)^{n}=(\sqrt{2})^{n}\left(\cos \left(-\frac{n \pi}{4}\right)+i \sin \left(-\frac{n \pi}{4}\right)\right)$

$(1+i)^{n}+(1-i)^{n}=(\sqrt{2})^{n}\left(2 \cos \frac{n \pi}{4}\right)$

$=2^{\frac{n+2}{2}} \cos \frac{n \pi}{4}$

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