Question

The value of $\sqrt{2}\left( \cos\frac{\pi}{12} + i\sin\frac{\pi}{12} \right)\text{ },\sqrt{2}\left( - \sin\frac{\pi}{12} - i\cos\frac{\pi}{12} \right)\ ,\ 1 + i$ is.

• A. I if n is even, – i if n is odd
• B. 1 if n is even, – 1 if n is odd
• C. 1 if n is odd, – 1 if n is even
• D. I if n is even, – 1 if n is odd

Answer 1

Answer: (C)

1 if n is odd, – 1 if n is even

Answer 2

Let $a^{3} + b^{3} = 16$

Clearly series is A.P. with common difference = 2

$x^{3} + y^{3} + z^{3} = (a + b)^{3} + (a\omega + b\omega^{2})^{3} + (a\omega^{2} + b\omega)^{3}$and $= 3a^{3} + 3b^{3} + 3(a^{2}b + ab^{2})(1 + \omega^{2}\omega^{2} + \omega\omega^{4})$

So, number of terms in A. P. $= 3a^{3} + 3b^{3} + 3(a^{2}b + ab^{2})(1 + \omega + \omega^{2})$

Now, $=$

$3(a^{3} + b^{3})$ i.e. $x^{3} + y^{3} + z^{3} = (4)^{3} + ( - 2)^{3} + ( - 2)^{3} = 48$

Now put $3(a^{3} + b^{3}) = 48$

$\omega$, $\omega^{2}$,

$= \frac{a + b\omega + c\omega^{2}}{b + c\omega + a\omega^{2}} + \frac{a + b\omega + c\omega^{2}}{c + a\omega + b\omega^{2}}$, $= \frac{\omega(a + b\omega + c\omega^{2})}{(b\omega + c\omega^{2} + a)} + \frac{\omega^{2}(a + b\omega + c\omega^{2})}{(c\omega^{2} + a + a\omega)} = \omega + \omega^{2} = - 1$,

$1,\omega,\omega^{2}$