Question

solution dena .......​

Answer 1

Step-by-step explanation:

hope it helps you

.......................

Answer 2

$\:\:\:\: \: \: \large\mathfrak{\underline{\huge\mathcal{\bf{\boxed{\boxed{\huge\mathcal{~~QUESTION~~}}}}}}}$

The value of......

$( \frac{i + \sqrt{3} }{2} ) {}^{2019} + ( \frac{i - \sqrt{3} }{2} ) {}^{2019}$

$\:\:\:\: \: \: \: \: \: \: \: \: \: \: \large\mathfrak{\underline{\huge\mathcal{\bf{\boxed{\huge\mathcal{!!ANSWER!!}}}}}}$

$<font color=red><marquee behavior=alternate>Solution</marquee></font>$

$= ( \frac{i + \sqrt{3} }{2} ) {}^{2019} + ( \frac{i - \sqrt{3} }{2} ) {}^{2019} \\ = ( \frac{ - i( - 1 + \sqrt{3}i) }{2} ) {}^{2019} + ( \frac{ - i( - 1 - \sqrt{3} i}{2} ) {}^{2019} \: \\ = ( \frac{( - i) {}^{2019} ( - 1 + \sqrt{3}i) {}^{2019} }{2 {}^{2019} } ) + ( \frac{( - i) {}^{2019} ( - 1 - \sqrt{3} i) {}^{2019} }{2 {}^{2019} } ) \: \\ = ( \frac{i( - 1 + \sqrt{3}i) {}^{2019} }{2 {}^{2019} } ) + ( \frac{i ( - 1 - \sqrt{3} i) {}^{2019} }{2 {}^{2019} } ) \: \\ = \frac{i}{2 {}^{2019} } (( - 1 + \sqrt{3} i) {}^{2019} + ( - 1 - \sqrt{3} i) {}^{2019} ) \\ now \: \: \\ 2w = ( - 1 + \sqrt{3}i) \: \: and \: \: 2w { \: }^{2} = ( - 1 - \sqrt{3}i) \\ and \: \: \: \: w {}^{3} = 1 \\ \\ therefore \: \: \: \\ = \frac{i}{2 {}^{2019} } ((2w) {}^{2019} + (2w {}^{2}) {}^{2019} ) \\ = \frac{i}{2 {}^{2019} } (2 {}^{2019} (w) {}^{2019} +2 {}^{2019} (w {}^{2}) {}^{2019} ) \: \\ = \frac{i}{2 {}^{2019} } \times 2 {}^{2019} ((w {}^{3} ) {}^{673} + (w {}^{3} ) {}^{2 \times 673} ) \\ = i(1 {}^{673} + 1 {}^{2 \times 673} ) \\ = i(1 + 1) \\ = 2i$

option (c)2i

$<font color=green><marquee direction="down">hope this help you$