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TS EAMCET 5 May 2018 Shift 1 Solved Paper

Section: Mathematics

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Question : 49 of 160

Marks: +1, -0

If z +

\frac{1}{z}

\frac{(z^{20}+1)(z^{40}+1)(z^{60}+1)}{z^{60}}

=

- 2
2
1
- 1

Solution:

The given equation is expressed as,

z+\frac{1}{z}=1

z^{2}-z+1=0

z = \frac{1 \pm \sqrt{3} i}{2}

=-\omega

and

-\omega^{2}

The given function is,

\frac{(z^{20}+1)(z^{40}+1)(z^{60}+1)}{z^{60}}

If

z=-\omega

then,

\frac{((-\omega)^{20}+1)((-\omega)^{40}+1)((-\omega)^{60}+1)}{(-\omega)^{60}}

=2(\omega^{3}+\omega^{2}+\omega+1)

=2

If

z=-\omega^{2}

then,

\frac{((-\omega^{2})^{20}+1)((-\omega^{2})^{40}+1)((-\omega^{2})^{60}+1)}{(-\omega^{2})^{60}}

=2(\omega^{3}+\omega^{2}+\omega+1)

= 2

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