KVPY 2020-21 SB SX Exam Previous Papers

Question : 2 of 120

Marks: +1, -0

Consider a regular 10 -gon with its vertices on the unit circle. With one vertex fixed, draw straight lines to the other 9 vertices. Call them

{L}_{1}, {L}_{2}, \....., {L}_{9}

and denote their lengths by

l_{1}, l_{2}, \....., l_{9}

respectively. Then the product

l_{1} l_{2}, \......, l_{9}

Solution:

Let

\α={e}^{({i} {2 \π}/{10})}={e}^{{i} {\π}/{5}}

Now,

z^{10}-1=(z-1)(z-\α) \......(z-\α^{9})

....(1)
so,

l_{1} l_{2} \.....l_{9}=|1-\α||1-\α^{2}|

\......|1-\α^{9} \|

=|(1-\α)(1-\α^{2}) \......(1-\α^{9})|

=|\lim ↙{z \→1} {z^{10}-1}/{z-1}|=10

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