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Question: If 1, a1, a2, a3 ....... and a8 are nine, ninth roots of unity (taken in counter-clockwise sequence)...

If 1, a1, a2, a3 ....... and a8 are nine, ninth roots of unity (taken in counter-clockwise sequence) then

|(2 – a1)(2 – a3)(2 – a5)(2 – a7)| is equal to

A

255\sqrt{255}

B

511\sqrt{511}

C

1023\sqrt{1023}

D

15

Answer

511\sqrt{511}

Explanation

Solution

Sol. (x – 1)(x – a1) (x – a2) ….(x – a8) º x9 – 1

\ (2 – a1) (2 – a2) ...... (2 – a8) = 29 – 1

Now since 2 – a1 and 2 – a8 are conjugates of each other

\ |2 – a1| = |2 – a8|

Similarly |2 – a2| = |2 – a7| , |2 – a3| = |2 – a6| and

|2 – a4| = |2 – a5|

2–a1)(2–a3)(2–a5)(2–a7)∣=(29−12 – a1)(2 – a3)(2 – a5)(2 – a7)| = (\sqrt{2^{9} - 1}

=511= \sqrt{511}