If n is a positive integer and \omega\neq 1 is a cube root of unity, the number of possible values of |e^{\sum^{n}_{k=0}(n/k)\omega^k}| is

KVPY SX SB Exam 20141 Question Paper

Question : 82 of 120

Marks: +1, -0

If n is a positive integer and

ω≠ 1

is a cube root of unity, the number of possible values of

|e^{∑↖{n}↙{k=0}(n/k)ω^k}|

Solution:

{∑↖{n}↙{k=0}} {^{n}C_k} = ^n C_0 + ^n C_1ω +…..+ ^n C_nω^n

(1 + ω)^ n = (-ω^ 2 )^ n

= (-1)^n ω ^{2n}

=|e^{(-\cos{4π}/3-i\sin{4π}/3)^n}|

=|e^{ -\cos{nπ}/3+i\sin{nπ}/3 } |

=|e^{\cos{nπ}/3}|

can have values
=

\{e^1, e^{1\/2}, e^{–1\/2}, e^{–1}\}

Four values.

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