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#### Please solve RD Sharma class 12 chapter Determinants exercise multiple choise question 13 maths textbook solution

Correct option (a)

Hint:

Simply solve this determinant.

Given:

$\Delta=\left|\begin{array}{ccc} 1 & \omega^{n} & \omega^{2 n} \\ \omega^{2 n} & 1 & \omega^{n} \\ \omega^{n} & \omega^{2 n} & 1 \end{array}\right|$

We have to find $\Delta$

Solution:

$\Delta=\left|\begin{array}{ccc} 1 & \omega^{n} & \omega^{2 n} \\ \omega^{2 n} & 1 & \omega^{n} \\ \omega^{n} & \omega^{2 n} & 1 \end{array}\right|$

Applying C1→C1+C2+C3

$\Rightarrow \Delta=\left|\begin{array}{ccc} 1+\omega^{n}+\omega^{2n} & \omega^{n} & \omega^{2 n} \\ 1+\omega^{n}+\omega^{2n} & 1 & \omega^{n} \\ 1+\omega^{n}+\omega^{2n} & \omega^{2 n} & 1 \end{array}\right|$

$\Rightarrow \Delta=\left|\begin{array}{ccc} 0 & \omega^{n} & \omega^{2 n} \\ 0 & 1 & \omega^{n} \\ 0 & \omega^{2 n} & 1 \end{array}\right|$                            $[\because 1+\omega ^{n}+\omega ^{2n}=0]$

$\Rightarrow \Delta=0$