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  4. Let x= cos (π/3)+i sin (π/3) and Δ=| 1 x x2 x2 1 x 1 x2 1 | then numerical value of Δ is

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Q. Let $x=\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}$ and $\Delta=\begin{vmatrix} 1 & x & x^2 \\ x^2 & 1 & x \\ 1 & x^2 & 1 \end{vmatrix}$ then numerical value of $\Delta$ is

Determinants

Solution:

Use $C_3 \rightarrow C_3-x C_2$ and $C_2 \rightarrow C_2-x C_1$.
$\Delta =\begin{vmatrix}1 & 0 & 0 \\x^2 & 1-x^3 & 0 \\1 & x^2-x & 1-x^3
\end{vmatrix}$
$ =\left(1-x^3\right)^2=[1-\cos \pi-i \sin \pi]^2=8$
[De Moivre's Theorem]