Question

The value of the determinant $\left|\begin{array}{ccc}1& a& a^2\\ \cos (n-1)x& \cos nx& \cos (n+1)x\\ \sin (n-1)x& \sin nx& \sin (n+1)x\end{array}\right|$ is zero if

• A. $x=n\pi$
• B. $x=n\pi /2$
• C. $x=(2n+1)\pi /2$
• D. $x=\frac{1+a^2}{2a}n\epsilon I$

Answer 1

The correct option is A $x=n\pi$

$\left|\begin{array}{ccc}1& a& a^2\\ \cos (n-1)x& \cos nx& \cos (n+1)x\\ \sin (n-1)x& \sin nx& \sin (n+1)x\end{array}\right|=0$
$C_1\to C_2-2cosx{C}_2+C_3$
$\left|\begin{array}{ccc}1-2a\cos x+a^2& a& a^2\\ 0& \cos nx& \cos (n+1)x\\ 0& \sin nx& \sin (n+1)x\end{array}\right|$
$(1-2a\cos x+a^2)\sin x=0$
$\Rightarrow \sin x=0or\cos x=\frac{(1+a^2)}{2a}$
$x=n\pi ,n\in I$.