Question

The determinant
$\Delta =\left|\begin{array}{ccc}{a}^2& a& 1\\ \cos \left(nx\right)& \cos (n+1)x& \cos (n+2)x\\ \sin \left(nx\right)& \sin (n+1)x& \sin (n+2)x\end{array}\right|$
is independent of

• A. $n$
• B. $a$
• C. $x$
• D. None of these

Answer 1

Answer: (A)

$n$

Applying $C_1\to C_1+C_3-2\cos x{C}_2$ we get

$\Delta =\left|\begin{array}{ccc}{a}^2+2a\cos x+1& a& 1\\ 0& \cos (n+1)x& \cos (n+2)x\\ 0& \sin (n+1)x& \sin (n+2)x\end{array}\right|$

$[\because \cos nx+\cos (n+2)x=2\cos (n+1)x\cos xand\sin nx+\sin (n+2)x=2\sin (n+1\left)x\cos x\right]$

Expanding along with $C_1$ we get

$\Delta =(a^2-2a\cos x+1)\left[\cos \right(n+1\left)x\sin \right(n+2)x-\sin (n+1\left)x\cos \right(n+2\left)x\right]$

$=(a^2-2a\cos x+1)\sin \left[\right(n+2)x-(n+1)x]$

$=(a^2-2a\cos x+1)\sin x$

which is independent of $n$