The value of the determinant|(a^2,a,1)(cos nx, cos(n+1)x,cos(n+2)x)(sin nx,sin(n+1)x,sin(n+2)x|

Question

The value of the determinant  

\(\begin{vmatrix} a^2 & a & 1 \\[0.3em] cos\,nx & cos(n+1)x & cos(n+2)x \\[0.3em] sin\,nx &sin(n+1)x &sin(n+2)x \end{vmatrix}\) is independent of

A. n 

B. a 

C. x 

D. none of these

Answer 1

A. n

Let us solve the determinant.

\(\begin{vmatrix} a^2 & a & 1 \\[0.3em] cos\,nx & cos(n+1)x & cos(n+2)x \\[0.3em] sin\,nx &sin(n+1)x &sin(n+2)x \end{vmatrix}\)

We know that, 

Determinant of 3 × 3 matrix is given as,

By trigonometric identity, we have 

sin (α – β) = sin α cos β – cos α sin β 

So, we can write

Note that, 

The result has ‘a’ as well as ‘x’, 

But doesn’t contain ‘n’. 

Thus,

The determinant is independent of n.