List of top Mathematics Questions on Evaluation of Determinants

Obtain the value of \( \Delta = \begin{vmatrix} 1+x & 1 & 1 \\ 1 & 1+y & 1 \\ 1 & 1 & 1+z \end{vmatrix} \) in terms of \( x, y \) and \( z \). Further, if \( \Delta = 0 \) and \( x, y, z \) are non-zero real numbers, prove that \( x^{-1} + y^{-1} + z^{-1} = -1 \).

If \( \begin{vmatrix} -1 & -2 & 5 \\ -2 & a & -1 \\ 0 & 4 & 2a \end{vmatrix} = -86 \), then the sum of all possible values of \( a \) is

One of the values of \( x \) for which \( \begin{vmatrix} \cos x & \sin x \\ -\cos x & \sin x \end{vmatrix} = 1 \) is

If \( \Delta_1 = \begin{vmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{vmatrix} \) and \( \Delta_2 = \begin{vmatrix} 0 & 2 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 6 \end{vmatrix} \), then