List of top Mathematics Questions on Properties of Determinants asked in KEAM

The roots of the equation \( \begin{vmatrix} x-1 & 1 & 1 \\ 1 & x-1 & 1 \\ 1 & 1 & x-1 \end{vmatrix} = 0 \) are:

If \( \begin{pmatrix} 1 & 2 & 4 \\ 1 & 3 & 5 \\ 1 & 4 & a \end{pmatrix} \) is singular, then the value of \( a \) is:

The value of the determinant \( \begin{vmatrix} 1 & 1 & 1 \\ p & q & r \\ p & q & r+1 \end{vmatrix} \) is equal to:

If \( A = \begin{vmatrix} 8 & 27 & 125 \\ 2 & 3 & 5 \\ 1 & 1 & 1 \end{vmatrix} \), then the value of \( A^2 \) is equal to:

If \( A = \begin{bmatrix} x & 1 & -x \\ 0 & 1 & -1 \\ x & 0 & 7 \end{bmatrix} \) and \( \det(A) = \begin{vmatrix} 3 & 0 & 1 \\ 2 & -1 & 2 \\ 0 & 0 & 3 \end{vmatrix} \), then the value of \( x \) is:

The coefficient of \( x^2 \) in the expansion of the determinant \( \begin{vmatrix} x^2 & x^3+1 & x^5+2 \\ x^3+3 & x^2+x & x^3+x^4 \\ x+4 & x^3+x^5 & 2^3 \end{vmatrix} \) is:

The value of the determinant \[ \begin{vmatrix} \cos^2 54^\circ & \cos^2 36^\circ & \cot 135^\circ \\ \sin^2 53^\circ & \cot 135^\circ & \sin^2 37^\circ \\ \cot 135^\circ & \cos^2 25^\circ & \cos^2 65^\circ \end{vmatrix} \] is equal to

If \( A = \begin{pmatrix} x & x-1 \\ 2x & 1 \end{pmatrix} \) and if \( \det A = -9 \), then the values of \(x\) are

If \[ \begin{vmatrix} 0 & 3 & 2b \\ 2 & 0 & 1 \\ 4 & -1 & 6 \end{vmatrix} \] is singular, then the value of \(b\) is

The roots of the equation \[ \begin{vmatrix} 1+x & 3 & 5 \\ 2 & 2+x & 5 \\ 2 & 3 & x+4 \end{vmatrix} = 0 \] are

If \( \Delta = \begin{vmatrix}1& 2& 3 \\ 2& 3& 5 \\ 3& 6& 12\end{vmatrix} \) and \( \Delta' = \begin{vmatrix}4& 8& 15 \\ 3& 6& 12 \\ 2& 3& 5\end{vmatrix} \), then

The value of the determinant $\begin{vmatrix}\sin ^{2} 36^{\circ} & \cos ^{2} 36^{\circ} & \cot 135^{\circ} \\ \sin ^{2} 53^{\circ} & \cot 135^{\circ} & \cos ^{2} 53^{\circ} \\ \cot 135^{\circ} & \cos ^{2} 25^{\circ} & \cos ^{2} 65^{\circ}\end{vmatrix}$ is