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

If \[ \begin{vmatrix} a & a^2 & 1+a^3 b & b^2 & 1+b^3 c & c^2 & 1+c^3 \end{vmatrix} = 0 \] and vectors \( (1,a,a^2), (1,b,b^2), (1,c,c^2) \) are non-coplanar, then the product \( abc \) is ___.

If \(A+B+C=\pi\), then \[ \begin{vmatrix} \sin(A+B+C) & \sin B & \cos C -\sin B & 0 & \tan A \cos(A+B) & -\tan A & 0 \end{vmatrix} \] is equal to

If \( \begin{vmatrix} x+y+2z & x & y z & y+z+2x & y z & x & z+x+2y \end{vmatrix} = k(x+y+z)^3 \), then the value of \( k \) is

If $a, b, c$ are cube roots of unity, then \[ \begin{vmatrix} e^a & e^{2a} & e^{3a}-1 e^b & e^{2b} & e^{3b}-1 e^c & e^{2c} & e^{3c}-1 \end{vmatrix} \] is equal to

The value of the determinant \[ \begin{vmatrix} (a^x + a^{-x})^2 & (a^x - a^{-x})^2 & 1 (b^x + b^{-x})^2 & (b^x - b^{-x})^2 & 1 (c^x + c^{-x})^2 & (c^x - c^{-x})^2 & 1 \end{vmatrix} \] is

The least positive integer \(n\) for which \(n!<\left(\frac{n+1}{2}\right)^n\) holds is

The value of \[ \left| \begin{array}{ccc} (a+1)^2 & (b+1)^2 & (c+1)^2 (a-1)^2 & (b-1)^2 & (c-1)^2 \end{array} \right| \] is:

If \( x, y, z \) are all positive and are the \( p \)th, \( q \)th and \( r \)th terms of a geometric progression respectively, then the value of the determinant \( \begin{vmatrix} \log x & p & 1 \\ \log y & q & 1 \\ \log z & r & 1 \end{vmatrix} \) equals

The value of \[ \begin{vmatrix} 1 & a & b + c \\ 1 & b & c + a \\ 1 & c & a + b \end{vmatrix} \] is:

If $A$ is a square matrix of order 3 and $|A| = 5$, then $|adj A|$ is: