If $\rm \Delta = \begin{vmatrix} a & b & c\\ d & e & f\\ g & h & i \end{vmatrix}$ then what is $\rm \begin{vmatrix} 3d + 5g & 4a + 7g & 6g\\ 3e + 5h & 4b + 7h & 6h\\ 3f + 5i & 4c + 7i & 6i \end{vmatrix}$ equal to?

For what values of k is the system of equations 2k²x + 3y - 1 = 0, 7x - 2y + 3 = 0, 6kx + y + 1 = 0 consistent?

What is the value of the following determinant?

$\begin{vmatrix} \cos \rm C & \tan \rm A & 0\\ \sin \rm B & 0 & -\tan \rm A\\ 0 & \sin \rm B & \cos \rm C \end{vmatrix}$