IPMAT Indore (MCQ) PYQs

Suppose $\left|\begin{array}{lll}a & a^{2} & a^{3}-1 \\ b & b^{2} & b^{3}-1 \\ c & c^{2} & c^{3}-1\end{array}\right|=0$, where $\mathrm{a}, \mathrm{b}$ and $\mathrm{c}$ are distinct real numbers. If ${a}=3$, then the value of $abc$ is:

If $A = \begin{bmatrix} x_1 & x_2 & 7 \\ y_1 & y_2 & y_3 \\ z_1 & 8 & 3 \end{bmatrix}$ is a matrix such that the sum of all three elements along any row, column or diagonal are equal to each other, then the value of determinant of A is:

A $2 \times 2$ matrix is filled with four distinct integers randomly chosen from the set $\{1,2,3,4,5,6\}$. Then the probability that the matrix generated in such a way is singular is