If $\begin{vmatrix} m! & (m+1)! & (m+2)! \\ (m+1)! & (m+2)! & (m+3)! \\ (m+2)! & (m+3)! & (m+4)! \end{vmatrix} = 2k$, where $m$ is a positive integer then which of the following are factors of $k$?

A. $m!$

B. $(m+1)!$

C. $(m+2)!$

D. $(m+3)!$

Choose the correct answer from the options given below:

Match List–I with List–II

Choose the correct answer from the options given below

If $\begin{bmatrix} -1 & 1 & 0 \\ a & b & 1 \\ 1 & 2 & 1 \end{bmatrix}$ is a singular matrix, then the relation between $a$ and $b$ is:

If $\begin{vmatrix} (a-x)^2 & (a-y)^2 & (a-z)^2 \\ (b-x)^2 & (b-y)^2 & (b-z)^2 \\ (c-x)^2 & (c-y)^2 & (c-z)^2 \end{vmatrix} = \lambda(a-b)(b-c)(c-a) \cdot (x-y)(y-z)(z-x)$

then the value of $\lambda$ is:

If $A = \begin{bmatrix} 3 & 0 & -1 \\ 2 & 3 & 0 \\ 0 & 4 & 1 \end{bmatrix}$, then which of the following is/are correct ?

A. $|A| = 1$

B. $A$ is singular matrix.

C. $|adj A| = 1$

D. $|adj.(adj A)| = 1$

Choose the correct answer from the options given below:

If $A=\begin{bmatrix}x+4 & 3\\2 & 1\end{bmatrix}, B=\begin{bmatrix}5 & 2x-1\\2 & x\end{bmatrix}$ and $\lvert adj(AB) \rvert=32$, then the value of x is:

If $\begin{vmatrix} 3x & 4 \\ 7 & x \end{vmatrix} = \begin{vmatrix} 6 & 3 \\ 2 & 1 \end{vmatrix}$ then :