Let $X = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}$ and $A = \begin{bmatrix} -1 & 2 & 3 \\ 0 & 1 & 6 \\ 0 & 0 & -1 \end{bmatrix}$. For $k \in N$,if $X^{T} A^{k} X = 33$,then $k$ is equal to:

Let $M$ and $N$ be two $3 \times 3$ matrices such that $MN = NM$. Further,if $M \neq N^2$ and $M^2 = N^4$,then:
$(A)$ determinant of $(M^2 + MN^2)$ is $0$
$(B)$ there is a $3 \times 3$ non-zero matrix $U$ such that $(M^2 + MN^2)U$ is the zero matrix
$(C)$ determinant of $(M^2 + MN^2) \geq 1$
$(D)$ for a $3 \times 3$ matrix $U$,if $(M^2 + MN^2)U$ equals the zero matrix then $U$ is the zero matrix

If $\Delta _1 = \left| \begin{array}{ccc} b^5c^6(c^3 - b^3) & a^4c^6(a^3 - c^3) & a^4b^5(b^3 - a^3) \\ b^2c^3(b^6 - c^6) & ac^3(c^6 - a^6) & ab^2(a^6 - b^6) \\ b^2c^3(c^3 - b^3) & ac^3(a^3 - c^3) & ab^2(b^3 - a^3) \end{array} \right|$ and $\Delta _2 = \left| \begin{array}{ccc} a & b^2 & c^3 \\ a^4 & b^5 & c^6 \\ a^7 & b^8 & c^9 \end{array} \right|$,then $\Delta _1 \Delta _2$ is equal to

If $\begin{bmatrix} x & 4 & -1 \end{bmatrix} \begin{bmatrix} 2 & 1 & 0 \\ 1 & 0 & 2 \\ 0 & 2 & 4 \end{bmatrix} \begin{bmatrix} x \\ 4 \\ -1 \end{bmatrix} = 0$,then $x=$

Let $p$ be a non-singular matrix such that $I + p + p^2 + .... + p^n = O$ (where $O$ denotes the null matrix and $I$ denotes the identity matrix),then $p^{-1} = $

If $ab + bc + ca = 0$ and $\begin{vmatrix} a - x & c & b \\ c & b - x & a \\ b & a & c - x \end{vmatrix} = 0$,then one of the values of $x$ is