Let A be a $3 \times 3$ matrix such that

$$ \mathrm{A}^{\mathrm{T}}\left[\begin{array}{l} 1 \\ 0 \\ 1 \end{array}\right]=\left[\begin{array}{l} 5 \\ 2 \\ 2 \end{array}\right], \mathrm{A}^{\mathrm{T}}\left[\begin{array}{l} 0 \\ 0 \\ 1 \end{array}\right]=\left[\begin{array}{l} 3 \\ 1 \\ 1 \end{array}\right], \mathrm{A}\left[\begin{array}{l} 1 \\ 0 \\ 1 \end{array}\right]=\left[\begin{array}{l} 3 \\ 4 \\ 4 \end{array}\right] \text { and } \mathrm{A}\left[\begin{array}{l} 0 \\ 0 \\ 1 \end{array}\right]=\left[\begin{array}{l} 1 \\ 3 \\ 1 \end{array}\right] . $$

If $\operatorname{det}(A)=1$, then $\operatorname{det}\left(\operatorname{adj}\left(A^2+A\right)\right)$ is equal to:

Consider the system of linear equations in $x, y, z$ :

$$ \begin{aligned} & x+2 y+t z=0 \\ & 6 x+y+5 t z=0 \\ & 3 x+t^2 y+f(t) z=0 \end{aligned} $$

where $f: \mathbb{R} \rightarrow \mathbb{R}$ is a differentiable function. If this system has infinitely many solutions for all $t \in \mathbb{R}$, then $f$

If the system of equations :

$$ \begin{aligned} & x+y+z=5 \\ & x+2 y+3 z=9 \\ & x+3 y+\lambda z=\mu \end{aligned} $$

has infinitely many solutions, then the value of $\lambda+\mu$ is :

Let $A=\left[\begin{array}{ccc}1 & 2 & 7 \\ 4 & -2 & 8 \\ 3 & 8 & -7\end{array}\right]$ and $\operatorname{det}(A-\alpha I)=0$, where $\alpha$ is a real number. If the largest possible value of $\alpha$ is $p$, then the circle $(x-p)^2+(y-2 p)^2=320$, intersects the co-ordinate axes at