Let $a, b \in \mathbb{C}$. Let $\alpha, \beta$ be the roots of the equation $x^2+a x+b=0$. If $\beta-\alpha=\sqrt{11}$ and $\beta^2-\alpha^2=3 i \sqrt{11}$, then $\left(\beta^3-\alpha^3\right)^2$ is equal to:

Let the sum of the first $n$ terms of an A.P. be $3 n^2+5 n$. Then the sum of squares of the first 10 terms of the A.P. is:

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 $f$