The value of $\mathrm{\lambda }$ if ${\mathrm{a}\mathrm{x}}^4\mathrm{ }+\mathrm{ }{\mathrm{b}\mathrm{x}}^3\mathrm{ }+\mathrm{ }{\mathrm{c}\mathrm{x}}^2\mathrm{ }+\mathrm{ }50\mathrm{x}\mathrm{ }+\mathrm{ }\mathrm{d}=\mathbf{ }\left|\mathrm{ }\begin{array}{lll}{\mathrm{x}}^3-14{\mathrm{x}}^2& -\mathrm{x}& 3\mathrm{x}+\mathrm{\lambda }\\ 4\mathrm{x}+1& 3\mathrm{x}& \mathrm{x}-4\\ -3& 4& 0\end{array}\mathrm{ }\right|$ is

The remainder when the value of the determinant $\left|\begin{array}{lll}{2014}^{2014}& {2015}^{2015}& {2016}^{2016}\\ {2017}^{2017}& {2018}^{2018}& {2019}^{2019}\\ {2020}^{2020}& {2021}^{2021}& {2022}^{2022}\end{array}\right|$ is divided by $5$ is

Let $A$ be a $2\times 2$ real matrix with entries from $\left\{0, 1\right\}$ and $\left|A\right|\ne 0$. Consider the following two statements;

$\left(P\right)$ If $A\ne I_2$, then $\left|A\right|=-1$

$\left(Q\right)$ If $\left|A\right|=1$, then $tr\left(A\right)=2$

Where $I_2$ denotes $2\times 2$ identity matrix and $tr\left(A\right)$ denotes the sum of the diagonal entries of $A$. Then