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Let \( \mathrm{x} \in \mathrm{R} \) and let \( \mathrm{P}=\left[\begin{array}{lll}1 & 1 & 1 \\ 0 & 2 & 2 \\ 0 & 0 & 3\end{array}\right], \mathrm{Q}=\left[\begin{array}{lll}2 & \mathrm{x} & \mathrm{x} \\ 0 & 4 & 0 \\ \mathrm{x} & \mathrm{x} & 6\end{array}\right] \)
and \( \mathrm{R}=\mathrm{PQP}{ }^{-1} \).
Then which of the following options is/are correct?
(1) For \( \mathrm{x}=1 \), there exists a unit vector \( \alpha \hat{\mathrm{i}}+\beta \hat{\mathrm{j}}+\gamma \hat{\mathrm{k}} \)
(
or which \( \mathrm{R}\left[\begin{array}{l}\alpha \\ \beta \\ \gamma\end{array}\right]=\left[\begin{array}{l}0 \\ 0 \\ 0\end{array}\right] \)
(2) There exists a real number \( x \) such that \( \mathrm{PQ}=\mathrm{QP} \)
(3) \( \operatorname{det} \mathrm{R}-\operatorname{det}\left[\begin{array}{ccc}2 & \mathrm{x} & \mathrm{x} \\ 0 & 4 & 0 \\ \mathrm{x} & \mathrm{x} & 5\end{array}\right]+8 \), for all \( \mathrm{x} \in \mathrm{R} \)
(4) For \( x=0 \), if \( R\left[\begin{array}{l}1 \\ a \\ b\end{array}\right]-6\left[\begin{array}{l}1 \\ a \\ b\end{array}\right] \), then \( a+b=5 \)
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