MCQ
If $ \displaystyle \begin{vmatrix}\text{a} &\text{amp; }\text{b} &\text{amp; 0}\\ 0 &\text{amp; a} &\text{amp; b}\\\text{b}&\text{amp; a}&\text{amp; 0}\end{vmatrix}=0,$ then the order is:
- ✓3 × 3
- B2 × 3
- C2 × 2
- DNone of these
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Then which of the following options is/are correct?
$(1)$ For $x =1$, there exists a unit vector $\alpha \hat{ i }+\beta \hat{ j }+\gamma \hat{ k }$ for which $R \left[\begin{array}{l}\alpha \\ \beta \\ \gamma\end{array}\right]=\left[\begin{array}{l}0 \\ 0 \\ 0\end{array}\right]$
$(2)$ There exists a real number $x$ such that $P Q=Q P$
$(3)$ $\operatorname{det} R=\operatorname{det}\left[\begin{array}{lll}2 & x & x \\ 0 & 4 & 0 \\ x & x & 5\end{array}\right]+8$, for all $x \in R$
$(4)$ 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 $a+b=5$