Question

Let $\mathrm{x} \in \mathbb{R}$ and let $\mathrm{P}=\left[\begin{array}{lll}1 & 1 & 1 \ 0 & 2 & 2 \ 0 & 0 & 3\end{array}\right]$, $\mathrm{Q}=\left[\begin{array}{ccc}2 & \mathrm{x} & \mathrm{x} \ 0 & 4 & 0 \ \mathrm{x} & \mathrm{x} & 6\end{array}\right]$ and $\mathrm{R}=\mathrm{PQP}^{-1}$. Then which of the following options is/are correct ?

Accepted Answer

(C) $\operatorname{det} \mathrm{R}=\operatorname{det}\left[\begin{array}{ccc}2 & \mathrm{x} & \mathrm{x} \ 0 & 4 & 0 \ \mathrm{x} & \mathrm{x} & 5\end{array}\right]+8$, for all $\mathrm{x} \in \mathbb{R}$; (D) For $\mathrm{x}=0$, if $\mathrm{R}\left[\begin{array}{l}1 \ \mathrm{a} \ \mathrm{b}\end{array}\right]=6\left[\begin{array}{l}1 \ \mathrm{a} \ \mathrm{b}\end{array}\right]$, then $\mathrm{a}+\mathrm{b}=5$

Answer

(A) For $\mathrm{x}=1$, there exists a unit vector $\alpha \hat{\mathrm{i}}+\hat{\mathrm{j}} \hat{\mathrm{j}}+\gamma \hat{\mathrm{k}}$ for which $\mathrm{R}\left[\begin{array}{l}\alpha \ \beta \ \gamma\end{array}\right]=\left[\begin{array}{l}0 \ 0 \ 0\end{array}\right]$

Answer

(B) There exists a real number x such that $\mathrm{PQ}=\mathrm{QP}$

JEE Advanced · 2019 · 27 May 2019 · Paper 2 · Matrices

Solution

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