JEE Advanced 2023 — Paper 2 — Question 4

$\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-5 \hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=3 \hat{\mathrm{i}}+6 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}, \overrightarrow{\mathrm{c}}=\frac{17}{5} \hat{\mathrm{i}}+\frac{16}{5} \hat{\mathrm{j}}+7 \hat{\mathrm{k}}, \overrightarrow{\mathrm{d}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}$ $\overrightarrow{\mathrm{PQ}}=2 \hat{i}+4 \hat{\mathrm{j}}+8 \hat{k}, \overrightarrow{\mathrm{PR}}=\frac{12}{5} \hat{\mathrm{i}}+\frac{6}{5} \hat{\mathrm{j}}+12 \hat{k}, \overrightarrow{\mathrm{PS}}=\hat{\mathrm{i}}-\hat{\mathrm{j}}+6 \hat{k}$ $[\overrightarrow{\mathrm{PQ}} \overrightarrow{\mathrm{PR}} \overrightarrow{\mathrm{PS}}]=\frac{1}{5}\left[\begin{array}{ccc}2 & 4 & 8\\ 12 & 6 & 60\\ 1 & -1 & 6\end{array}\right]=0$ A is incorrect $|\vec{b} \times \overrightarrow{\mathrm{d}}|=\left|\begin{array}{ccc}\hat{\mathrm{i}} & \hat{j} & \hat{k}\\ 3 & 6 & 3\\ 2 & 1 & 1\end{array}\right|=3 \hat{i}+3 \hat{j}-9 \hat{k}$ $|\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{d}}|^{2}=9+9+81=99$ D option is incorrect

$$\begin{aligned} & \frac{\overrightarrow{\mathrm{b}}+2 \overrightarrow{\mathrm{~d}}}{3}=\frac{7 \hat{\mathrm{i}}+8 \hat{\mathrm{j}}+5 \hat{k}}{3}=\frac{21 \hat{i}+24 \hat{j}+15 \hat{k}}{9} \\& \frac{5 \overrightarrow{\mathrm{c}}+4 \overrightarrow{\mathrm{a}}}{9}=\frac{\overrightarrow{\mathrm{b}}+2 \overrightarrow{\mathrm{~d}}}{3} \end{aligned}$$

$B$ is correct.