JEE 2023 — Mathematics PYQ

\begin{aligned}
 & d=\lambda(\vec{b}\times\vec{c}) & 
\begin{bmatrix}
\cdot\cdot\vec{a}.\vec{d}=18
\end{bmatrix} \\
 & \vec{a}.\vec{d}=\lambda[\vec{a}.(\vec{b}\times\vec{c})]\Longrightarrow18=\lambda[\vec{a}\vec{b}\vec{c}] \\
 & \Rightarrow18=\lambda
\begin{vmatrix}
2 & 3 & 4 \\
1 & -2 & -2 \\
-1 & 4 & 3
\end{vmatrix} \\
 & \Rightarrow18=9\lambda\Rightarrow\lambda=2 \\
 & \Rightarrow\vec{d}=2(2\hat{i}-\hat{j}+2\hat{k}) \\
 & \mid\vec{a}\times\vec{d}\mid^{2}=a^{2}d^{2}-(\vec{a}.\vec{d})^{2} \\
 & =29\times36-(18)^{2}=18(58-18)=18\times40=720
\end{aligned}