JEE 2023 — Mathematics PYQ

Solution:

Vector identity ka use karke: $(\vec{a}+\vec{b})\times (\vec{a}\times \vec{b})=(\vec{a}\cdot \vec{b}+|\vec{b}{|}^2)\vec{a}-(|\vec{a}{|}^2+\vec{a}\cdot \vec{b})\vec{b}$.

Solve karne par equation banti hai: $2(|\vec{a}\times \vec{b}{|}^2)(\vec{a}\times \vec{b})=\dots$ (Vector Triple Product properties).

Given LHS ko simplify karne par humein milta hai: $2(|\vec{a}{|}^2|\vec{b}{|}^2-(\vec{a}\cdot \vec{b}{)}^2)$ ke terms.

Comparing coefficients: $\lambda =1$.

Value find karni hai: $|\lambda (-2\vec{a}\times \vec{b}){|}^2=4|\vec{a}\times \vec{b}{|}^2$.

$\lambda =1$ ke liye, $\vec{a}\times \vec{b}=(4-3\lambda )\hat{i}-(2\lambda +3)\hat{j}+(-{\lambda }^2-2)\hat{k}=\hat{i}-5\hat{j}-3\hat{k}$.

$|\vec{a}\times \vec{b}{|}^2=1^2+(-5{)}^2+(-3{)}^2=1+25+9=35$.

$4\times 35=140$.

Sahi Option: (3)