[shahkb4]

Let $\overrightarrow{\mathrm{a}}=2 \hat{\mathrm{i}}-5 \hat{\mathrm{j}}+5 \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{b}}=\hat{\mathrm{i}}-\hat{\mathrm{j}}+3 \hat{\mathrm{k}}$. If $\overrightarrow{\mathrm{c}}$ is a vector such that $2(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{c}})+3(\overrightarrow{\mathrm{~b}} \times \overrightarrow{\mathrm{c}})=\overrightarrow{0}$ and $(\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}) . \overrightarrow{\mathrm{c}}=-97$, then $|\overrightarrow{\mathrm{c}} \times \hat{\mathrm{k}}|^{2}$ is equal to
[JEE MAIN 24 Jan 2026 S II]
(1) 193
(2) 233
(3) 218
(4) 205

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[shahkb4]

Solution

\[ \begin{aligned} & 2 \vec{a} \times \vec{c}+3 \vec{b} \times \vec{c}=0 \Rightarrow(2 \vec{a}+3 \vec{b}) \times \vec{c}=0 \\ & \Rightarrow \vec{c} \| 2 \vec{a}+3 \vec{b} \Rightarrow \vec{c}=\lambda(2 \vec{a}+3 \vec{b}) \\ & \Rightarrow \vec{c}=\lambda(2(2 \hat{i}-5 \hat{j}+5 \hat{k})+3(\hat{i}-\hat{j}+3 \hat{k})) \\ & \Rightarrow \vec{c}=\lambda(7 \hat{i}-13 \hat{j}+19 \hat{k}) \\ & (\vec{a}-\vec{b}) \cdot \vec{c}=(\hat{i}-4 \hat{j}+2 \hat{k}) \cdot \lambda(7 \hat{i}-13 \hat{j}+19 \hat{k})=-97 \\ & \Rightarrow \lambda(7+52+38)=-97 \Rightarrow \lambda=-1 \\ & \vec{c}=-7 \hat{i}+13 \hat{j}-19 \hat{k} \\ & \vec{c} \times \hat{k}=-7 \hat{i} \times \hat{k}+13 \hat{j} \times \hat{k}-19 \hat{k} \times \hat{k} \\ & \Rightarrow \vec{c} \times \hat{k}=7 \hat{j}+13 \hat{i}-0 \\ & |\vec{c} \times \hat{k}|^{2}=49+169=218 \text { Answer }(3) \end{aligned} \]

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