CBSE 2022 · Region 1 · Set 1 · Q8 · 3 marks
Find the vector equation of the plane passing through the intersection of the planes $\displaystyle \overrightarrow{\mathrm{r}} \cdot(2 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}-3 \hat{\mathrm{k}})=7$ and $\displaystyle \overrightarrow{\mathrm{r}} \cdot(2 \hat{\mathrm{i}}+5 \hat{\mathrm{j}}+3 \hat{\mathrm{k}})=9$ and through the point ($\displaystyle 2$, $\displaystyle 1$, $\displaystyle 3$).
Marking-scheme solution
Equation of the plane is \(\displaystyle \vec{r} \cdot(2 \hat{i}+2 \hat{j}-3 \hat{k})-7+\lambda[\vec{r} \cdot(2 \hat{i}+5 \hat{j}+3 \hat{k})-9]=0\)
\[\begin{aligned}
& \text { or } \vec{r} \cdot[(2+2 \lambda) \hat{i}+(2+5 \lambda) \hat{j}+(-3+3 \lambda) \hat{k}]-7-9 \lambda \\
& \quad(2,1,3) \text { lies on it } \\
& \therefore 2(2+2 \lambda)+1(2+5 \lambda)+3(-3+3 \lambda)-7-9 \lambda=0 \\
& \quad 9 \lambda=10 \Rightarrow \lambda=10 / 9
\end{aligned}
\]
Equation is \(\displaystyle \vec{r} \cdot(38 \hat{i}+68 \hat{j}+3 \hat{k})=153\)
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CBSE Class 12 Mathematics past-paper question from the 2022board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.