Between the following two statements: Statement I : Let $\vec{a}=\hat{i}+2 \hat{j}-3 \hat{k}$ and $\vec{b}=2…

Between the following two statements: Statement I : Let $\vec{a}=\hat{i}+2 \hat{j}-3 \hat{k}$ and $\vec{b}=2 \hat{i}+\hat{j}-\hat{k}$. Then the vector $\vec{r}$ satisfying $\vec{a} \times \vec{r}=\vec{a} \times \vec{b}$ and $\vec{a} \cdot \vec{r}=0$ is of magnitude $\sqrt{10}$.
Statement II : In a triangle $A B C, \cos 2 A+\cos 2 B+\cos 2 C \geq-\frac{3}{2}$.
  1. Statement I is incorrect but Statement II is correct.
  2. Both Statement I and Statement II are correct.
  3. Statement I is correct but Statement II is incorrect.
  4. Both Statement I and Statement II are incorrect.

Solution

$\begin{aligned} & \bar{a}=\hat{i}+2 \hat{j}-3 \hat{k} \\ & \bar{a}=2 \hat{i}+\hat{j}-\hat{k} \\ & \bar{a} \times \bar{r}=\bar{a} \times \bar{b} ; \bar{a} \cdot \bar{r}=0 \\ & \Rightarrow \bar{a} \times(\bar{r}-\bar{b})=\overline{0} \\ & \Rightarrow \bar{a}=\lambda(\bar{r}-\bar{b}) \\ & \bar{a} \cdot \bar{a}=\lambda(\bar{a} \cdot \bar{r}-\bar{a} \cdot \bar{b}) \\ & 14=-7 \lambda \Rightarrow \lambda=-2 \\ & \frac{-\bar{a}}{2}=\bar{r}-\bar{b} \Rightarrow \bar{r}=\bar{b}-\frac{\bar{a}}{2} \\ & =\frac{2 \bar{b}-\bar{a}}{2}=\frac{3 \hat{i}+\hat{k}}{2}\end{aligned}$
Statement (I) is incorrect $\begin{aligned} & \cos 2 \mathrm{~A}+\cos 2 \mathrm{~B}+\cos 2 \mathrm{c} \geq-\frac{3}{2} \\ & 2 \mathrm{~A}+2 \mathrm{~B}+2 \mathrm{C}=2 \pi \\ & \cos 2 \mathrm{~A}+\cos 2 \mathrm{~B}+\cos 2 \mathrm{C} \\ & =-1-4 \cos \mathrm{A} \cdot \cos \mathrm{B} \cdot \cos \mathrm{C} \\ & \geq-1-4 \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \\ & =-\frac{3}{2} \end{aligned}$
Statement (II) is correct.

Asked in: JEE Main 2024 (09 Apr Shift 2)

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