STD 12 - 1. relation and function — Mathematics STD 12 Science — Question

Statement $-I$ : Let $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-3 \hat{\mathrm{k}}$ and $\vec{b}=2 \hat{i}+\hat{j}-\hat{k}$. Then the vector $\vec{r}$ satisfying $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{r}}=\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{r}}=0$ is of magnitude $\sqrt{10}$.

Statement $-II$ : In a triangle $A B C, \cos 2 A+\cos 2 B$ $+\cos 2 \mathrm{C} \geq-\frac{3}{2}$

$\vec{a}=3 \hat{i}+\hat{j}-\hat{k},$

$\vec{b}=\hat{i}+b_2 \hat{j}+b_3 \hat{k},  b_2, b_3 \in R ,$

$\vec{c}=c_1 \hat{i}+c_2 \hat{j}+c_3 \hat{k},  c_1, c_2, c_3 \in R$

be three vectors such that $b_2 b_3>0, \vec{a} \cdot \vec{b}=0$ and

$\left(\begin{array}{ccc}0 & -c_3 & c_2 \\ c_3 & 0 & -c_1 \\ -c_2 & c_1 & 0\end{array}\right)\left(\begin{array}{l}1 \\ b_2 \\ b_3\end{array}\right)=\left(\begin{array}{c}3-c_1 \\ 1-c_2 \\ -1-c_3\end{array}\right)$.

Then, which of the following is/are TRUE?

$(A)$ $\overrightarrow{ a } \cdot \overrightarrow{ c }=0$

$(B)$ $\vec{b} \cdot \vec{c}=0$

$(C)$ $|\vec{b}|>\sqrt{10}$

$(D)$ $|\vec{c}| \leq \sqrt{11}$