Question

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

• A. Both Statement-I and Statement-II are incorrect
• B. Statement-I is incorrect but Statement-II is correct
• C. Both Statement-I and Statement-II are correct
• D. Statement-I is correct but Statement-II is incorrect

Answer 1

Answer: (B)

Statement-I is incorrect but Statement-II is correct

Answer 2

Analyzing Statement-I:
We are given:
$\mathbf{a} = \hat{i} + 2\hat{j} - 3\hat{k}$, $\mathbf{b} = 2\hat{i} + \hat{j} - \hat{k}$
We are looking for a vector $\mathbf{r}$ satisfying two conditions: 1. $\mathbf{a} \times \mathbf{r} = \mathbf{a} \times \mathbf{b}$ 2. $\mathbf{a} \cdot \mathbf{r} = 0$
Step 1: Find $\mathbf{a} \times \mathbf{b}$.
The cross product $\mathbf{a} \times \mathbf{b}$ is calculated as follows:
$\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\\ 1 & 2 & -3 \\\ 2 & 1 & -1 \end{vmatrix}$
Expanding the determinant:
$\mathbf{a} \times \mathbf{b} = \hat{i} \left(2(-3) - (1)(-1)\right) - \hat{j} \left(1(-1) - 2(-1)\right) + \hat{k} \left(1(1) - 2\right)$ $\mathbf{a} \times \mathbf{b} = \hat{i}(2 - (-3)) - \hat{j}(1 - (-6)) + \hat{k}(1 - 4)$ $\mathbf{a} \times \mathbf{b} = 5\hat{i} - 5\hat{j} - 3\hat{k}$ Thus, $\mathbf{a} \times \mathbf{b} = 5\hat{i} - 5\hat{j} - 3\hat{k}$.
Step 2: Solve $\mathbf{a} \times \mathbf{r} = \mathbf{a} \times \mathbf{b}$.
The vector $\mathbf{r}$ must satisfy $\mathbf{a} \times \mathbf{r} = 5\hat{i} - 5\hat{j} - 3\hat{k}$. We can find $\mathbf{r}$ by using the conditions on $\mathbf{r}$, but after solving, it turns out that the magnitude of $\mathbf{r}$ does not match the given value $\sqrt{10}$.
Thus, Statement-I is incorrect.

Analyzing Statement-II:
We are given a triangle ABC with the inequality:
$\cos 2A + \cos 2B + \cos 2C \geq -\frac{3}{2}$ This is a known trigonometric inequality for the angles of a triangle. By using standard trigonometric identities and properties of angles in a triangle, it is established that:
$\cos 2A + \cos 2B + \cos 2C \geq -\frac{3}{2}$ Thus, Statement-II is correct.

Conclusion:
Since Statement-I is incorrect and Statement-II is correct, the correct answer is:
Answer: (2) Statement-I is incorrect but Statement-II is correct.