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

The Correct Option is B

1. Analysis of Statement-I:

We are given vectors \( \vec{a} = \hat{i} + 2\hat{j} - 3\hat{k} \) and \( \vec{b} = 2\hat{i} + \hat{j} - \hat{k} \). The problem states that vector \( \vec{r} \) satisfies both:

• \( \vec{a} \times \vec{r} = \vec{a} \times \vec{b} \)
• \( \vec{a} \cdot \vec{r} = 0 \)

1. Analysis of Statement-II:

In any triangle \( \triangle ABC \), we have the identity:

\[\cos 2A + \cos 2B + \cos 2C = 1 - 4 \sin A \sin B \sin C\]

The minimum value of \(\sin A \sin B \sin C\) is 0 (when any angle is \( 0 \) or \( 180^\circ\) which makes it degenerate), making the minimum sum:

\[1 - 4(0) = 1\]

Using AM-GM inequality for angles strictly less than \( 180^\circ \), \( \cos 2A + \cos 2B + \cos 2C \geq -\frac{3}{2} \). This is true for any non-degenerate triangle.

Hence, Statement-II is correct.

1. Conclusion: Statement-I is incorrect while Statement-II is correct. Therefore, the correct answer is:

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.