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:

Given vectors \( \vec{a} = \hat{i} + 2\hat{j} - 3\hat{k} \) and \( \vec{b} = 2\hat{i} + \hat{j} - \hat{k} \). Vector \( \vec{r} \) must satisfy the following two conditions:

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

1. Analysis of Statement-II:

For any triangle \( \triangle ABC \), the identity is:

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

The minimum value of \(\sin A \sin B \sin C\) is 0 for degenerate triangles. This implies the minimum sum is:

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

For non-degenerate triangles (angles strictly less than \( 180^\circ \)), the AM-GM inequality yields \(\cos 2A + \cos 2B + \cos 2C \geq -\frac{3}{2}\). Statement-II is correct.

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

Statement-I is incorrect but Statement-II is correct