Question

Among the statements :
I: If the given determinants are equal, then \( \cos^2\alpha + \cos^2\beta + \cos^2\gamma = \frac{3}{2} \), and
II: If the polynomial determinant equals \( px + q \), then \( p^2 = 196q^2 \), identify the truth value.

• A. both are true
• B. only I is true
• C. both are false
• D. only II is true

Hint:

To find \( q \) in a determinant equal to \( px+q \), simply substitute \( x=0 \) into the original determinant.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This problem requires expanding determinants. Statement I deals with a specific trigonometric identity, while Statement II involves solving for coefficients in a linear polynomial resulting from a determinant.
Step 2: Detailed Explanation:
Statement I: Expanding the LHS: \( 1 - (\cos^2\alpha + \cos^2\beta + \cos^2\gamma) + 2\cos\alpha\cos\beta\cos\gamma \). Expanding the RHS: \( 2\cos\alpha\cos\beta\cos\gamma \). Equating gives \( \cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1 \). Hence, Statement I is false.
Statement II: Using row transformations \( R_2 \to R_2 - 2R_1 \) and \( R_3 \to R_3 - R_1 \), the quadratic terms \( x^2 \) vanish. The expansion results in \( px + q \). Computing \( p \) and \( q \) confirms the relationship \( p^2 = 196q^2 \).
Step 3: Final Answer:
Only Statement II is true.