Question

For some \( a, b \), let \( f(x) = \left| \begin{matrix} a + \frac{\sin x}{x} & 1 & b \\ a & 1 + \frac{\sin x}{x} & b \\ a & 1 & b + \frac{\sin x}{x} \end{matrix} \right| \), where \( x \neq 0 \), \( \lim_{x \to 0} f(x) = \lambda + \mu a + \nu b \). Then \( (\lambda + \mu + \nu)^2 \) is equal to:

• A. \( 25 \)
• B. \( 16 \)
• C. \( 36 \)
• D. \( 9 \)

Hint:

When working with matrices and limits, compute the determinant for the limit case, and then solve for the coefficients by comparing it to the given expression.

Answer 1

The Correct Option is D

First, compute the determinant of the matrix as \( x \to 0 \) and then take the limit to find the value of \( \lambda + \mu + \nu \). 
The limit and determinant calculation gives the value 3 for \( \lambda + \mu + \nu \), so squaring this gives 9. 
Final Answer: \( (\lambda + \mu + \nu)^2 = 9 \).