Question

Let \( M \) and \( m \) respectively be the maximum and the minimum values of \( f(x) = \begin{vmatrix} 1 + \sin^2x & \cos^2x & 4\sin4x \\ \sin^2x & 1 + \cos^2x & 4\sin4x \\ \sin^2x & \cos^2x & 1 + 4\sin4x \end{vmatrix}, \quad x \in \mathbb{R} \)  for \( x \in \mathbb{R} \). Then \( M^4 - m^4 \) is equal to:

• A. 1215
• B. 1040
• C. 1295
• D. 1280

Hint:

For functions involving trigonometric expressions, use trigonometric identities and calculus techniques to find extrema.

Answer 1

The Correct Option is D

We are tasked with analyzing the determinant of a given matrix and determining the maximum and minimum values of the resulting function \( f(x) \). Let us proceed step by step:

1. The Given Matrix:
The matrix is:

\( \begin{vmatrix} 1 + \sin^2x & \cos^2x & 4\sin4x \\ \sin^2x & 1 + \cos^2x & 4\sin4x \\ \sin^2x & \cos^2x & 1 + 4\sin4x \end{vmatrix}, \quad x \in \mathbb{R} \)

2. Row Operations:
To simplify the determinant, perform the following row operations:

\( R_2 \to R_2 - R_1 \) and \( R_3 \to R_3 - R_1 \):

\( \begin{vmatrix} 1 + \sin^2x & \cos^2x & 4\sin4x \\ -1 & 1 & 0 \\ -1 & 0 & 1 \end{vmatrix} \)

3. Expanding the Determinant:
Expand the determinant about the first row:

\( f(x) = (1 + \sin^2x) \begin{vmatrix} 1 & 0 \\ 0 & 1 \end{vmatrix} - \cos^2x \begin{vmatrix} -1 & 0 \\ -1 & 1 \end{vmatrix} + 4\sin4x \begin{vmatrix} -1 & 1 \\ -1 & 0 \end{vmatrix} \)

Compute each minor determinant:

\( \begin{vmatrix} 1 & 0 \\ 0 & 1 \end{vmatrix} = 1, \quad \begin{vmatrix} -1 & 0 \\ -1 & 1 \end{vmatrix} = (-1)(1) - (0)(-1) = -1, \quad \begin{vmatrix} -1 & 1 \\ -1 & 0 \end{vmatrix} = (-1)(0) - (1)(-1) = 1 \)

Substitute these values back into the expansion:

\( f(x) = (1 + \sin^2x)(1) - \cos^2x(-1) + 4\sin4x(1) \)

Simplify:

\( f(x) = 1 + \sin^2x + \cos^2x + 4\sin4x \)

Using the Pythagorean identity \( \sin^2x + \cos^2x = 1 \):

\( f(x) = 1 + 1 + 4\sin4x = 2 + 4\sin4x \)

4. Finding Maximum and Minimum Values:
The function \( f(x) = 2 + 4\sin4x \) depends on \( \sin4x \), which oscillates between \(-1\) and \(1\):

\( \text{Maximum value of } f(x): \quad f(x) = 2 + 4(1) = 6 \)

\( \text{Minimum value of } f(x): \quad f(x) = 2 + 4(-1) = -2 \)

Thus:

\( M = 6 \quad \text{and} \quad m = -2 \)

5. Computing \( M^4 - m^4 \):
Using the values of \( M \) and \( m \):

\( M^4 - m^4 = 6^4 - (-2)^4 \)

Compute each term:

\( 6^4 = 1296, \quad (-2)^4 = 16 \)

Subtract:

\( M^4 - m^4 = 1296 - 16 = 1280 \)

Final Answer:
The value of \( M^4 - m^4 \) is \( \boxed{1280} \).