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:
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:
Show Hint
- 1215
- 1040
- 1295
- 1280
The Correct Option is D
Solution and Explanation
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 \( \boxed{1280} \).
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