Key Concepts and Formulas

This problem primarily involves the following mathematical concepts:

1. Properties of Determinants:

   • Row/Column Operations: The value of a determinant remains unchanged if we apply the operation $R_i \rightarrow R_i + k R_j$ (or $C_i \rightarrow C_i + k C_j$) where $R_i$ and $R_j$ are rows (or $C_i$ and $C_j$ are columns). This property is crucial for simplifying determinants by introducing zeros.
   • Expansion of a $3 \times 3$ Determinant: For a matrix $\begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}$, its determinant can be expanded along the first row as $a(ei - fh) - b(di - fg) + c(dh - eg)$. Expanding along any row or column yields the same result.

2. Trigonometric Identities:

   • The fundamental identity: $\sin^2 x + \cos^2 x = 1$.

3. Range of Trigonometric Functions:

   • For any real number $\theta$, the sine function satisfies $-1 \le \sin \theta \le 1$. This property is used to find the maximum and minimum values of the function.

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Step-by-Step Solution

Step 1: Simplify the Determinant using Row Operations

We are given the function $f(x)$ as a determinant: $f(x)=\left|\begin{array}{ccc} 1+\sin ^2 x & \cos ^2 x & 4 \sin 4 x \\ \sin ^2 x & 1+\cos ^2 x & 4 \sin 4 x \\ \sin ^2 x & \cos ^2 x & 1+4 \sin 4 x \end{array}\right|$

Our goal is to simplify this determinant. A common strategy is to introduce zeros into rows or columns using row/column operations, which significantly eases the expansion process. Observe that the first two columns have similar terms in different rows. We can perform row operations to simplify these terms.

Let's apply the following row operations:

1. $R_2 \rightarrow R_2 - R_1$
2. $R_3 \rightarrow R_3 - R_1$

Why these operations? These operations are chosen because they will create simple entries (including zeros) in the second and third rows, especially in the first two columns. For example, in the second row, the first element will become $\sin^2 x - (1+\sin^2 x) = -1$, and the second element will become $(1+\cos^2 x) - \cos^2 x = 1$. Similarly for the third row. These operations do not change the value of the determinant.

Let's apply them: For $R_2 \rightarrow R_2 - R_1$:

• $(R_2)_{1} = \sin^2 x - (1+\sin^2 x) = -1$
• $(R_2)_{2} = (1+\cos^2 x) - \cos^2 x = 1$
• $(R_2)_{3} = 4 \sin 4x - 4 \sin 4x = 0$

For $R_3 \rightarrow R_3 - R_1$:

• $(R_3)_{1} = \sin^2 x - (1+\sin^2 x) = -1$
• $(R_3)_{2} = \cos^2 x - \cos^2 x = 0$
• $(R_3)_{3} = (1+4 \sin 4x) - 4 \sin 4x = 1$

After applying these operations, the determinant becomes: $f(x)=\left|\begin{array}{ccc} 1+\sin ^2 x & \cos ^2 x & 4 \sin 4 x \\ -1 & 1 & 0 \\ -1 & 0 & 1 \end{array}\right|$

Step 2: Expand the Simplified Determinant

Now that we have a simplified determinant with zeros, we can expand it. We can expand along any row or column. Expanding along $R_1$ (as in the original solution) or $R_2$ or $R_3$ (due to the presence of zeros) would be efficient. Let's expand along $R_1$:

$f(x) = (1+\sin^2 x) \cdot \text{Cofactor}(R_1, C_1) - (\cos^2 x) \cdot \text{Cofactor}(R_1, C_2) + (4 \sin 4x) \cdot \text{Cofactor}(R_1, C_3)$

Where $\text{Cofactor}(R_i, C_j) = (-1)^{i+j} \times \text{Minor}(R_i, C_j)$.

• The minor for $(1+\sin^2 x)$ is $\left|\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right| = (1)(1) - (0)(0) = 1$.
• The minor for $(\cos^2 x)$ is $\left|\begin{array}{cc} -1 & 0 \\ -1 & 1 \end{array}\right| = (-1)(1) - (0)(-1) = -1$.
• The minor for $(4 \sin 4x)$ is $\left|\begin{array}{cc} -1 & 1 \\ -1 & 0 \end{array}\right| = (-1)(0) - (1)(-1) = 0 - (-1) = 1$.

Substituting these values into the expansion: $f(x) = (1+\sin^2 x)(1) - (\cos^2 x)(-1) + (4 \sin 4x)(1)$ $f(x) = 1+\sin^2 x + \cos^2 x + 4 \sin 4x$

Why this simplification? We use the fundamental trigonometric identity $\sin^2 x + \cos^2 x = 1$. $f(x) = 1 + (\sin^2 x + \cos^2 x) + 4 \sin 4x$ $f(x) = 1 + 1 + 4 \sin 4x$ $f(x) = 2 + 4 \sin 4x$ This is the simplified form of the function $f(x)$.

Step 3: Determine the Maximum (M) and Minimum (m) Values of $f(x)$

We have $f(x) = 2 + 4 \sin 4x$. To find the maximum and minimum values of $f(x)$, we need to consider the range of the sine function. We know that for any real value $\theta$ (in our case, $\theta = 4x$): $-1 \le \sin \theta \le 1$ $-1 \le \sin 4x \le 1$

Now, we build up the expression for $f(x)$ using this inequality: First, multiply by 4: $-4 \le 4 \sin 4x \le 4$ Next, add 2 to all parts of the inequality: $2 - 4 \le 2 + 4 \sin 4x \le 2 + 4$ $-2 \le f(x) \le 6$

From this inequality, we can identify the maximum and minimum values of $f(x)$:

• Maximum value, $M = 6$
• Minimum value, $m = -2$

Step 4: Calculate $M^4 - m^4$

Finally, we need to calculate the value of $M^4 - m^4$: $M^4 - m^4 = (6)^4 - (-2)^4$

Calculate the powers:

• $6^4 = 6 \times 6 \times 6 \times 6 = 36 \times 36 = 1296$
• $(-2)^4 = (-2) \times (-2) \times (-2) \times (-2) = 4 \times 4 = 16$

Substitute these values: $M^4 - m^4 = 1296 - 16$ $M^4 - m^4 = 1280$

Thus, the value of $M^4 - m^4$ is 1280.

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Tips for Students & Common Mistakes to Avoid

1. Prioritize Simplification: Always look for opportunities to simplify determinants using row or column operations before expanding. This reduces the number of terms and the chances of calculation errors. Operations like $R_i \rightarrow R_i + kR_j$ (or $C_i \rightarrow C_i + kC_j$) are powerful because they do not change the determinant's value.
2. Trigonometric Identities are Key: Keep fundamental trigonometric identities like $\sin^2 x + \cos^2 x = 1$ handy. They often appear in these problems to simplify expressions.
3. Range of Functions: When finding maximum/minimum values, remember the basic ranges of trigonometric functions. For $\sin x$ and $\cos x$, the range is $[-1, 1]$. Be careful when multiplying or adding constants to these inequalities.
4. Powers of Negative Numbers: Pay attention to signs when calculating powers of negative numbers. An even power of a negative number results in a positive number (e.g., $(-2)^4 = 16$), while an odd power results in a negative number.
5. Careful Expansion: When expanding a determinant, remember the alternating signs of the cofactors. If expanding along $R_1$, the signs are + - +.

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Summary and Key Takeaway

This problem is an excellent illustration of how different mathematical topics—matrices and determinants, trigonometry, and functions—can be integrated into a single question. The most efficient path to the solution involves:

1. Strategically applying row/column operations to simplify the determinant.
2. Using trigonometric identities