Let $M$ and $m$ respectively be the maximum and the minimum values of $\begin{vmatrix} 1 + \sin^2 x & \cos^2 x & 4 \sin 4x \\ 1 + \sin^2 x & \cos^2 x & 4 \sin 4x \\ \sin^2 x & \cos^2 x & 1 + 4 \sin 4x \end{vmatrix}, x \in \mathbf{R}$. Then $M^4 - m^4$ is equal to:

Step 1: Simplify the Determinant

First, let's denote the given determinant as $D$. We observe that the first two rows are identical. A fundamental property of determinants states that if two rows (or columns) of a matrix are identical, the determinant of that matrix is zero. However, in this case, the first two rows are not identical. Let's re-examine the problem statement carefully. Ah, I see a typo in the problem statement. Assuming the problem intended for the first two rows to be identical, the determinant would be zero. But if we proceed with the given matrix, we can simplify it using row operations.

Step 2: Apply Row Operations

To simplify the determinant, we can perform row operations. Subtract the first row ($R_1$) from the second row ($R_2$) and also from the third row ($R_3$). This operation does not change the value of the determinant.

Step 3: Evaluate the Determinant

After performing the row operations, the second row becomes all zeros. A property of determinants states that if any row or column of a matrix consists entirely of zeros, then the determinant of the matrix is zero. Therefore, the value of the determinant $D$ is $0$ for all $x \in \mathbf{R}$.

Step 4: Determine Maximum and Minimum Values

Since the determinant $D$ is always $0$, its maximum value $M$ and minimum value $m$ are both $0$.

Step 5: Calculate $M^4 - m^4$

Finally, we substitute the values of $M=0$ and $m=0$ into the expression $M^4 - m^4$ to get the final answer.

More from this chapter

• Evaluate the determinant 1 & 0 & 4 3 & 5 & -1 0 & 1 & 2 (i) with the help of second row. (ii) with the help of third column.
• If A = [[1, 2], [3, 4]] find det(A) and adj(A).
• The value of 1 & 2 & 3 4 & 5 & 6 7 & 8 & 9 is:
• Using Cramer's rule, show that the following system of linear equations is consistent and hence solve it: x + y + z = 1, 2x + 3y + 2z = 2, x + y + 2z = 4.
• Let a R and A be a matrix of order 3 3 such that det(A)=-4 and A+I= 1 & a & 1 2 & 1 & 0 a & 1 & 2, where I is the 3 3 identity. If det ((a+1)adj((a-1)A))=2^m3^n
• Let A be a 3 × 3 matrix such that |adj (adj(adj A))| = 81. If S = n Z: |adj(adj A)|^(((n-1)²)/(2)) = |A|³n² - 5n - 4, then Σ_n S | A^(n² + n) | is equal to:
• Find the equation of the line joining A(1,3) and B(0,0) using determinants and find k if D(k, 0) is a point such that area of triangle ABD is 3 sq units.
• If the system of linear equations 3x + y + β z = 3, 2x + α y - z = -3, x + 2y + z = 4 has infinitely many solutions, then the value of 22β - 9α is:

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