Question

If \[ \Delta_r = \begin{vmatrix} 2r - 1 & mC_r & 1 \\ m^2 - 1 & 2^m & m + 1 \\ \sin^2(m^2) & \sin^2(m) & \sin^2(m + 1) \end{vmatrix}, \] then the value of \[ \sum_{r=0}^{m} \Delta_r \text{ is:} \]

• A. 1
• B. 0
• C. 2
• D. None of these

Hint:

When summing over determinants with patterns or symmetries, look for simplifications using properties of determinants or known identities.

Answer 1

The Correct Option is B

Step 1: Simplifying the Determinant.
The determinant \( \Delta_r \) is a 3x3 matrix. To solve for \( \sum_{r=0}^{m} \Delta_r \), we need to evaluate the sum of the determinants for each value of \( r \). The entries depend on \( r \) and \( m \), and we need to simplify the expression.
Step 2: Using Properties of Determinants.
We can use properties of determinants (such as expansion along rows or columns) to simplify the matrix. However, after simplification and summing over all values of \( r \), the result for \( \sum_{r=0}^{m} \Delta_r \) is found to be zero.
Conclusion.
The correct answer is (2) \( 0 \), as the sum of the determinants evaluates to zero.