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Let \(\Delta_r\) denote the determinant \[ \begin{vmatrix} 2r-1 & {^mC_r} & 1\\ m^2-1 & 2^m & m+1\\ \sin^2(m^2) & \sin^2(m) & \sin^2(m+1) \end{vmatrix}. \] Then the value of \(\sum_{r=0}^{m} \Delta_r\) is

1. 0
2. 4
3. 3
4. 1

Correct answer: 0

Solution

The determinant \\Delta_r is structured such that its rows exhibit linear dependencies when summed over the range of r from 0 to m, leading to a total sum of zero. This is due to the properties of determinants where linear combinations of rows can result in a determinant of zero.

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