Let m be a positive integer and $\Delta_r=\left|\begin{array}{ccc}2 r-1 & m C_r & 1 \\ m^2-1 & 2^m & m+1 \\ \sin ^2\left(m^2\right) & \sin ^2(m) & \sin \left(m^2\right)\end{array}\right|$. Then the value of $\sum\limits_{r=0}^m \Delta_r$ is given by

0

Using concept of summation of determinant.

We get R₁, R₂ are identical so $\sum\limits_{r=0}^m \Delta_r$ is zero.

Hence (1) is the correct answer.