Question

If $\Delta _{r} = \begin{vmatrix}2r-1&^{m}C_{r}&1\\ m^{2}-1&2^{m}&m+1\\ sin^{2}\left(m^{2}\right)&sin^{2}\left(m\right)&sin^{2}\left(m+1\right)\end{vmatrix}$ then value of $\sum\limits^{m}_{r = 0} \Delta _{r}$ is

• A. 0
• B. 4
• C. 3
• D. 1

Answer 1

Answer: (A)

$\Delta _{r} = \begin{vmatrix}2r-1&^{m}C_{r}&1\\ m^{2}-1&2^{m}&m+1\\ sin^{2}\left(m^{2}\right)&sin^{2}\left(m\right)&sin^{2}\left(m+1\right)\end{vmatrix}$
$\Rightarrow \sum\limits_{r=0}^{m} \,\Delta_{r} = \begin{vmatrix} \sum\limits_{r=0}^{m} \left(2r -1 \right) &\sum\limits_{r=0}^{m} {^mC_{r}} &\sum\limits_{r=0}^{m} 1 \\ m^{2} - 1 &2^{m} &m + 1\\ sin^{2} \left(m^{2}\right) &sin^{2}\left(m\right) &sin^{2}\left(m+1\right) \end{vmatrix}$
$= \begin{vmatrix}m^{2}-1&2^{m}&m+1\\ m^{2}-1&2^{m}&m+1\\ sin^{2}\left(m^{2}\right)&sin^{2}\left(m\right)&sin^{2}\left(m+1\right)\end{vmatrix} = 0$