If m is a positive integer, `Delta_r=|{:(2r-1,""^mC_r,1),(m^2-1,2^m,m+1),(sin^2(m^2),sin^2(m),sin^2(m+1)):}|,"then value of " sum_(r=0)^(m)Delta_r" is"`

Options

A. 1
B. 0
C. 2
D. -1

Explanation

As variables (r ) is in only lst row
`:." "underset(r=0)overset(m)sumDeltar=|{:(underset(r=0)overset(m)(sum)r-1,underset(r=0)overset(m)(sum)""^mC_r,underset(r=0)overset(m)(sum)1),(m^2-1,2^m,m+1),(sin^2(m^2),sin^2m,sin^2(m+1)):}|`
`=|{:(m^2-1,2^m,m+1),(m^2-1,2^m,m+1),(sin^2m^2,sin^2m,sin^2(m+1)):}|=0["as "R_1" and "R_2" are identical"]`
Hence option (2) correct.