If `D_k=|1nn2k n^2+n+2n^2+n2k-1n^2n^2+n+2|a n d` `sum_(k=1)^n D_k=48 ,t h e nne q u a l s` 4 (b) 6 (c) 8 (d) none of these

Options

A. 4
B. 6
C. 8
D. none of these

Answer

A

Explanation

We have,
`underset(k=1)overset(n)sum D_(k) = 48`
`rArr |(underset(k=1)overset(n)sum 1,n,n),(underset(k=1)overset(n)sum 2k,n^(2) + n +2,n^(2) +n),(underset(k=1)overset(n)sum 2k-1,n^(2),n^(2) + n +2)| = 48`
`rArr |(n,n,n),(n(n+1),n^(2) + n +2,n^(2) + n),(n^(2) ,n^(2),n^(2) + n +2)|`
`rArr |(n,0,0),(n^(2) + n,2,0),(n^(2),0,n +2)| = 48 " " [("Applying" C_(2) rarr C_(2) - C_(1)),(C_(3) rarr C_(3) - C_(1))]`
`rArr n (2n +4) = 48`
`rArr n^(2) + 2n - 24 = 0`
`rArr (n +6) (n-4) = 0 rArr n = 4 " " [ :' n + 6 != 0]`