Let `K` be a positive real number and `A=[2k-1 2sqrt(k)2sqrt(k)2sqrt(k)1-2k-2sqrt(k)2k-1]a n dB=[0 2k-1sqrt(k)1-2k0 2-sqrt(k)-2sqrt(k)0]` . If det `(a d jA)+det(a d jB)=10^6,t h e n[k]` is equal to. [Note: `a d jM` denotes the adjoint of a square matix `M` and `[k]` denotes the largest integer less than or equal to `K` ].

Answer

D

Explanation

`abs(A) = (2k-1)(-1+4k^(2))+ 2sqrt(k) (2sqrt(k)+4ksqrt(k))`
`+ 2 sqrt(k)(4ksqrt(k)+2sqrt(k)) (2k-1) (4k^(2)-1)`
`+ 4k + 8k^(2) + 8k^(2) + 4k`
`=(2k-1)(4k^(2)-1)+8k+16k^(2)`
`= 8k^(3) - 4k^(2)-2k + 1 + 8k + 16k^(2)`
`=8k^(3) + 12k^(2) + 6k +1`
`abs(B) = 0 ` is skew-symmetric matrix of odd order.
`rArr (8k^(3) + 12k^(2) + 6k+1)^(2) = (10^(3))^(2)`
`rArr (2k+1)^(3) = 10^(3)`
`rArr 2k +1 =10`
`rArr k = 4.5`
`rArr [k] = 4`