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Let K be a positive real number and A=[(...

Let K be a positive real number and `A=[(2k-1,2sqrt(k),2sqrt(k)),(2sqrt(k),1,-2k),(-2sqrt(k),2k,-1)]` and `B=[(0,2k-1,sqrt(k)),(1-2k,0,2),(-sqrt(k),-2sqrt(k),0)]`. If det (adj A) + det (adj B) `=10^(6)`, then `[k]` is equal to ______ .
[Note : adj M denotes the adjoint of a square matrix M and `[k]` denotes the largest integer less than or equal to `k`.]

Text Solution

Verified by Experts

The correct Answer is:
D
`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`
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