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Let d in R , " and " A = [{:(-2, 4+d, (...

Let ` d in R , " and " A = [{:(-2, 4+d, (sin theta) - 2),(2, sintheta+4, 2d),(5, 2sintheta-d, (-sintheta) +2+2d):}]` . If the minimum value of det(A) is 16 for all `theta in [ 0, 2pi ] , ` then a value of d is ,

A

`2 (sqrt(2) + 2)`

B

`-5`

C

`-7`

D

`2(sqrt(2) + 1) `

Text Solution

Verified by Experts

The correct Answer is:
B
`R_3 to R_3 + R_1 - R_2 `
`A = 2 [{:(-2, 4+d,(sin theta) - 2),(1,(sintheta)+2, d), (1, 0, 0):}]`
` A = 2 [d (4 + d) - (sin^2 theta - 4) ]`
` 2 [(d + 2)^2 -( sin ^2 theta ]`
`A_("min") = 2 [ (d + 2)^2 - 1 ] = 16`
` d + 2 pm 3`
` d = 1, -5`
d =-5
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