Let `d in R, and A[{:(,-2,4+d,(sin theta-2)),(,1,(sin theta)+2,d),(,5,(2sin theta)d,(-sin theta)+2+2d):}]=theta in [0,2pi]` If the minimum value of det(A) is B. Then the value of d is:

Options

A. `2 (sqrt(2) + 2)`
B. `-5`
C. `-7`
D. `2(sqrt(2) + 1) `

Answer

B

Explanation

`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