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(sqrt2+2)`
B. `-5`
C. `-7`
D. `2(sqrt2+1)`

Answer

B

Explanation

`A=abs({:(-2,4+d,(sintheta)-2),(1,(sintheta)+2,d),(5,(2sintheta-d),-sintheta+2+2d):})`
`R_(3)toR_(3)+R_(1)-2R_(2)`
`A=abs({:(-2,4+d,sintheta-2),(1,sintheta)+2,d),(1,0,0):})`
`A=d(4+d)-(sin^(2)theta-4)`
`(d+2)^(2)-sin^(2)theta`
`A_("min")=(d+2)^(2)-1=8`
`d+2=pm3`
`d=1,-5`