Question

If `A= [{:(1, "sin"theta,1), (-"sin"theta, 1, "sin" theta),(-1, -"sin"theta, 1):}], " then for all "theta in ((3pi)/(4), (5pi)/(4))` det (A) lies in the interval
A. `((3)/(2),3]`
B. `[(5)/(2),4)`
C. `(0,(3)/(2)]`
D. `(1,(5)/(2)]`

Answer 1

Correct Answer - D
Given matrix `A= [{:(1,"sin"theta,1), (-"sin"theta, 1,"sin"theta), (-1, -"sin"theta, 1):}]`
`rArr "det"(A) = |A| = |{:(1,"sin"theta,1), (-"sin"theta, 1,"sin"theta), (-1, -"sin"theta, 1):}|`
`=1(1+"sin"^(2)theta)-"sin"theta(-"sin" theta + "sin"theta)+1("sin"^(2)theta + 1)`
`rArr |A| = 2(1+"sin"theta) " "...(i)`
As we know that, for `theta in ((3pi)/(4), (5pi)/(4))`
`"sin"theta in (-(1)/(sqrt(2)), (1)/(sqrt(2))`
`"sin"^(2)theta in [0, (1)/(2)) rArr 1+"sin"^(2)theta in [0+1, (1)/(2) + 1)`
`rArr 1+"sin"^(2) theta in [1, (3)/(2))`
`rArr 2(1+"sin"^(2)theta) in [2, 3) rArr |A| in [2, 3) sub ((3)/(2), 3]`