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If A= [{:(1, "sin"theta,1), (-"sin"theta...

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

`[(5)/(2), 4)`

B

`((3)/(2), 3]`

C

`(0, (3)/(2)]`

D

`(1, (5)/(2)]`

Text Solution

Verified by Experts

The correct Answer is:
B
`|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] = 2[1+sin^(2) theta]`
For `theta in ((3pi)/(4), (5pi)/(4)), sin theta in ((-1)/(sqrt(2)),(1)/(sqrt(2))) sin^(2) theta in [0, (1)/(2))`
`[1+sin^(2) theta] in [1, (3)/(2))` and `2[1+sin^(2) theta] in [2,3)`
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