The value of `theta` lying between `theta=0a n dtheta=pi/2` and satisfying the equation `|1+sin^2thetacos^2theta4sin4thetasin^2theta1+cos^2theta4sin4thetasin^2thetacos^2theta1+4sin4theta|=0a r e` `(7pi)/(24)` (b) `(5pi)/(24)` (c) `(11pi)/(24)` (d) `pi/(24)`

Options

A. `7pi//24`
B. `5pi//24`
C. `11pi//24`
D. `pi//24`

Answer

A, C

Explanation

Given, `|{:(,1+sin^(2)theta,cos^(2) theta,4sin4 theta),(,sin^(2)theta,1+cos^(2)theta,4sin4theta),(,sin^(2)theta,cos^(2)theta,1+4sin 4theta):}|=0`
Applying `R_(3) to R_(3) - R_(1) and R_(2) to R_(2)-R_(1)` we get
`|{:(,1+sin^(2)theta, cos^(2)theta, 4sin4theta),(,-1,1,0),(,-1,0,1):}|=0`
Applying `C_(1) to C_(1) +C_(2)` we get
`|{:(,2,cos^(2)theta,4sin4theta),(,0,1,0),(,-1,0,1):}|=0`
`rArr 2+4sin 4 theta =0`
`rArr sin 4theta=(1)/(2)`
`rArr 4 theta = npi + ( -1)^(n)(-(pi)/(6))`
`rArr theta = (npi)/(4) + (-1)^(n+1)((pi)/(24))`