`" If "|{:(sin theta cos phi,,sin theta sin phi,,cos theta),(cos theta cos phi,, cos theta sin phi,,-sin theta),(-sin theta sin phi,,sin theta cos phi,,theta):}|`then

Options

A. `Delta` is independent of theta
B. `Delta` is independent of `phi`
C. `Delta `is a constant
D. `[(dDelta)/d (theta)]_(theta=pi//2)=0`

Answer

B, D

Explanation

applying `C_(1) to C_(1) - (cot phi) C_(2) ` we get
`Delta =|{:(0,,sin 0 sin phi,,cos 0),(0,,cos 0 sin phi,,-sin 0),(-sin 0 // sin phi,,sin 0 cos phi ,,0):}|`
`=-(sin 0)/(sin phi) [-sin phi sin^(2) 0 - cos^(2) 0 sin phi]`
[expanding along `C_(1)`]
=sin 0 which is independent of `phi`
`" Also " .(dDelta)/(d0) =cos 0 rArr (dDelta)/(d0) ]_(0=pi//2) = cos (pi//2) =0`