`" if " f(0) = |{:(sin 0 ,,cos 0,,sin 0),(cos0,,sin0,,cos0),(cos0,, sin 0,,sin 0):}|` then

Options

A. f(0)= 0 has exactly 2 real solutions in `[0 ,pi]`
B. f(0) =0 has exactly 3 real solutions in `[0 ,pi]`
C. range of function `(f(0))/(1-sin 2 0) " is " [-sqrt(2) , sqrt(2)]`
D. range of fucntion .`(f(0))/(sin 2 0-1) " is " [-3,3]" is " [-3,3]`

Answer

A, C

Explanation

`f(0) =sin^(3) 0 +cos^(3) 0 -cos 0 sin 0 (sin0 + cos 0)`
`=(sin 0 + cos 0 )^(3) -4 sin 0 cos 0 (sin 0 + sin 0)`
`=(sin 0 +cos 0) [1-sin 2 0]`
`Now f(0)=0`
`rArr tan 0 =-1 " or " sin 20 =1 `
`rArr f(0) =0` has 2 real solutions in `[0,pi]`