Number of values of `theta` lying I [0,100`pi`] for which the system of equations (sin 3`theta`) x-y+z=0, (cos 2`theta`) x+4y +3z=0, 2x+ 7y+7z =0 has non-trivial solution is `"____"`

Answer

201

Explanation

The system has a non-trivial solution if
`Delta = |{:(sin 3 0 ,,-1,,1),(cos 2 0 ,,4,,3),(2,,7,,7):}| =0`
`rArr (28 -21) sin 3 0 -(-7-7) cos 3 0 +2 (-3-4)=0`
`rArr sin 3 0 + 2cos 2 0 -2 =0`
`rArr 3 sin 0 -4 sin^(3) 0+ 2 -4 sin^(2) 0 -2 =0`
`rArr 4 sin^(3) 0 +4 sin^(2) 0 -3 sin 0 =0`
`rArr sin 0 (4 sin ^(2) 0 +4 sin 0-3 =0`
`rArr sin 0 (2 sin 0 +3 ) (2 sin 0 -1) =0`
So, sin 0 = 0 , `1//2 ` since sin `0 ne -3//2`
Now sin 0 =0 is for `0 , pi , 2pi ......, 100 pi ` total 101 values
sin 0 `=1//2 `for 0 = `pi//3` and `2pi//3`
So there wil be 100 values lying in `[0, 100pi]`
Therefore total number of values of 0 is 20]