Consider the system of linear equations in `x , ya n dz :` `""(sin3theta)x-y+z=0(cos2theta)x+4y+3z=0` `3x+7y+7z=0` Which of the following can be the value of `theta` for which the system has a non-trivial solution `npi+(-1)^npi/6,AAn in Z` `npi+(-1)^npi/3,AAn in Z` `npi+(-1)^npi/9,AAn in Z` none of these

Options

A. `npi+(-1)^(n) pi//6, AA n in Z`
B. `n pi+(-1)^(n) pi//3, AA n in Z`
C. `npi+(-1)^(n) pi//9, AA n in Z`
D. none of these

Answer

A

Explanation

Since the system has a non-trivial solution, the determinant of coefficients is 0. Thus,
`|(sin 3 theta,-1,1),(cos 2 theta,4,3),(2,7,7)|=0`
or `sin 3 theta (28-21)-cos 2 theta(-7-7)+2(-3-4)=0`
or `sin 3 theta+2 cos 2 theta-2 =0`
or `(3 sin theta-4sin^(3) theta)+2(1-2 sin^(2) theta)-2=0`
or `4 sin^(3) theta+4 sin^(2) theta-3 sin theta=0`
or `sin theta(2 sin theta-1) (2 sin theta+3) =0`
or `sin theta=0, 1/2`
`:. theta=n pi or theta= npi +(-1)^(n) pi//6, AA n in Z`