Question

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

Answer 1

As given equations has a non-trivial solution,
`:. |[sin3theta,-1,1],[cos2theta,4,3],[2,7,7]| = 0`
`=>sin3theta(28-21)-cos2theta(-7-7) +2(-3-4) = 0`
`=>7sin3theta + 14cos2theta -14 =0`
`=>sin3theta +2cos2theta -2 = 0`
`=>3sintheta - 4sin^3theta +2- 4sin^2theta -2 =0`
`=>-sintheta(4sin^2theta + 4sintheta -3) = 0`
`=>sintheta(4sin^2theta + 6sintheta - 2sintheta -3) = 0`
`=>sintheta(2sintheta(2sintheta+3) - 1(2sintheta +3)) = 0`
`=>sintheta(2sintheta- 1)(2sintheta +3) = 0`
`=>sintheta = 0 or sin theta = 1/2 or sin theta = -3/2`
`sin theta` can not be less than `-1`, so `sin theta != -3/2`.
`:. sintheta = 0 or sin theta = 1/2`
`theta = npi or theta = npi+(-1)^n(pi/6).`