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

`theta = n pi, npi + (-1)^(n)(pi)/(6), n in Z`

Explanation

The systems of equations has non-trivial solution, if `Delta = 0`
`rArr |{:("sin" 3 theta, -1, 1), ("cos" 2theta, 4, 3), (2, 7, 7):}| =0`
Expanding along `C_(1)`, we get
`"sin"3theta *(28-21) -"cos" 2 theta (-7-7) + 2 (-3-4) = 0`
`rArr 7 "sin" 3 theta + 14"cos" 2 theta-14 = 0`
`rArr "sin" 3 theta + 2 "cos" 2 theta -2 =0`
`rArr 3 "sin" theta - 4"sin"^(3) theta + 2(1-2 "sin"^(2)theta) -2 =0`
`rArr "sin" theta (4"sin"^(2) theta + 4 "sin" theta -3) =0`
`rArr "sin" theta (2"sin" theta-1) (2"sin" theta + 3) = 0`
`rArr "sin" theta = 0, "sin" theta = (1)/(2)`
`["neglecting sin"theta =-3//2]`
`rArr theta= n pi, npi + (-1)^(n) (pi)/(6), n in Z`