Given,
S be the set of all values of θ∈[−π,π] for which the system of linear equations
x+y+3z=0
−x+(tanθ)y+7z=0
x+y+(tanθ)z=0
has non-trivial solution,
So, by condition of non-trivial solution we get,
∣1−111tanθ137tanθ∣=0
⇒(tan2θ−7)−(−tanθ−7)+3(−1−tanθ)=0
⇒tan2θ+(1−3)tanθ−3=0
⇒(tanθ+1)(tanθ−3)=0
⇒tanθ=3,−1
⇒θ=3π,3−2π,4−π,43π
Hence,
π120∑θ=120(31−32−41+43)=20