Question

The set of equations: \lambda x-y+(\cos \theta) z=0,3 x+y+2 z=0,(\cos \theta) x+y+2 z=0,0 \leq \theta<2 \piλx−y+(cosθ)z=0,3x+y+2z=0,(cosθ)x+y+2z=0,0≤θ<2π
has non-trivial solution(s):

Answer 1

The set of equation has non-trivial solution(s) for only one value of λ and all values of θ.

Given:

λx - y + cos 3 = 0

3x + y + 2z = 0

cos x + y + 2z = 0

θ ∈ [0, 2π]

Step-by-step explanation:

Δ = 0

$\left|\begin{array}{ccc}\lambda & -1 & \cos \theta \\3 & 1 & 2 \\\cos \theta & 1 & 2\end{array}\right|=0$

21 + 3 cos θ - 2 cos θ - cos² θ - 2λ + 6 = 0

Δ = cos θ - cos² θ + 6 = 0

cos² θ - cos θ - 6 = 0

⇒ (cos θ - 3) (cos θ + 2) = 0