The set of equations λx−y+(cosθ)z=0, 3x+y+2z=0 and (cosθ)x+y+2z=0 where 0≤θ<2π has nontrivial solutions,

312cosθ12λ−1cosθ=3(cosθ+2)−1(cos2θ−2λ) +2(−cosθ−λ) =3cosθ+6−cos2θ+2λ−2cosθ −2λ =cosθ−cos2θ+6>0

The set of equations λx−y+(cosθ)z=0, 3x+y+2z=0 and (cosθ)x+y+2z=0 where 0≤θ<2π has nontrivial solutions,

312cosθ12λ−1cosθ=3(cosθ+2)−1(cos2θ−2λ) +2(−cosθ−λ) =3cosθ+6−cos2θ+2λ−2cosθ −2λ =cosθ−cos2θ+6>0

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The set of equations $λ x - y + (\cos θ) z = 0, 3 x + y + 2 z = 0$ and $(\cos θ) x + y + 2 z = 0$ where $0 \leq θ < 2 π$ has nontrivial solutions,

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for no value of $θ$

for all values of $λ, θ$

for all values of $λ$ and only two values of $θ$

for only one value of $λ$ and all values of $θ$

answer is A.

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Detailed Solution

$|\begin{matrix} 3 & 1 & 2 \\ \cos θ & 1 & 2 \\ λ & - 1 & \cos θ \end{matrix}| = 3 (\cos θ + 2) - 1 (\cos^{2} θ - 2 λ) + 2 (- \cos θ - λ) \begin{array}{l} = 3 \cos θ + 6 - \cos^{2} θ + 2 λ - 2 \cos θ - 2 λ \\ = \cos θ - \cos^{2} θ + 6 > 0 \end{array}$