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Q.

If the system of linear equations (sin3 $θ$ ) x + y + z = 0, (cos2 $θ$ ) x + 4y + 3z = 0, 2 x + 7y + 7z = 0 has a non-trivial solution then the values of $θ$ are

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a

$nπ, nπ + (- 1)^{n} \frac{π}{3}, n \in Z$

b

$nπ, nπ + (- 1)^{n} \frac{π}{6}, n \in Z$

c

$nπ, nπ + (- 1)^{n} \frac{π}{2}, n \in Z$

d

$nπ, nπ + (- 1)^{n} \frac{π}{4}, n \in Z$

answer is B.

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Detailed Solution

$\begin{array}{l} |\begin{array}{ccc} \sin 3 θ & - 1 & 1 \\ \cos θ & 4 & 3 \\ 2 & 7 & 7 \end{array}| = 0 \\ (\sin θ) (28 - 21) (7 \cos θ - 6) + 1 (7 \cos θ - 8) = 0 \\ 7 \sin 3 θ + 7 \cos 2 θ - 6 + 7 \cos θ - 8 = 0 \Rightarrow 14 \cos 2 θ + 7 \sin 3 θ - 14 = 0 \\ \sin θ = \frac{1}{2}, \sin θ = \frac{- 3}{2} \\ ∴ θ = n \prod + (- 1)^{n} \frac{Π}{6} \end{array}$

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