The value of θ for which the system of equations
(sin 3 θ ) x – 2y + 3z = 0 cos 2 θ ) x + 8y – 7z = 0
2x + 14y – 11z = 0 has a non-trivial solution is -
Text Solution
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The system of equations has a non-trivial solution if and only if 
= 0
Applying R 2 → R 2 + 4R 1 , R 3 → R 3 + 7R 1 , we get
= 0
Expanding along C 2 , we get
2(cos 2 θ + 4 sin 3 θ ) – (2 + 7 sin 3 θ ) = 0
⇒ 2 – 2 cos 2 θ – sin 3 θ = 0 ⇒ 4 sin 2 θ – (3 sin θ – 4 sin 3 θ ) = 0 ⇒ sin θ (4 sin 2 θ + 4 sin θ – 3) = 0 ⇒ sin θ (2 sin θ – 1) (2 sin θ + 3) = 0
⇒ sin θ = 0 or sin θ = 1/2.
[ sin θ = –3/2 is non possible]
∴ For, θ = n π the system of equations has a non-trivial
solution.
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