The number of values of θ∈ $$0$$,πfor which the system of equations $$x+3y+7z=0$$, −$$x+4y+7z=0$$, $$sin3$$θ$$x+cos2$$θ$$y+2z=0has$$ a $$non-trivial$$ solution is

Question

The number of values of θ∈ $$0$$,πfor which the system of equations $$x+3y+7z=0$$, −$$x+4y+7z=0$$, $$sin3$$θ$$x+cos2$$θ$$y+2z=0has$$ a $$non-trivial$$ solution is

Infinity Learn · Accepted Answer

System has $$non-trivial$$  solution⇒Δ$$=0$$ ⇒$$137$$−$$147sin3$$θ$$cos2$$θ$$2=0$$ ⇒$$18-7cos2$$θ$$-3-2-7sin3$$θ$$+7-cos2$$θ$$-4sin3$$θ$$=0$$ ⇒$$14-14cos2$$θ$$-7sin3$$θ$$=0$$ ⇒$$sin3$$θ$$+2cos2$$θ$$-2=0$$ ⇒$$3sin$$θ$$-4sin3$$θ$$+21-2sin2$$θ$$-2=0$$ ⇒$$sin$$θ$$4sin2$$θ$$+4sin$$θ$$-3=0$$ ⇒$$sin$$θ$$4sin2$$θ$$+6sin$$θ$$-2sin$$θ$$-3=0$$ ⇒$$sin$$θ$$2sin$$θ$$-12sin$$θ$$+3=0$$ ⇒$$sin$$θ$$=0$$ ,  $$2sin$$θ$$-1=0$$⇒$$sin$$θ$$=12$$ ∵θ∈$$0$$,π⇒θ$$=$$π$$6$$,$$5$$$$π$$$$6$$ ⇒The number of values of θ is $$2$$

The number of values of $θ \in (0, π)$ for which the system of equations $x + 3 y + 7 z = 0$ $, - x + 4 y + 7 z = 0$ , $(\sin 3 θ) x + (\cos 2 θ) y + 2 z = 0$ has a non-trivial solution is

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The number of values of θ∈ 0,πfor which the system of equations x+3y+7z=0, −x+4y+7z=0, sin3θx+cos2θy+2z=0has a non-trivial solution is

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The number of values of $θ \in (0, π)$ for which the system of equations $x + 3 y + 7 z = 0$ $, - x + 4 y + 7 z = 0$ , $(\sin 3 θ) x + (\cos 2 θ) y + 2 z = 0$ has a non-trivial solution is