If the system of equations $x+(\sqrt{2}\sin\alpha)y+(\sqrt{2}\cos\alpha)z=0$, $x+(\cos\alpha)y+(\sin\alpha)z=0$, $x+(\sin\alpha)y-(\cos\alpha)z=0$ has a non-trivial solution, then $\alpha \in \left(0, \frac{\pi}{2}\right)$ is equal to

JEE Main · 2024 · 04 Apr 2024 · Shift 1 · Determinants

If the system of equations

x + (2 sin α) y + (2 cos α) z = 0

x + (cos α) y + (sin α) z = 0

x + (sin α) y - (cos α) z = 0

has a non-trivial solution, then

α \in (0, \frac{π}{2})

is equal to

Answer: (C)

A non-trivial solution requires the coefficient determinant to vanish:

111 2 sin α cos α sin α 2 cos α sin α - cos α = 0.

Expanding along the first column gives

- 1 + 2 sin 2 α - 2 cos 2 α = 0 \Rightarrow sin 2 α - cos 2 α = \frac{1}{2} .

That is

2 sin (2 α - \frac{π}{4}) = \frac{1}{2}

, so

sin (2 α - \frac{π}{4}) = \frac{1}{2}

.
Taking

2 α - \frac{π}{4} = \frac{π}{6}

gives

2 α = \frac{5 π}{12}

, hence

α = \frac{5 π}{24} \in (0, \frac{π}{2}) .

Option C.

JEE Main · 2024 · 04 Apr 2024 · Shift 1 · Single correct option