Unit vectors $\vec a$ and $\vec b$ are perpendicular to each other and the unit vector $\vec c$ is inclined at angle θ to both $\vec a$ and $\vec b$. If $\vec c=m(\vec a + \vec b)+n(\vec a × \vec b)$ are real, then

$\frac{\pi}{4}≤θ≤\frac{3\pi}{4}$

$\vec a$ is perpendicular to $\vec b$ ⇒ $\vec a.\vec b = 0$ and $|\vec a×\vec b|=1$

Hence $\vec c=m(\vec a+\vec b)+n(\vec a×\vec b) ⇒\vec a.\vec c=m\,\,\vec a.\vec a= ma^2=m⇒m=cosθ$

Also $\vec c.\vec c=[m(\vec a+\vec b)+n(\vec a×\vec b)].[m(\vec a+\vec b)+n(\vec a×\vec b)]=2m^2+n^2$

$n^2=1-2m^2=1-2cos^2θ=-cos2θ⇒-cos2θ≥0$

$⇒\frac{\pi}{4}≤θ≤\frac{3\pi}{4}$

Hence (A) is the correct answer.