$y=3\cos \theta+5\sin (\theta-\large\frac{\pi}{6})$
$y=3\cos \theta+5(\sin \theta\cos\large\frac{\pi}{6}$$-\cos\theta\sin \large\frac{\pi}{6})$
$\Rightarrow 3\cos\theta+5(\large\frac{\sqrt 3}{2}$$\sin \theta-\large\frac{1}{2}$$\cos\theta)$
$\Rightarrow \large\frac{1}{2}$$(5\sqrt 3\sin \theta-\cos \theta)$
Multiplying and dividing by
$\sqrt{5\sqrt 3)^2+1^2}=\sqrt{76}$
$\Rightarrow \large\frac{1}{2}$$\sqrt{76}(\large\frac{5\sqrt 3}{\sqrt{76}}$$\sin \theta-\large\frac{1}{\sqrt{76}}$$\cos\theta)$
$\Rightarrow \sqrt{19}(\cos\alpha.\sin \theta-\sin \alpha\cos\theta)$
Where $\alpha=\tan-(\large\frac{1}{5\sqrt 3})$
Now maximum value of $\sin(\theta-\alpha)=1$
$\therefore Y_{max}=\sqrt{19}$
Now minimum value of $\sin(\theta-\alpha)=-1$
$\therefore Y_{min}=-\sqrt{19}$
Hence (A) is the correct answer.