The vector $\vec{a}(x)=\cos x \hat{i}+\sin x \hat{j}$ and $\vec{b}(x)=x \hat{i}+\sin x \hat{j}$ are collinear for 

unique value of x, $\pi / 6<x<\pi / 3$

Since $\vec{a}$ and $\vec{b}$ are collinear, for some $\lambda$, we can write $\vec{a}=\lambda \vec{b}$

$\Rightarrow \cos x \hat{i}+\sin x \hat{j}=\lambda(x \hat{i}+\sin x \hat{j})$

$\Rightarrow \cos x=x \lambda \text { and } \lambda=1 \Rightarrow \cos x=x$

Here we will get only one unique value of x which belongs to $\left(\frac{\pi}{6}, \frac{\pi}{3}\right)$

Hence (2) is correct answer.