C

Step 1: Identify the initial velocities of both particles. For Particle 1, using the angle 30 degrees, its horizontal component is \( u_1 = u \cos(30^{\circ}) = \frac{u\sqrt{3}}{2} \). For Particle 2, with angle 60 degrees, its horizontal component is \( u_2 = u \cos(60^{\circ}) = \frac{u}{2} \).

Step 2: The horizontal distance covered by both particles when the gap becomes zero needs to be equal. Let \( t \) be the time. The distance for Particle 1 is \( D_1 = u_1 t = \frac{u\sqrt{3}}{2} t \) and for Particle 2 it is \( D_2 = u_2 t = \frac{u}{2} t \).

Step 3: Set the relation for distance: \( D_2 - D_1 = x \)
This leads to:
\( \frac{u}{2} t - \frac{u\sqrt{3}}{2} t = x \)

Step 4: Rearranging, we get: \( t(u/2 - u\sqrt{3}/2) = x \)
Therefore, \( t = \frac{x}{(u/2)(1 - \sqrt{3})} \).

Hence the correct answer is Option C.