C

To determine the variation of tension in the string as the particle moves in a vertical circular path, we analyze the forces acting on the particle at different positions.
Step 1: At the topmost point of the circular path (when the angle \( \theta = 0 \)), the centripetal force required is provided by the weight of the particle and the tension in the string. The equation is given by:
\[ T + mg = \frac{mv^2}{r} \]
where \( T \) is the tension, \( m \) is the mass, \( g \) is the acceleration due to gravity, and \( v \) is the tangential velocity.
Step 2: At the lowest point of the circular path (when the angle \( \theta = \pi \)), the forces acting on the particle are only the tension and its weight acting downward:
\[ T - mg = \frac{mv^2}{r} \]
Step 3: As the particle moves from the top to the bottom, it gains speed due to gravitational potential energy converting to kinetic energy. Thus, the tension will vary inversely as the speed changes as the particle moves along the path.
Step 4: The tension is at its minimum at the top of the circular path and at its maximum at the bottom. Therefore, a graph plotting tension against angle would show a curve increasing from zero tension at the top to a maximum at the bottom, then decreasing again as it reaches the next top. The final form would be similar to option C in the provided graphs.
Conclusion: The correct plot of tension against the angle \( \theta \) will resemble option C.