In case of vertical circular motion of a particle by a thread of length \( r \), if the tension in the thread is zero at an angle \(30^\circ\) as shown in the figure, the velocity at the bottom point (A) of the vertical circular path is ( \( g \) = gravitational acceleration ). 
Find speed given to particle at lowest point so that tension in string at point A becomes zero. 


Match the following:
In the following, \( [x] \) denotes the greatest integer less than or equal to \( x \). 
Choose the correct answer from the options given below:
For x < 0:
f(x) = ex + ax
For x ≥ 0:
f(x) = b(x - 1)2