Minimum time period in a compound pendulum is obtained when:
(A) l = ± K / 2
(B) l = ± K
(C) l = ± K / √2
(D) l = 0

The time period of a compound pendulum depends on its length (l) and the acceleration due to gravity (g). The formula for the time period (T) of a compound pendulum is given by:

T = 2π√(l/g)

Where:
T = Time period
l = Length of the pendulum
g = Acceleration due to gravity

To find the minimum time period, we need to minimize the expression for T. To do this, we need to minimize the value inside the square root, which is (l/g). Since g is a constant (approximately 9.81 m/s² on the surface of the Earth), we should minimize the length (l) of the pendulum.

Among the options you provided:
A. l = ± K/2
B. l = ± K
C. l = ± K/√2
D. l = 0

The option that minimizes the length (l) the most is option D, where l = 0. However, this option is not physically meaningful because a pendulum with zero length cannot exist. Therefore, we cannot choose option D.

Option C, where l = ± K/√2, minimizes the length the least among the given options. Therefore, the minimum time period is obtained when l = ± K/√2. So, the correct answer is:

C. l = ± K/√2