**Effective Column Length with Different Restraints**

The effective column length is an important parameter in structural analysis and design, as it affects the stability and buckling behavior of the column. The effective length factor, commonly denoted as K, is used to determine the effective length of a column based on its boundary conditions or restraints at the ends.

In this case, the column is properly restrained at one end in both position and direction, while it is imperfectly restrained at the other end in both position and direction. Let's analyze the effect of these restraints on the effective column length.

**Proper Restraint at One End**
When a column is properly restrained at one end, it means that the column is completely fixed in both position and direction. This type of restraint provides maximum resistance to lateral deflection and rotation.

**Imperfect Restraint at the Other End**
When a column is imperfectly restrained at the other end, it means that the column is partially restrained in both position and direction. This type of restraint allows some lateral deflection and rotation at the end of the column.

**Determining the Effective Column Length**
To determine the effective column length, we need to calculate the effective length factor, K. The effective length factor depends on the type of restraint at each end of the column.

For a column with one end properly restrained and the other end imperfectly restrained, the effective length factor can be taken as:

K = 0.75

The effective column length (Leff) can then be calculated by multiplying the actual column length (L) by the effective length factor (K):

Leff = K * L = 0.75 * L

Therefore, the effective column length is 0.75 times the actual column length.

**Answer: A) 0.75L**

This means that the effective column length is 0.75 times the actual column length when one end is properly restrained and the other end is imperfectly restrained in both position and direction.