The degree of statical indeterminacy of a structure refers to the number of unknown reactions or internal forces that cannot be determined by the equations of static equilibrium alone. It is an important concept in structural analysis to understand the behavior of a structure under loads.

In this particular case, we have a beam that is fixed at one end and vertically supported at the other end. Let's analyze the degree of statical indeterminacy for this beam.

1. Identify the support conditions:
- Fixed support at one end: This support prevents both translation and rotation of the beam.
- Vertical support at the other end: This support prevents only vertical translation of the beam.

2. Determine the number of unknown reactions:
- Fixed support at one end: This support generates three unknown reactions - vertical reaction, horizontal reaction, and moment reaction.
- Vertical support at the other end: This support generates one unknown reaction - vertical reaction.

3. Apply the equations of static equilibrium:
- For a two-dimensional structure like this beam, the equations of static equilibrium are:
- Sum of vertical forces = 0
- Sum of horizontal forces = 0
- Sum of moments = 0

4. Analyze the equations of static equilibrium:
- Sum of vertical forces: With one vertical reaction at the vertical support, we can determine this equation.
- Sum of horizontal forces: Since there are no horizontal external forces, this equation is automatically satisfied.
- Sum of moments: The fixed support at one end generates a moment reaction, but the vertical support at the other end does not generate any moment reaction. So, this equation is not applicable.

5. Conclusion:
- From the analysis, we can see that we have one unknown reaction (vertical reaction at the vertical support) and one equation (sum of vertical forces) to solve for it.
- Therefore, the degree of statical indeterminacy is 1.

So, the correct answer is option 'A' - 1.