A beam rests on a hinged support at $B$ and a spring support at $C.$ A concentrated load $P(10N)$ acts at the end of the beam at $A.$ Span $AB$ has $5$ m length, the span $BC$ has $10$ m length, and the spring has $10k\frac{N}{m}$ spring constant $(k).$ Young's modulus $(E)$ of the beam is $200$ GPa.

Provide answers to the following questions:

a$)$ Calculate reactions at support $B$ and $C$ and obtain internal bending moment in the whole range of the beam

b$)$ Determine the displacement $({}_A)$ and indicate the displacement direction at $A$ by equating the work done during loading to the strain energy

c$)$ Determine the displacement $({}_A)$ and indicate the displacement direction at $A$ by using Castigliano's theorem

Note: Disregard beam weight and shear deformation. Assume that no torsional force occurs in the beam.