A uniformly loaded propped cantilever beam and its free body diagram are shown below. The reactions are

The given propped cantilever beam can be assumed to be consisting of two types of loads. First part is simple cantilever with UDL and the second part is cantilever beam with point load of R₂ at end.

Balancing the forces we get

R₁ + R₂ = qL            ----------(1) 

Balancing the deflection at end point as net deflection at the end is zero.

Deflection at B due to the UDL alone

${\delta }_B=\frac{q{L}^4}{8EI}$

Deflection at B due to R₂ alone, 

${\delta }_B^{{}^{\prime }}=\frac{R_2{L}^3}{3EI}$

$\therefore {\delta }_B={\delta }_B^{{}^{\prime }}$

$\Rightarrow \frac{q{L}^4}{8EI}=\frac{R_2{L}^3}{3EI}\Rightarrow R_2=\frac{3qL}{8}$

From 1) 

$R_1=qL-R_2$

$=qL-\frac{3qL}{8}=\frac{5qL}{8}$

$Moment,M=R_{2}L-qL\times \frac{L}{2}$

$=\frac{3q{L}^2}{8}-\frac{q{L}^2}{2}=-\frac{q{L}^2}{8}$