Castigliano's Theorem for Deflections

Q. A simply supported beam of length 2L is subjected to a moment M at the mid-point x=0 as shown in figure. The deflection in the domain $0\le x\le L$ is
$W=\frac{-Mx}{12EIL}(L-x)(x+c)$
where E is the Young's modulus, I is the area moment of inertia and C is a constant (to be determined).
The slope at the center $x=0$ is

1. $\frac{ML}{\left(2EI\right)}$
2. $\frac{ML}{\left(3EI\right)}$
3. $\frac{ML}{\left(6EI\right)}$
4. $\frac{ML}{\left(12EI\right)}$

Q. A frame is subjected to a load P as shown in the figure. The frame has a constant flexural rigidity EI. The effect of axial load is neglected. The deflection at point A due to the applied load P is

1. $\frac{1}{3}\frac{P{L}^3}{EI}$
2. $\frac{2}{3}\frac{P{L}^3}{EI}$
3. $\frac{P{L}^4}{EI}$
4. $\frac{4}{3}\frac{P{L}^3}{EI}$

Q. For the cantilever beam shown in the given figure, which one of the following pairs is not correctly matched ?

1. $Moh{r}^{\prime }s:AreaofBMD\times DistanceTheoremofcentroidofBMDfromB$
2. $Castiglian{o}^{\prime }s:\underset{0}{\overset{L}{\int }}W{x}^{2}dxTheorem$
3. $Conjugate:Shearforceatthefixedbeamendofconjugatebeam$
4. $Successive:\int \int -Wxdxintegration$

Q. A plane frame PQR (fixed at P and free at R) is shown in the figure. Both members (PQ and QR) have length L, and flexural rigidity, EI. Neglecting the effect of axial stress and transverse shear, the horizontal deflection at free end R, is

1. $\frac{2F{L}^3}{3EI}$
2. $\frac{4F{L}^3}{3EI}$
3. $\frac{F{L}^3}{3EI}$
4. $\frac{5F{L}^3}{3EI}$