A $$3$$ $$m$$ long simply supported beam of uniform cross section is subjected to a uniformly distributed load of $$w = 20$$ $$kN/m$$ in the central $$1$$ $$m$$ as shown in the figure.

If the flexural rrigidity $$(EI)$$ of the beam is $$30 \times {10^6}\,\,N$$ - $${m^2},$$ the maximum slope (expressed in radians) of the deformed beam is

Two beams $$PQ$$ (fixed at $$P$$ and with a roller support at $$Q,$$ as shown in Figure $$I,$$ which allows vertical movement) and $$XZ$$ (with a hinge at $$Y$$) are shown in the Figures $$I$$ and $$II$$ respectively. The spans of $$PQ$$ and $$XZ$$ are L and $$2L$$ respectively. Both the beams are under the action of uniformly distributed load $$(W)$$ and have the same flexural stiffness, $$EI$$ (where, $$E$$ and $$I$$ respectively denote modulus of elasticity and moment of inertia about axis of bending). Let the maximum deflection and maximum rotation be $${\delta _{\max 1}}$$ and $${\theta _{\max 1}},$$ respectively, in the case of beam $$PQ$$ and the corresponding quantities for the beam $$XZ$$ be $${\delta _{\max 2}}$$ and $${\theta _{\max 2}},$$ respectively.

Which one of the following relationships is true?

A propped cantilever of span $$L$$ carries a vertical concentrated load at the mid-span. If the plastic moment capacity of the section is $${M_P}$$, the magnitude of the collapse load is

A plane truss with applied loads is shown in the figure.

The members which do not carry any force are