Let $$\,\,f = {y^x}.$$ What is $$\,\,{{{\partial ^2}f} \over {\partial x\partial y}}\,\,$$ at $$x=2,$$ $$y=1$$?

A circular disk of radius $$'R'$$ rolls without slipping at a velocity $$v$$. The magnitude of the velocity at point $$'P'$$ (see figure) is

Consider a truss $$PQR$$ loaded at $$P$$ with a force $$F$$ as shown in the figure. The tension in the member $$QR$$ is

A straight rod length $$L(t),$$ hinged at one end freely extensible at the other end, roates through an angle $$\theta \left( t \right)$$ about the hinge. At time $$t,$$ $$L(t)=1m,$$ $$L(t)=1m/s,$$ $$\theta \left( t \right) = \pi /4$$ rad and $$\mathop \theta \limits^ \cdot \left( t \right) = 1\,\,rad/s.$$ The magnitude of the velocity at the other end of the rod is