What should be the value of $$\lambda $$ such that the function defined below is continuous at $$x = {\pi \over 2}$$?
$$f\left( x \right) = \left\{ {\matrix{ {{{\lambda \,\cos x} \over {{\pi \over 2} - x}},} & {if\,\,x \ne {\pi \over 2}} \cr {1\,\,\,\,\,\,\,\,\,\,,} & {if\,\,x = {\pi \over 2}} \cr } } \right.$$

What is the value of the definite integral? $$\,\,\int\limits_0^a {{{\sqrt x } \over {\sqrt x + \sqrt {a - x} }}dx\,\,} $$?

For a body completely submerged in a fluid, the centre of gravity (G) and centre of Buoyancy (O) are known. The body is considered to be in stable equilibrium is

Consider a bar of diameter $$'D'$$ embedded in a large concrete block as shown in the adjoining figure, with a pull out force $$P$$ being applied. Let $${\sigma _b}$$ and $${\sigma _{st}},$$ be the bond strength (between the bar and concrete) and the tensile strength of the bar, respectively. If the block is held in position and it is assumed that the material of the block does not fail, which of the following options represents the maximum value of $$P?$$