The quadratic approximation of $$f\left( x \right) = {x^3} - 3{x^2} - 5\,\,$$ at the point $$x=0$$ is

An assembly made of a rigid arm A-B-C hinged at end A and supported by an elastic rope C-D at end C is shown in the figure. The members may be assumed to be weightless and the lengths of the respective members are as shown in the figure.

Under the action of a concentrated load P at C as shown, the magnitude of tension developed in the rope is

The hardness of a ground water sample was found to be 420 mg/L as $$CaCO_3$$. A softener containing ion exchange resins was installed to reduce the total hardness to 75 mg/L as $$CaCO_3$$ before supplying to 4 households. Each household gets treated water at a rate of 540 L/day. If the efficiency of the softener is 100%, the bypass flow rate (expressed in L/day) is .

A sample of water has been analyzed for common ions and results are presented in the form of a bar diagram as shown. The non-carbonate hardness (expressed in mg/L as $${\mathrm{CaCO}}_3$$) of the sample is