$$X$$ and $$Y$$ are two random independent events. It is known that $$P(X)=0.40$$ and $$\,P\left( {X \cup {Y^C}} \right) = 0.7.$$ Which one of the following is the value of $$P\left( {X \cup Y} \right)$$ $$?$$

Consider the following linear system $$$x+2y-3z=a$$$ $$$2x+3y+3z=b$$$ $$$5x+9y-6z=c$$$
This system is consistent if $$a,b$$ and $$c$$ satisfy the equation

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

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