The area (in square units) of the triangle formed by the points with polar coordinates $(1, 0)$,$(2, \frac{\pi}{3})$,and $(3, \frac{2\pi}{3})$ is

If $\int \frac{e^x-1}{e^x+1} d x=f(x)+c$,then $f(x)$ is equal to

The products formed in the reaction of phenol with $Br_2$ dissolved in $CS_2$ at $0^{\circ} C$ are

The instantaneous values of alternating current and voltage in a circuit are given as $i = \frac{1}{\sqrt{2}} \sin(100\pi t) \text{ A}$ and $e = \frac{1}{\sqrt{2}} \sin(100\pi t + \frac{\pi}{3}) \text{ V}$. The average power in Watts consumed in the circuit is:

The atomic numbers of elements $A$,$B$,$C$ and $D$ are $Z-1$,$Z$,$Z+1$ and $Z+2$,respectively. If $B$ is a noble gas,choose the correct statements from the following:
$(1)$ $A$ has high electron affinity
$(2)$ $C$ exists in $+2$ oxidation state
$(3)$ $D$ is an alkaline earth metal

Which reagent is commonly used in titration to determine the hardness of water?