Half life of the zero order reaction is:

The correct answer is Option (4) → Directly proportional to initial concentration of the reactant

The half-life of a reaction is the time in which the concentration of a reactant is reduced to one half of its initial concentration. 

For a zero-order reaction, rate constant is given by equation

$ k=\frac{[\text{R}]_0-[\text{R}]}{t} $

At

$ t=t_{\frac12} $

, and

$ [\text{R}]=\frac12[\text{R}]_0 $ 

The rate constant at half life becomes

$ k=\frac{[\text{R}]_0-1/2[\text{R}]_0}{t_{1/2}} $

$ t_{1/2}=\frac{[\text{R}]_0}{2k} $ 

It is clear that half life for a zero-order reaction is directly proportional to the initial concentration of the reactants and inversely proportional to the rate constant.