b
\(Time\, period\, of\, spring - block \,system,\)

\(T=2 \pi \sqrt{\frac{m}{k}}\)

For given spring, \(T \propto \sqrt{m}\)

\(\frac{T_{1}}{T_{2}} =\sqrt{\frac{m_{1}}{m_{2}}}\)

\(\text { Here, } T_{1} =3 \mathrm{s}, m_{1}=m, T_{2}=5 \mathrm{s}, m_{2}=m+1, m=?\)

\( \frac{3}{5}=\sqrt{\frac{m}{m+1}} \text { or } \frac{9}{25}=\frac{m}{m+1}\) 

\(25 m=9 m+9 \Rightarrow 16 m=9\)

\(\therefore m=\frac{9}{16} \mathrm{kg}\)