$(a)\;\sqrt{2}\qquad(b)\;\large\frac{1}{ \sqrt{2}}\qquad(c)\;\large\frac{1}{3 \sqrt{2}}\qquad(d)\;None$

Answer : (b) $\;\large\frac{1}{ \sqrt{2}}$

Explanation :

Fraction of nuclei which remain undecayed is

$f=\large\frac{N}{N_{0}}=\large\frac{N_{0} e^{- \lambda t}}{N_{0}}$

$=e^{- \lambda t}$

$=e^{-(\large\frac{ln 2}{T})\;(\large\frac{T}{2})}$

$=e^{\large\frac{1}{ln \sqrt{2}}}=\large\frac{1}{\sqrt{2}}\;.$

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