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(C) For element $X$, the half-life is given by $(t_{1/2})_X = \frac{0.693}{\lambda_X}$.For element $Y$, the mean life is given by $(	au)_Y = \frac{1}

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$Y$ decays faster than $X$

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(C) For element $X$, the half-life is given by $(t_{1/2})_X = \frac{0.693}{\lambda_X}$.For element $Y$, the mean life is given by $(	au)_Y = \frac{1}{\lambda_Y}$.Given that $(t_{1/2})_X = (	au)_Y$, we have $\frac{0.693}{\lambda_X} = \frac{1}{\lambda_Y}$.This implies $\lambda_X = 0.693 \lambda_Y$, which means $\lambda_X The decay rate is given by $R = \lambda N$. Since initially $N_X = N_Y$ and $\lambda_Y > \lambda_X$, the decay rate of $Y$ is greater than that of $X$ $(R_Y > R_X)$.Therefore, $Y$ decays faster than $X$.

The half-life period of element $X$ is same as the mean life time of element $Y$. Assume initially $X$ and $Y$ have same number of atoms. Then

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