KVPY Sample Paper KVPY Stream-SX Model Paper-25

  • question_answer
    Half lives of two isotopes X and Y of a material are known to be \[2\times {{10}^{9}}\]years and \[4\times {{10}^{9}}\] respectively. If a planet was formed with equal number of these isotopes, then the current age of planet, given that currently the material has 20% of X and 80% of Y by number, will be:

    A) \[2\times {{10}^{9}}\]years

    B) \[4\times {{10}^{9}}\]years

    C) \[6\times {{10}^{9}}\]years

    D) \[8\times {{10}^{9}}\]years

    Correct Answer: D

    Solution :

    \[{{N}_{x}}=\frac{{{N}_{0}}}{2}\cdot {{e}^{-{{\lambda }_{1}}t}}=0.2{{N}_{0}}\]
    \[{{N}_{y}}=\frac{{{N}_{0}}}{2}\cdot {{e}^{-{{\lambda }_{2}}t}}=0.8{{N}_{0}}\]
    \[{{e}^{({{\lambda }_{1}}-{{\lambda }_{2}})}}=4\]    \[\Rightarrow \] \[t=8\times {{10}^{9}}\]years

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