An expression for a dimensionless quantity $P$ is given by $P=\frac{\alpha}{\beta} \log _{e}\left(\frac{ kt }{\beta x }\right)$; where $\alpha$ and $\beta$ are constants, $x$ is distance ; $k$ is Boltzmann constant and $t$ is the temperature. Then the dimensions of $\alpha$ will be

  • [JEE MAIN 2022]
  • A
    $[ M ^{0} L ^{-1} T ^{0} ]$
  • B
    $[ ML ^{0} T ^{-2}]$
  • C
    $[ MLT ^{-2}]$
  • D
    $[ ML ^{2} T ^{-2}]$

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A book with many printing errors contains four different formulas for the displacement $y$ of a particle undergoing a certain periodic motion:

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  • [KVPY 2015]