For each positive real number $\lambda$. Let $A_\lambda$ be the set of all natural numbers $n$ such that $|\sin (\sqrt{n+1})-\sin (\sqrt{n})|<\lambda$. Let $A_\lambda^c$ be the complement of $A_\lambda$ in the set of all natural numbers. Then,

  • [KVPY 2016]
  • A

    $A_{1 / 2}, A_{1 / 3}, A_{25}$ are all finite sets

  • B

    $A_{1 / 3}$ is a finite set but $A_{1 / 2}, A_{25}$ are infi,nite sets

  • C

    $A_{12}^c, A_{13}^c, A_{25}^c$ are all finites sets

  • D

    $A_{1 / 3}, A_{2 / 5}$ are finite sets and $A_{1 / 2}$ is an infinite set

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