text: # Which of the following is not a binary operation on $R$
  
  $\begin{array}{1 1}(1)a^{\ast} b=ab&(2)a^{\ast} b=a-b\\(3)a^{\ast} b=\sqrt{ab}&(4)a^{\ast} b=\sqrt{a^{2}+b^{2}}\end{array}$
  
  $a * b =\sqrt {ab}$ is not a binary operation on R .
  Since for $a,-b \in R$
  $a * (-b) =\sqrt {a(-b)}=\sqrt {-ab} \neq R$
  $\sqrt {-ab}$ is imaginary
  Hence 3 is the correct answer.
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