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Q.
Let $N$ be the sum of the numbers appeared when two fair dice are rolled and let the probability that $N-2, \sqrt{3 N}, N+2$ are in geometric progression be $\frac{k}{48}$, Then the value of $k$ is
$n(s)=36$
Given : $N -2, \sqrt{3 N }, N +2$ are in G.P.
$ 3 N =( N -2)( N +2)$
$ 3 N = N ^2-4 $
$ \Rightarrow N ^2-3 N -4=0$
$ ( N -4)( N +1)=0 \Rightarrow N =4 \text { or } N =-1 \text { rejected }$
$( Sum =4) \equiv\{(1,3),(3,1),(2,2)\} $
$ n ( A )=3$
$ P ( A )=\frac{3}{36}=\frac{1}{12}=\frac{4}{48} \Rightarrow k =4$