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Question

The sum of two postive quantities is equal to 2n. The probability that their product is not less than 34 times their greatest product is k8 then find k

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Solution

Let one of the quantities be x. Thus the other is 2nx.
Then product will be greatest when they are equal i.e., each is n in which case the product is n2.
By the given condition
x(2nx)34n2
8nx4x23n24x28nx+3n20
(2x3n)(2xn)0n2x32n.
favourable number of cases =3n2n2
Hence required probability =n2n=12

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