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The sum of two positive quantities is eq...

The sum of two positive quantities is equal to `2ndot` the probability that their product is not less than 3/4 times their greatest product is `3//4` b. `1//4` c. `1//2` d. none of these

Text Solution

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Let one of the quantities be x.
Then the other quantity will be 2n - x.
Their product will be greatest when they are equal, i.e., each is n. In this case, the product is `n^(2)`.
According to the proposition.
`x(2n - x) ge (3)/(4) n^(2)`
`implies 4x^(2) - 8nx + 3n^(2) le 0`
`implies (2x - 3n) (2x-n) le0`
`implies (n)/(2) le x le (3)/(2)n`
So, number of favourable cases is equivalents to the length of the interval `[n//2, 3n//2]`, which is n.
Also, `x in (0, 2n)`.
So, total number of cases is equivalent to length of the interval (0, 2n).
`therefore` Required probability = `(n)/(2n) = (1)/(2)`
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