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The integral value of for which the root...

The integral value of for which the root of the equation `m x^2+(2m-1)x+(m-2)=0` are rational are given by the expression [where `n` is integer] `n^2` b. `n(n+2)` c. `n(n+1)` d. none of these

A

`n^(2)`

B

`n(n + 2)`

C

`n(n + 1)`

D

none of these

Text Solution

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The correct Answer is:
3
Discriminant `D = (2m - 1)^(2) - 4(m - 2)m = 4m + 1` must be perfect square. Hence,
`4m + 1 = k^(2)`, for some `k in I`
`rArr m = ((k - 1)(k + 1))/(4)`
Clearly, k must be odd. Let k = 2n + 1.
`therefore m = (2n(2n + 2))/(4) = n(n + 1), n in I`
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