if a `in` R and equation `(a-2)(x-[x])^(2)+2(x-[x])+a^(2)=0` (where [x] represent G.I.F) has no integral solution and has exactly one solution in the interval (12,13) then a lies in `(alpha,beta)` the value of `alpha^(2)+3beta^(2)` is

Answer

1

Explanation

let `x-[x]=t`
`f(t)=(a-2)t^(2)+2t+a^(2)=0` ..(i)
case (i) `ane2` exactly one root of equation (i) lies in (0,1)
`f(0),f(1)lt0` ltbr. `a^(2)(a^(2)+a)lt0`
`a in (-1,0)`
case (ii) if `a=2` the `t=2`
`implies` no root
Hence `a in (-1,0)`
`alpha=-1`
`beta=0`