Let `x^(2)-(m-3)x+m=0(m epsilonR)` be a quadratic equation. Find the value of `m` for which
(i) both the roots are smaller than 2.
(ii) both the roots are greater than 2.
(iii) one root is smaller than 2 and the other root is greater than 2.
(iv) exactly one root lies in the interval (1,2).
(v) both the roots lie in the interval (1,2).
(vi) one root is greater than 2 and the other root is smaller than 1.
(vii) atleast one root lie in the interval (1,2).
(viii) atleast one root is greater than 2.

Explanation

Here `a=1,b=2(m-1)` and `c=m+5`
`:.D=b^(2)-4ac=4(m-1)^(2)-4(m+5)`
`=4(m^(2)-3m-4)`
`:.D=4(m-4)(m+1)` and here `a=1 ge0`
(i) `b=0` and `D gt0`
`implies2(m-1)=0` and `4(m-4)(m+1)gt0`
`impliesm=1` and `m epsilon (-oo, 1)uu(4,oo)`
`:.m epsilon phi` [null set]
(ii) `a=c` and `Dge0`
`implies 1=m+5` and `4(m-4)(m+1)ge0`
`implies m=-4` and `m epsilon(-oo,-1]uu[4,oo)`
`:.m=-4`
(iii) `agt0, clt0` and `Dgt0`
`implies1`gt0,m+5lt0` and `4(m-4)(m+1)gt0`
`impliesmlt-5` and `m epsilon (-oo,-1)uu(4,oo)`
`:.m epsilon(-oo,-5)`
(iv) `agt0, blt0, cgt0` and `Dge0`
`implies1 gt0,2(m-1)lt0,m+5gt0`
and `4(m-4)(m+1)ge0`
`impliesmlt1,mgt-5` and `m epsilon (-oo,-1]uu[4,oo)`
`impliesm epsilon (-5,-1]`
(v) `agt0, bgt0,cgt0` and `Dge0`
`implies1gt0,2(m-1)gt0,m+5gt0`
and `4(m-4)(m+1)ge0` ltbr. `impliesmgt1,mgt=5` and `m epsilon (-oo,-1]uu[4,00O`
`:.m epsilon[4,oo)`
(vi) Either one root is positive or both roots are positive
i.e (c) `uu` (d)
`implies m epsilon (-oo,-5)uu(-5,-1]`
(vii) Either one root is negative or both roots are negative
i.e. (c) `uu` (e)
`impliesm epsilon (-oo,-5)uu[4,oo)`