Column I contain rational algebraic expressions and ColumnII contains possible integers which lie in their range. Match the entries of Column I with one or more entries of the elements of Column II.

Answer

`(A)to(r,s),(B)to(p,q,r,s,t),(C)to(p,q,t)`

Explanation

`Ato(r,s),Bto(p,q,r,s,t),Cto(p,q,t)`
(A) We have `y=(x^(2)-2x+4)/(x^(2)+2x+4)`
`impliesx^(2)(y-1)+2(y+1)x+4(y-1)=0`
As `x epsilonR` we get
`Dge0`
`=4(y+1)^(2)-16(y-1)^(2)ge0`
`implies3y^(2)-10y+3le0`
`implies(y-3)(3y-1)le0`
`implies1/3leyle3`
(B) We have `y=(2x^(2)+4x+1)/(x^(2)+4x+2)`
`impliesx^(2)(y-2)+4(y-1)x+2y=1=0`
As ` x epsilon R` we get
`Dge0`
`implies16(y-1)^(2)-4(y-2)(2y-1)ge0`
`implies4(y-1)^(2)-(y-2)(2y-1)ge0`
`implies2y^(2)-3yh+2ge0`
`impliesy^(2)-3/2y+1ge0`
`implies(y-3/4)^(2)+7/16ge0`
`:.y epsilonR`
(C) We have `y=(x^(2)-3x+4)/(x-3)`
`impliesx^(2)-(3+y)x+3y+4=0`
As ` x epsionR` we get
`Dge0implies(3+y)^(2)-4(3y+4)ge0`
`impliesy^(2)-6y-7ge0implies(y+1)(y-7)ge0`
`impliesy epsilon (-oo,-1]uu[7,oo)`