Match the following for lists:

Answer

`a to r; b to r; c to q; d to p `

Explanation

`ator,btor,ctoq,d""to""p`
`(m-2)x^(2)-(8-2m)x-(8-3m)=0` has roots of opposite signs. The product of roots is
`-(8-3m)/(m-2)lt0`
or `(3m-8)/(m-2)lt0`
or `2ltmlt8//3`
(b) Exactly one root of equation `x^(2)-m(2x-8)-15=0` lies in interval (0,1).
`f(0)f(1)lt0`
`implies(0-m(-8)-14)(1-m(-6)-15)lt0`
`implies(8m-15)(6m-14)lt0`
`implies15//8ltmlt7//3`
(c) `x^(2)+2(m+1)x+9m-5=0` has both roots negative. Hence, sum of roots is
`-2(m+1)ltormgt-1" "(1)`
Product of roots is
`9m-5gt0impliesmgt5//9" "(2)`
Discriminant,
`Dge0implies4(m+1)^(2)-4(9m-5)ge0`
`impliesm^(2)-7m+6ge0`
`impliesmle1ormge0" "(3)`
Hence, for (1), (2),and (3), we get
`m""in((5)/(9),1]uu[6,oo)`
(d) `f(x)=x^(2)+2(m-1)x+m+5=0` has one root less than 1 and the other root greater than 1. Hence,
`f(1)lt0`
`implies1+2(m-1)+m+5lt0`
`impliesmlt-4//3`