Entries of column I are to be matched with one or more entries of column II.

Explanation

(A)`rto(r,s,t),(B)to(p,q,r),(C)to(r,s,t)`
(A)Let `f(x)=ax^(2)+bx+c`
Then `f(1)=a+b+c=-c[ :' a+b+2c=0]`
and `f(0)=c`
`:.f(0)f(1)=-c^(2)lt0[:'c!=0]`
`:.` Equation `f(x)=0` has a root in (0,1).
`:.f(x)` has a root in (0,2) as well as in`(-1,1)(r)`
(B)Let `f'(x)=ax^(2)+bx+c`
`:.f(x)=(ax^(3))/3+(bx^(2))/2+cx+d`
`:.f(0)=d`
and `f(-1)=-a/3+b/2+c=d=-((2a-3b+6c)/6)+d`
`=0+d=d[:'2a-3b+6c=0]`
Hence `f(0)=(f-1)`
Hence `f'(x)=0` has atleast one root in `(-1,0)(q)`
`:.f(x)=0` has a root in `(-2,0)(p)` as well as `(-1,1)(r)`
(C)Let `f(x)=int(1+cos^(8)x)(x^(2)+bx+c)dx`
Given `f(1)-f(0)=f(2)-f(0)`
`impliesf(1)=f(2)`
`impliesf'(x)=0` has atleast one root in (1,2).
`implies(1+cos^(8)x)(ax^(2)+bx+c)=0` has atleast one root in (1,2).
`impliesax^(2)+bx+c=0` has atleast one root in (0,2) (t)