algebra-4

as the roots of the eqn are positive.. hence product = r/p >0
==> r and p are of same sign
and -q/p >0
==> q and p are of opposite sign

CASE I p>0
... now... p>0 ==> r>0 and q<0 .. so when x lies between the roots i.e. when A<x<B then f(x) = px² + qx + r is <0

So. the limit when A<n<B = -1
and when x is outside the roots then f(x) >0 so the limit is 1

Now.. option (b) is not clear whether n lies between or outside the roots so discarded

Option (d) .. here p=1 i.e. >0 and n>A but it doesnt say whether n>B or n<B so discarded again

CASE II p<0

then here .. q>0 and r<0 ..
when x lies between the roots .. the f(x) > 0

So. option (a) is correct..

Again option (c) is not clear whether n lies between or outside the roots so discarded

So answer is (a)

px²+qx+r=p(x-A)(x-B)

lim x→n |p||(x-A)(x-B)|/p(x-A)(x-B) =1

|p|/p x(-1)=1 , if n lies between A and B --------------------1

|p|=-p

p is negative, p<0-----------------2

so from 1 and 2 option a