(i) , (ii) , (iii)

Given that

Let . Then,

or

where

Let and

.
(i) For . That means (1) should have at least one solution in . From (i), it is obvious that . Now represents a straight line. It should meet the

curve , at least once in .

If , Then ; if , then . Hence, the required range is .
(ii) For at least one positive solution, . That means graphs of and should meet at least once in . If , both the curves touch each other at ( 1,4 ). Hence, the required range is .
(iii) In this case both graphs should meet at least once in . For both the curves touch, hence, the required range is .

──────────────────────────────────────────────────────────────────────────────────────────