Consider the quadratic equation `(c - 5)x^(2) - 2cx + (c - 4) = 0, c ne 5`. Let S be the set of all integral values of c for which one root of the equation lies in the interval (0, 2) and its other root lies in the interval (2, 3). Then the number of elements in S is a. 11 b. 18 c. 10 d. 12

Explanation

`f(x) = (c-3)x^2 - 2cx + ( c- 2) = 0 , c ne 3`
Condition for 1 root in (0,2)
` f(0) . (f(2) lt 0 rArr (c-2) (4 (c-3) ) - 4c + c - 2 lt 0`
`rArr (c - 2) (c - 14) lt 0 " " ….(i)`
` (c - 14) (4c - 29) lt 0 " " …(ii)`
By (i) or (ii)
` 29/4 lt c lt 14`
` S = { 8,9,10,11,12,13}`
Total elements = 6