Consider the quadratic equation c - 5 x 2 - 2 c x + c - 4 = 0 , c ≠ 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

Step-by-step solution

Case - I c - 5 > 0 ⇒ c > 5       . . . i Then, the graph of the quadratic is shown below Hence, we have f 0 > 0 ⇒ c - 4 > 0 ⇒ c > 4       . . . i i And f 2 < 0 ⇒ 4 c - 5 - 4 c + c - 4 < 0 ⇒ 4 c - 20 - 4 c + c - 4 < 0 ⇒ c < 24       . . . i i i And f 3 > 0 ⇒ 9 c - 5 - 6 c + c - 4 > 0 ⇒ 9 c - 45 - 6 c + c - 4 > 0 ⇒ 4 c - 49 > 0 ⇒ c > 49 4