question_answer

Consider the quadratic equation \[(c-5){{x}^{2}}-2cs+(c-4)=0,\] \[c\,\ne \,5.\] Let S be the set of all integral values of g 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) 18                                

B)  12

C) 10                                

D) 11

Correct Answer: D

Solution :

Case-1,              \[c-5>0\]		?(i)
\[f(0)>0\]
\[c-4>0\]		?(ii)
\[f(c)<0\]
\[4(c-5)-4c+c-4<0\]
\[c<24\]
\[f(2)>0\]
\[c-4>0\]
\[9(c-5)-6c+c-4>0\]
\[4c-49>0\]\[\Rightarrow \]\[c>\frac{49}{4}\]
Here, (i) \[\cap \] (ii) \[\cap \] (iii) \[\cap \] (iv)
\[c\in \left( \frac{49}{4},24 \right)\]

Case-II, \[c-5<0\]	?(i)
\[f(0)<0\]
\[C>4\]		?(ii)
\[f(2)>0\] \[\Rightarrow \] \[c<24\]		?(iii)
\[f(3)<0\]\[\Rightarrow \]\[c<49\]		?.(iv)
4 \[\Rightarrow \]\[c\in \phi \]
\[c\in \left( \frac{49}{4},24 \right).\]