| \[(c-5){{x}^{2}}-2cx+(c-4)=0,c\ne 5\]. Let S be the set of all integral values of c for which one root of the equation lies in the 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) 12
B) 18
C) 10
D) 11
Correct Answer: D
Solution :
\[f(x)=(c-5){{x}^{2}}-2cx+(c-4)=0,\,\,c\ne 5\] \[f\left( 0 \right)\,\,f\left( 2 \right)<0\] \[\Rightarrow \,\,\,\left( c-4 \right)\left( c-24 \right)<0\] \[\Rightarrow \,\,c\in \left( 4,\,\,24 \right)\] ....... (i) \[f\left( 2 \right)\text{ }f\left( 3 \right)<0\] \[\Rightarrow \,\,\,\left( c-24 \right)\left( 4c-49 \right)<0\] \[\Rightarrow \,\frac{49}{4}<c<24\] Eq. (i) \[\cap \] (ii) \[\frac{49}{4}<c<24\] \[c\in \left\{ 13,\text{ }14,\text{ }.......\text{ }23 \right\}\] 11 elements
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