The greatest integral value of c so that both the roots of  the equation (c−5)x2−2cx+(c−4)=0 are positive, one  root is less than 2 and other root is lying between 2 and 3 is

We have, x2−2cc−5x+c−4c−5=0 Let f(x)=x2−2cc−5x+c−4c−5 So, f(0)>0,f(2)<0,f(3)>0 must be satisfied simultaneously.  Now, f(0)>0⇒c−4c−5>0-----(1)f(2)<0⇒c−24c−5<0-----(2) and f(3)>0⇒4c−49c−5>0-----(3) Hence, (1)∩(2)∩(3)⇒ c∈494,24So, the greatest integral value of c is 23.