question_answer

If the quadratic equation \[f(x)=p{{x}^{2}}-qx+r=0\] has two distinct roots in (0, 2) where p, q, \[r\in N\] and \[f(1)=-\,1\] then the minimum value of p is-

A) 1

B) 2

C) 4

D) none of these

Correct Answer: B

Solution :

\[f\,(1)=p-q+r=-\,1\]	?.(i)
\[f\,(0)=r>0\]	?.. (ii)
\[f\,(2)=4p-2q+r>0\]	?.. (iii)

\[(ii)+(iii)\Rightarrow 4p-2q+2r>0\]

\[2p-q+r>0\]

\[p+(p-q+r)>0\]               [using (i)]

\[p-1>0\]

\[\Rightarrow p>1\] \[\therefore p=2\]                                    \[(\because p\in N)\]