Question

If the roots of the quadratic equation $(4p-p^2-5)x^2-(2p-1)x+3p=0$ lie on either side of unity, then the number of integral values of $p$ is

• A. $2$
• B. $3$
• C. $4$
• D. $1$

Answer 1

The correct option is A $2$

Given equation
$(4p-p^2-5)x^2-(2p-1)x+3p=0$
We know that
$4p-p^2-5=-(p^2-4p+4)-1=-1-(p-2{)}^2\therefore 4p-p^2-5<0\forall p\in R$

According to the question,
$1$ must lie in between the roots.
So,
$f\left(1\right)>0$
$\Rightarrow 4p-p^2-5-2p+1+3p>0$
$\Rightarrow p^2-5p+4<0$
$\Rightarrow (p-1)(p-4)<0$
$\Rightarrow p\in (1,4)$
Intergral value of
$\therefore p=2,3$

Hence, the number of integral value of $p$ is $2$.