Question

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

Answer 1

Answer: 2

Note that coefficient of $x^2$ is $\left(4p-p^2-5\right)<0$ , Therefore, the
graph is concave downward.
According to the question, $1$ must lie between the roots.

Hence, $f(1)>0$
$\Rightarrow 4p-p^2-5-2p+1+3p>0$
$\Rightarrow -p^2+5p-4>0$
$\Rightarrow p^2-5p+4<0$
$\Rightarrow \left(p-4\right)\left(p-1\right)<0$
$\Rightarrow 1<p<4$
$\Rightarrow p\in \left\{2,3\right\}$