Question

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

Answer 1

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

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