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

1

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2

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3

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4

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Solution

The correct option is B $2$
$4 p - p^{2} - 5 < 0, \forall p \in R$
Therefore, graph of the quadratic curve is opening downward.
According to the question, $1$ must lie in between the roots.
So, $f (1) > 0$
$\Rightarrow 4 p - p^{2} - 5 - 2 p + 1 + 3 p > 0$
$\Rightarrow p^{2} - 5 p + 4 < 0$
$\Rightarrow (p - 1) (p - 4) < 0$
$\Rightarrow p \in (1, 4)$
$∴ p = 2, 3$

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