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 $2$
Given, $(4 p - p^{2} - 5) x^{2} - (2 p - 1) x + 3 p = 0$ has roots on either sides of unity.
Here, $a = 4 p - p^{2} - 5 = - (p - 2)^{2} - 1 < 0$
So,
$f (1) > 0 \Rightarrow 4 p - p^{2} - 5 - 2 p + 1 + 3 p > 0$
$\Rightarrow - p^{2} + 5 p - 4 > 0$
$\Rightarrow p^{2} - 5 p + 4 < 0$
$\Rightarrow (p - 4) (p - 1) < 0$
$\Rightarrow p \in (1, 4)$
Only $2$ integers, $2$ and $3$ satisfy this condition.
Hence, option 'B' is correct.

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