First slide

Theory of expressions

Question

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, the number of integral values of p is

Moderate

A

1
B

2
C

3
D

4

Solution

Since, the coefficient of $x^{2} = (4 p - p^{2} - 5) < 0$
Therefore, the graph is open downward.
According to the question, 1 must lie between the roots.
Hence,
$\begin{aligned} \Rightarrow f (1) & > 0 \\ \Rightarrow 4 p - p^{2} - 5 - 2 p + 1 + 3 p & > 0 \\ \Rightarrow - p^{2} + 5 p - 4 & > 0 \\ \Rightarrow p^{2} - 5 p + 4 & < 0 \\ \Rightarrow (p - 4) (p - 1) & < 0 \\ \Rightarrow 1 < p & < 4 \\ ∴ p & = 2, 3 \end{aligned}$
Hence, number of integral values of p is 2.

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