The value of p for which both the roots of th | vedclass

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(D) Let $f(x) = 4x^2 - 20px + (25p^2 + 15p - 66) = 0$ .....$(i)$The roots of $(i)$ are real if $D \ge 0$:$D = (-20p)^2 - 4(4)(25p^2 + 15p - 66) = 400p

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$(-\infty, -1)$

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(D) Let $f(x) = 4x^2 - 20px + (25p^2 + 15p - 66) = 0$ .....$(i)$The roots of $(i)$ are real if $D \ge 0$:$D = (-20p)^2 - 4(4)(25p^2 + 15p - 66) = 400p^2 - 16(25p^2 + 15p - 66) = 16(66 - 15p) \ge 0$$\Rightarrow 66 - 15p \ge 0$ $\Rightarrow p \le \frac{22}{5}$ .....$(ii)$For both roots to be less than $2$:$1)$ $D \ge 0 \Rightarrow p \le \frac{22}{5}$$2)$ $f(2) > 0 \Rightarrow 4(2)^2 - 20p(2) + 25p^2 + 15p - 66 > 0$$16 - 40p + 25p^2 + 15p - 66 > 0$ $\Rightarrow 25p^2 - 25p - 50 > 0$ $\Rightarrow p^2 - p - 2 > 0$$(p - 2)(p + 1) > 0 \Rightarrow p  2$ .....$(iii)$$3)$ Vertex $x_v Taking the intersection of $(ii), (iii),$ and $(iv)$:$p < -1$.

Column-$I$	Column-$II$
$A$. Imaginary roots	$P$. $a \in (-1, 4)$
$B$. One root less than $3$ and other greater than $3$	$Q$. $a \in (-\infty, -1)$
$C$. One root less than $1$ and other greater than $3$	$R$. $a \in (-\infty, -4/3)$

The value of $p$ for which both the roots of the equation $4x^2 - 20px + (25p^2 + 15p - 66) = 0$ are less than $2$ lies in:

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