If the roots of the equation $(p-3)x^2 + 2(p-3)x + 2p-5 = 0$ are real and distinct for $\alpha < p < \beta$ and $(\beta - \alpha)$ is maximum,then the extreme value of the quadratic expression $-(\alpha + \beta)x^2 + \alpha \beta x + (\alpha - \beta)$ is

Let $a_n, a_{n-1}, \ldots, a_1, a_0 \in \mathbb{C}$ and $f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0$ be a polynomial. If the polynomial $f(x)$ is monic,then:

If $\alpha \neq 0$ and $0$ are the roots of the equation $x^2 - 5kx + (6k^2 - 2k) = 0$,then $\alpha = $

If $\alpha$ is a multiple root of the equation $x^5-6x^4+11x^3-2x^2-12x+8=0$,then $3\alpha^2-2\alpha+1=$

Which of the following six statements are true about the cubic polynomial $P(x) = 2x^3 + x^2 + 3x - 2$?
$(i)$ It has exactly one positive real root.
$(ii)$ It has either one or three negative roots.
$(iii)$ It has a root between $0$ and $1$.
$(iv)$ It must have exactly two real roots.
$(v)$ It has a negative root between $-2$ and $-1$.
$(vi)$ It has no complex roots.

If $(1 - p)$ is a root of the quadratic equation $x^2 + px + (1 - p) = 0$,then its roots are: