Quadratic Equations - Algebra - Mathematics Previous Year Questions

With respect to the roots of the equation $3 x^{3}+b x^{2}+b x+3=0$, match the items of List I with those fo List II

$\alpha, \beta, \gamma, 2$ and $\varepsilon$ are the roots of the equation

$$\begin{aligned} & \alpha, \beta, \gamma+4 x^4-13 x^3-52 x^2+36 x+144=0 . \text { If } \\ & \alpha<\beta<\gamma<2<\varepsilon \text {, then } \alpha+2 \beta+3 \gamma+5 \varepsilon= \end{aligned}$$

If the equation $x^2-3 a x+a^2-2 a-k=0$ has different real roots for every rational number $a$, then $k$ lies in the interval

The number of all common roots of the equation $x^4-10 x^3+37 x^2-60 x+36=0$ and the transformed equation of it obtained by increasing any two distinct roots of it by 1 , keeping the other two roots fixed, is

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3-P x^2+Q x-R=0$ and $(\alpha-2)^2,(\beta-2)^2,(\gamma-2)^2$ are the roots of the equation $x^3-5 x^2+4 x=0$, then the possible least value of $P+Q+R$ is

The number of integral values of ' $a$ ' for which the quadratic equation $a x^2+a x+5=0$ cannot have real roots is

If the roots of the equation $32 x^3-48 x^2+22 x-3=0$ are in arithmetic progression, then the square of the common difference of the roots is