$\alpha$ is the maximum value of $1-2x-5x^2$ and $\beta$ is the minimum value of $x^2-2x+r$. If $5\alpha x^2+\beta x+6>0$ for all real values of $x$,then the interval in which $r$ lies is

Let $\lambda \in R$ and let the equation $E$ be $|x|^2 - 2|x| + |\lambda - 3| = 0$. Then the largest element in the set $S = \{x + \lambda : x \text{ is an integer solution of } E\}$ is $..........$

Let $\phi(x)=\frac{x}{(x^2+1)(x+1)}$. If $a, b$ and $c$ are the roots of the equation $x^3-3x+\lambda=0, (\lambda \neq 0)$,then $\phi(a) \phi(b) \phi(c) =$

Suppose $a, b$ denote the distinct real roots of the quadratic polynomial $x^2+20x-2020$ and suppose $c, d$ denote the distinct complex roots of the quadratic polynomial $x^2-20x+2020$. Then the value of $ac(a-c)+ad(a-d)+bc(b-c)+bd(b-d)$ is

If $\alpha$ is a root of the equation $x^2-x+1=0$,then $\left(\alpha+\frac{1}{\alpha}\right)^3+\left(\alpha^2+\frac{1}{\alpha^2}\right)^3+\left(\alpha^3+\frac{1}{\alpha^3}\right)^3+\left(\alpha^4+\frac{1}{\alpha^4}\right)^3=$

The equation $\left(x^4+1\right)=\frac{1}{a}(x+1)^4$ is a reciprocal equation: