The set of all solutions of the inequation $x^2 - 2x + 5 \leq 0$ in $R$ is

If $x^2+2px-2p+8>0$ for all real values of $x$,then the set of all possible values of $p$ is

Number of integral values of $m$ for which $\{x\}^2 + 5m\{x\} - 3m + 1 < 0$ for all $x \in \mathbb{R}$ is (where $\{.\}$ denotes the fractional part function).

What is the solution set of the inequality $x^2 + 2|x| - 15 \geq 0$?

If $x^2-4ax+5+a>0$ for all $x \in R$ whenever $a \in (\alpha, \beta)$,then $4\beta+\alpha=$

Let $\alpha$ and $\beta$ be the distinct roots of the equation $x^2 - (t^2 - 5t + 6)x + 1 = 0$,where $t \in \mathbb{R}$,and $a_n = \alpha^n + \beta^n$. Then the minimum value of $\frac{a_{2023} + a_{2025}}{a_{2024}}$ is