The integer $k$,for which the inequality $x^{2}-2(3k-1)x+8k^{2}-7>0$ is valid for every $x \in \mathbb{R}$,is

If $x^2-5x-14 > 0$ implies $x$ lies outside $[\alpha, \beta]$,then find the value of $\frac{\alpha}{\beta}$.

The solution set of the inequation $3^x+3^{1-x}-4 < 0$ is

If the expression $\left( mx - 1 + \frac{1}{x} \right)$ is non-negative for all positive real numbers $x$,then what must be the minimum value of $m$?

If $[x]$ denotes the greatest integer not exceeding $x$,then the values of $x$ satisfying $[x]^2-7[x]+12 \leq 0$ are

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