If $a$ and $b$ are the roots of the equation $x^2-7x-1=0$,then the value of $\frac{a^{21}+b^{21}+a^{17}+b^{17}}{a^{19}+b^{19}}$ is equal to $........$.

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

The set of values of $x$ for which the inequalities $x^2-3x-10 < 0$ and $10x-x^2-16 > 0$ hold simultaneously is:

Statement $(I)$: The set of solutions of $|x|^2 - 4|x| + 3 < 0$ is the interval $(-3, 3)$.
Statement $(II)$: If $x < 3$ or $x > 5$, then $x^2 - 8x + 15 > 0$.
Which of the above statements is (are) true?

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

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