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

Let $\alpha, \beta$ be the roots of the quadratic equation $x^2+\sqrt{6} x+3=0$. Then $\frac{\alpha^{23}+\beta^{23}+\alpha^{14}+\beta^{14}}{\alpha^{15}+\beta^{15}+\alpha^{10}+\beta^{10}}$ is

Let α,β be the roots of the equation $x^2+2 \sqrt{2} x-1=0$. . The quadratic equation, whose roots are $\alpha^4+\beta^4$ and $\frac{1}{10}\left(\alpha^6+\beta^6\right)$, is :

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

If 2 and 6 are the roots of the equation $\mathrm{ax}^2+\mathrm{bx}+1=0$, then the quadratic equation, whose roots are $\frac{1}{2 a+b}$ and $\frac{1}{6 a+b}$, is :

Let the sum of the maximum and the minimum values of the function $f(x)=\frac{2 x^2-3 x+8}{2 x^2+3 x+8}$ be $\frac{m}{n}$, where $\operatorname{gcd}(\mathrm{m}, \mathrm{n})=1$. Then $\mathrm{m}+\mathrm{n}$...

Let $\alpha, \beta \in N$ be roots of equation $x^2-70 x+\lambda=0$, where $\frac{\lambda}{2}, \frac{\lambda}{3} \notin \mathrm{~N}$. If $\lambda$ assumes the minimum possible value, then $\frac{(\sqrt{\alpha-1}+\sqrt{\beta-1})(\lambda+35)}{|\alpha-\beta|}$...

Let α,β be the roots of the equation$x^2-x+2=0$ with Im⁡(α)>Im⁡(β). Then $\alpha^6+\alpha^4+\beta^4-5 \alpha^2$ is equal to

Let S={x∈R:(√3+√2 )^x+(√3-√2 )^x=10}. Then the number of elements in S is :

If the set of all $a \in \mathbf{R}$, for which the equation $2 x^2+(a-5) x+15=3$ a has no real root, is the interval $(\alpha, \beta)$,...