If $\alpha, \beta$ are the roots of the equation, $x^2-x-1=0$ and $S_n=2023 \alpha^n+2024 \beta^n$, then
$2 \mathrm{~S}_{12}=\mathrm{S}_{11}+\mathrm{S}_{10}$
$\mathrm{S}_{12}=\mathrm{S}_{11}+\mathrm{S}_{10}$
$2 \mathrm{~S}_{11}=\mathrm{S}_{12}+\mathrm{S}_{10}$
$\mathrm{S}_{11}=\mathrm{S}_{10}+\mathrm{S}_{12}$
The number of integers $a$ in the interval $[1,2014]$ for which the system of equations $x+y=a$, $\frac{x^2}{x-1}+\frac{y^2}{y-1}=4$ has finitely many solutions is
Consider the equation ${x^2} + \alpha x + \beta = 0$ having roots $\alpha ,\beta $ such that $\alpha \ne \beta $ .Also consider the inequality $\left| {\left| {y - \beta } \right| - \alpha } \right| < \alpha $ ,then
If $|x - 2| + |x - 3| = 7$, then $x =$
The number of distinct real roots of the equation $x ^{7}-7 x -2=0$ is
The real roots of the equation ${x^2} + 5|x| + \,\,4 = 0$ are