The condition that ${x^3} – 3px + 2q$ may be divisible by a factor of the form ${x^2} + 2ax + {a^2}$ is

Let $x$ and $y$ be two $2-$digit numbers such that $y$ is obtained by reversing the digits of $x$. Suppose they also satisfy $x^2-y^2=m^2$ for some positive integer $m$. The value of $x+y+m$ is

If $x$ is real, then the value of ${x^2} – 6x + 13$ will not be less than

Let $\alpha $ and $\beta $ are roots of $5{x^2} – 3x – 1 = 0$ , then $\left[ {\left( {\alpha  + \beta } \right)x – \left( {\frac{{{\alpha ^2} + {\beta ^2}}}{2}} \right){x^2} + \left( {\frac{{{\alpha ^3} + {\beta ^3}}}{3}} \right){x^3} -……} \right]$ 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