The value of $a$ for which the sum of the squares of the roots of the equation $x^2-(a-2)x-(a+1)=0$ assumes the least value is

If $2$ and $3$ are two roots of the equation $2x^3 + mx^2 - 13x + n = 0$,then the values of $m$ and $n$ are respectively

If $\alpha, \beta, \gamma$ are the roots of $x^3+4x+1=0$,then the equation whose roots are $\frac{\alpha^2}{\beta+\gamma}, \frac{\beta^2}{\gamma+\alpha}, \frac{\gamma^2}{\alpha+\beta}$ is

If $\alpha$ and $\beta$ are the roots of $x^{2}-px+1=0$ and $\gamma$ is a root of $x^{2}+px+1=0,$ then $(\alpha+\gamma)(\beta+\gamma)$ is

Let $x, y$ be real numbers such that $x \neq y$ and $xy \neq 1$. If $ax + b \sec(\tan^{-1} x) = c$ and $ay + b \sec(\tan^{-1} y) = c$,then $\frac{x+y}{1-xy} =$

If $\alpha$ and $\beta$ are the roots of the equation $2x^2 - 35x + 2 = 0$,then the value of $(2\alpha - 35)^3 \cdot (2\beta - 35)^3$ is equal to