If $x$ is real,then the value of $\frac{x^2 + 34x - 71}{x^2 + 2x - 7}$ does not lie between

$E_1: a+b+c=0$,if $1$ is a root of $ax^2+bx+c=0$. $E_2: b^2-a^2=2ac$,if $\sin \theta, \cos \theta$ are the roots of $ax^2+bx+c=0$. Which of the following is true?

If $x$ is real,then the difference between the greatest and least values of $\frac{x^2-x+1}{x^2+x+1}$ is

The sum of all integral values of $k$ $(k \neq 0)$ for which the equation $\frac{2}{x-1}-\frac{1}{x-2}=\frac{2}{k}$ in $x$ has no real roots,is ..... .

For all real values of $x$,the minimum value of $\frac{1-x+x^2}{1+x+x^2}$ is

$A$ student,while solving a quadratic equation in $x$,copied its constant term incorrectly and obtained the roots as $5$ and $9$. Another student copied the constant term and the coefficient of $x^2$ of the same equation correctly as $12$ and $4$ respectively. If $s$,$p$,and $\Delta$ denote the sum of the roots,the product of the roots,and the discriminant of the correct equation respectively,then find the value of $\frac{\Delta}{3p+s}$.