If the equation $2x^2 - kx + x + 8 = 0$ has equal and real roots,then the value of $k$ will be .....

Let $E$ denote the set of all integers $a$ such that the point of intersection of the parabola $y = x^2 + 2ax + 2021$ with the $x$-axis has rational coordinates. The largest element of $E$ is

The product of all real roots of the equation $x^2 - |x| - 6 = 0$ is

If $\alpha$ satisfies the equation $\sqrt{\frac{x}{2x+1}} + \sqrt{\frac{2x+1}{x}} = 2$,then the roots of the equation $\alpha^2 x^2 + 4\alpha x + 3 = 0$ are

Suppose that $x$ and $y$ are positive numbers with $xy = \frac{1}{9}$,$x(y + 1) = \frac{7}{9}$,and $y(x + 1) = \frac{5}{18}$. The value of $(x + 1)(y + 1)$ is equal to:

The number of all possible positive integral values of $\alpha$ for which the roots of the quadratic equation $6x^2 - 11x + \alpha = 0$ are rational numbers is: