If the roots of the equation $(m - n)x^2 + (n - l)x + (l - m) = 0$ are equal,then which of the following is true for $l, m,$ and $n$?

Let $p(x) = x^2 - 5x + a$ and $q(x) = x^2 - 3x + b$,where $a$ and $b$ are positive integers. Suppose $\text{HCF}(p(x), q(x)) = x - 1$ and $k(x) = \text{LCM}(p(x), q(x))$. If the coefficient of the highest degree term of $k(x)$ is $1$,then the sum of the roots of $(x - 1) + k(x)$ is:

Let one root of $ax^2 + bx + c = 0$,where $a, b, c$ are integers,be $3 + \sqrt{5}$. Then the other root is:

Each of the roots of the equation $x^3-6x^2+6x-5=0$ are increased by $h$. If the new transformed equation does not contain the $x^2$ term,then $h$ is equal to:

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

One root of the given equation $2x^5 - 14x^4 + 31x^3 - 64x^2 + 19x + 130 = 0$ is