If the roots of the equation $\frac{x^2 - bx}{ax - c} = \frac{m - 1}{m + 1}$ are equal but opposite in sign,then the value of $m$ is

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:

If $\alpha$ and $\beta$ are the roots of the equation $x^2+x+1=0$,then the quadratic equation whose roots are $\alpha^{2023}$ and $\beta^{1012}$ is

Let $p, q, r \in \mathbb{R}$ and $r > p > 0$. If the quadratic equation $px^2 + qx + r = 0$ has two complex roots $\alpha$ and $\beta$,then what is the value of $|\alpha| + |\beta|$?

The roots of the equation $x^4 - 8x^2 - 9 = 0$ are

The positive root of $x^{2}-78.8=0$ after the first approximation by the Newton-Raphson method,assuming the initial approximation to the root is $14$,is: