If the roots of the equation $x^2 - 8x + (a^2 - 6a) = 0$ are real,then:

The sum of all the roots of the equation $(x-1)^{2}-5|x-1|+6=0$ is:

The number of real roots of the equation $\sqrt{\frac{x}{1-x}}+\sqrt{\frac{1-x}{x}}=\frac{13}{6}$ 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 $a, b$ are real numbers and $\alpha$ is a real root of $x^2 + 6x + 12 + 3 \sin(a + b\alpha) = 0$,then the value of $\cos(a + b\alpha)$ for the least positive value of $a + b\alpha$ is

The value of $m$ for which the equation $\frac{a}{x + a + m} + \frac{b}{x + b + m} = 1$ has roots equal in magnitude but opposite in sign is