The sum of all distinct roots of the equation $x^5-3x^4+5x^3-5x^2+3x-1=0$ is

The equation $(\cos p - 1) x^2 + (\cos p) x + \sin p = 0$ in the variable $x$ has real roots. Then $p$ can take any value in the interval

The quadratic equation whose roots are $\sin^2 18^{\circ}$ and $\cos^2 36^{\circ}$ is:

Let $\alpha, \beta$ be the roots of the quadratic equation $12x^{2}-20x+3\lambda=0$, where $\lambda \in \mathbb{Z}$. If $\frac{1}{2} \le |\beta-\alpha| \le \frac{3}{2}$, then the sum of all possible values of $\lambda$ is:

Let $S$ be the set of all non-zero real numbers $\alpha$ such that the quadratic equation $\alpha x^2 - x + \alpha = 0$ has two distinct real roots $x_1$ and $x_2$ satisfying the inequality $|x_1 - x_2| < 1$. Which of the following intervals is(are) a subset$(s)$ of $S$?
$(A) \left(-\frac{1}{2}, -\frac{1}{\sqrt{5}}\right)$
$(B) \left(-\frac{1}{\sqrt{5}}, 0\right)$
$(C) \left(0, \frac{1}{\sqrt{5}}\right)$
$(D) \left(\frac{1}{\sqrt{5}}, \frac{1}{2}\right)$

What is the minimum value of the expression $x^2 + 4y^2 + 3z^2 - 2x - 12y - 6z + 14$?