If $\alpha, \beta, \gamma, \delta$ are the roots of the equation $x^4-8x^3+11x^2+32x-60=0$ and $\alpha < \beta < \gamma < \delta$, then $4\alpha+3\beta+2\gamma+\delta=$

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:

If $a, b \in \{1, 2, 3\}$ and the equation $ax^{2} + bx + 1 = 0$ has real roots, then

Let $\theta$,$0 < \theta < \pi / 2$,be an angle such that the equation $x^2 + 4x \cos \theta + \cot \theta = 0$ has equal roots for $x$. Then $\theta$ in radians is

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

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3-Px^2+Qx-R=0$ and $(\alpha-2)^2, (\beta-2)^2, (\gamma-2)^2$ are the roots of the equation $x^3-5x^2+4x=0$,then the possible least value of $P+Q+R$ is