If $\alpha, \beta$ are the roots of $x^2-a(x-1)+b=0$, then the value of $\frac{1}{\alpha^2-a \alpha}+\frac{1}{\beta^2-a \beta}+\frac{2}{a+b}$ is:

Let $a_1, a_2, a_3, a_4$ be real numbers such that $a_1+a_2+a_3+a_4=0$ and $a_1^2+a_2^2+a_3^2+a_4^2=1$. Then,the smallest possible value of the expression $(a_1-a_2)^2+(a_2-a_3)^2+(a_3-a_4)^2+(a_4-a_1)^2$ lies in the interval

Let $\lambda \in R$ and let the equation $E$ be $|x|^2 - 2|x| + |\lambda - 3| = 0$. Then the largest element in the set $S = \{x + \lambda : x \text{ is an integer solution of } E\}$ is $..........$

Let $2 \sin^2 x + 3 \sin x - 2 > 0$ and $x^2 - x - 2 < 0$ ($x$ is measured in radians). Then $x$ lies in the interval

If $m$ is a root of the equation $(1 - ab)x^2 - (a^2 + b^2)x - (1 + ab) = 0$ and $m$ harmonic means are inserted between $a$ and $b$,then the difference between the last and the first of the means equals

$f(x)$ is a quadratic expression such that $f(x)$ is negative when $x \in \left(-\infty, -\frac{5}{3}\right) \cup (3, \infty)$ and positive when $x \in \left(-\frac{5}{3}, 3\right)$. $g(x)$ is another quadratic expression such that $g(x)$ is negative when $x \in \left(3, \frac{9}{2}\right)$ and positive when $x \in \mathbb{R} - \left[3, \frac{9}{2}\right]$. Then,the sign of $f(x)g(x)$ in $[0, 5]$ is