For what set of values of $p$ do the roots of the equation $3x^2 + 2x + p(p - 1) = 0$ have opposite signs?

If $S = \{a \in R : |2a - 1| = 3[a] + 2\{a\}\}$,where $[t]$ denotes the greatest integer less than or equal to $t$ and $\{t\}$ represents the fractional part of $t$,then $72 \sum_{a \in S} a$ is equal to:

If the roots of the equation $\frac{1}{x + p} + \frac{1}{x + q} = \frac{1}{r}$ are equal in magnitude but opposite in sign,then the product of the roots will be

Solve the equation $x^{2}-2x+\frac{3}{2}=0$.

The number of real roots of the polynomial equation $x^4-x^2+2x-1=0$ is

The number of positive integers $x$ satisfying the equation $\frac{1}{x} + \frac{1}{x+1} + \frac{1}{x+2} = \frac{13}{12}$ is: