The equation $\log_e x + \log_e(1 + x) = 0$ can be written as

Number of integral values of $m$ for which ${x}^2 + 5m{x} - 3m + 1 < 0$ $\forall x \in R$ is (where ${.}$ denotes the fractional part function).

The solution set of the equation $x^{\log_x(1 - x)^2} = 9$ is

Let $\alpha$ and $\beta$ be two roots of the equation $x^2 + 2x + 2 = 0$. Then,$\alpha^{15} + \beta^{15}$ is equal to:

The harmonic mean of the roots of the equation $(5 + \sqrt{2})x^2 - (4 + \sqrt{5})x + 8 + 2\sqrt{5} = 0$ is

If $\alpha$ and $\beta$ are the roots of $x^2 - 3x + a = 0$,where $a \in R$ and $\alpha < 1 < \beta$,then: