The equation $e^{4 x}+8 e^{3 x}+13 e^{2 x}-8 e^x+1=0, x \in R$ has:
two solutions and both are negative
no solution
four solutions two of which are negative
two solutions and only one of them is negative
The sum of the roots of the equation, ${x^2}\, + \,\left| {2x - 3} \right|\, - \,4\, = \,0,$ is
If $a, b, c, d$ and $p$ are distinct real numbers such that $(a^2 + b^2 + c^2)\,p^2 -2p\, (ab + bc + cd) + (b^2 + c^2 + d^2) \le 0$, then
The equation${e^x} - x - 1 = 0$ has
If the equation $\frac{1}{x} + \frac{1}{{x - 1}} + \frac{1}{{x - 2}} = 3{x^3}$ has $k$ real roots, then $k$ is equal to -
If $\alpha , \beta $ are the roots of the equation $x^2 - 2x + 4 = 0$ , then the value of $\alpha ^n +\beta ^n$ is