The number of real roots of the equation $5 + |2^x - 1| = 2^x(2^x - 2)$ is

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 -

The number of real solution of equation $(\frac{3}{2})^x =  -x^2 + 5x-10$ :-

The equation $x^2-4 x+[x]+3=x[x]$, where $[x]$ denotes the greatest integer function, has:

Let $m$ and $n$ be the numbers of real roots of the quadratic equations $x^2-12 x+[x]+31=0$ and $x ^2-5| x +2|-4=0$ respectively, where $[ x ]$ denotes the greatest integer $\leq x$. Then $m ^2+ mn + n ^2$ is equal to $..............$.

The number of real roots of the equation $\mathrm{e}^{4 \mathrm{x}}-\mathrm{e}^{3 \mathrm{x}}-4 \mathrm{e}^{2 \mathrm{x}}-\mathrm{e}^{\mathrm{x}}+1=0$ is equal to $.....$