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 

The number of distinct real roots of the equation $|\mathrm{x}+1||\mathrm{x}+3|-4|\mathrm{x}+2|+5=0$, is ...........

Let $f(x)=a x^2+b x+c$, where $a, b, c$ are integers, Suppose $f(1)=0,40 < f(6) < 50,60 < f(7) < 70$ and $1000 t < f(50) < 1000(t+1)$ for some integer $t$. Then, the value of $t$ is

The number of distinct real roots of the equation $|\mathrm{x}||\mathrm{x}+2|-5|\mathrm{x}+1|-1=0$ is....................

If $\alpha ,\beta$ are the roots of $x^2 -ax + b = 0$ and if $\alpha^n + \beta^n = V_n$, then -

The sum of all the solutions of the equation $(8)^{2 x}-16 \cdot(8)^x+48=0$ is :