The number of real solutions of the equation $x\left(x^2+3|x|+5|x-1|+6|x-2|\right)=0$ is

The equation${e^x} - x - 1 = 0$ has

Suppose $a, b, c$ are positive integers such that $2^a+4^b+8^c=328$. Then, $\frac{a+2 b+3 c}{a b c}$ is equal to

The number of the real roots of the equation $(x+1)^{2}+|x-5|=\frac{27}{4}$ is ....... .

Number of natural solutions of the equation $x_1 + x_2 = 100$ , such that $x_1$ and $x_2$ are not multiple of $5$

All the points $(x, y)$ in the plane satisfying the equation $x^2+2 x \sin (x y)+1=0$ lie on