Let $[x]$ denote the greatest integer less than or equal to $x$. The number of positive integer solutions of the equation

$$ \left[\frac{x}{19}\right]=\left[\frac{x}{20}\right] $$

are:

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ be two functions such that

$$ g \circ f(x)= \begin{cases}x^2, & \text { if } x \geq 0 \\ e^x-1, & \text { if } x<0\end{cases} $$

Then which of the following statements is correct?

$F(x)=\int_0^{e^x}\left(t^3+2 t^2-t-2\right) d t$, then for how many real numbers $x$ does $F^{\prime}(x)=0$ ?

Let $C$ be a circle which passes through $(-2,1)$ and $(0,3)$ and whose centre lies on the line $y-x=1$. Then which of the following points lies on $C$ ?