The value of the integral

$$ \int_1^{100} \frac{[x]}{x} d x $$

where $[x]$ is the greatest integer less than or equal to $x$ for any real number $x$, is

For arbitrary constants $\alpha, \beta$, the differential equation representing the family of curves $y=$ $(\alpha x+\beta) e^x$ is

A particle experiences an acceleration $\vec{a}=\alpha \vec{v}$, where $\vec{v}$ is the velocity of the particle and $\alpha$ is a constant. If the distances traveled by the particle in the time intervals $t_2-t_1$ and $t_3-t_1$ are $S_{12}$ and $S_{13}$, respectively, which of the following relations is true?

A point mass $m$ attached to a massless string is undergoing circular motion in a vertical plane. The length of the string is $R$ and the acceleration due to gravity is $g$. If the minimum value of the tension in the string is 2 mg , the maximum speed of this circular motion of the point mass is