Let $\mathrm{f}(x)=x\left[\frac{x}{2}\right]$, for $-10< x<10$, where $[t]$ denotes the greatest integer function. Then the number of points of discontinuity of $f$ is equal to

Let $\hat{a}$ and $\hat{b}$ be two unit vectors. If the vectors $\overline{\mathrm{c}}=\hat{\mathrm{a}}+2 \hat{\mathrm{~b}}$ and $\overline{\mathrm{d}}=5 \hat{\mathrm{a}}+4 \hat{\mathrm{~b}}$ are perpendicular to each other, then the angle between $\hat{a}$ and $\hat{b}$ is

Integrating factor of the differential equation $\frac{\mathrm{d} y}{\mathrm{~d} x}+y=\frac{1+y}{x}$ is

A service station manager sells gas at an average of ₹ 100 per hour on a rainy day, ₹ 150 per hour on a dubious day, ₹ 250 per hour on a fair day and ₹ $300$ on a clear sky. If weather bureau statistics show the probabilities of weather as follows, then his mathematical expectation is