If A(1,2),B(-1,4)  and if the axes are rotated through an angle $\frac{\mathrm{\pi }}{12}$ in anticlockwise sense, then the slope of the line $\mathrm{AB}$ with reference to the new axes is

$$\begin{gathered} \mathrm{Given} \mathrm{point} \mathrm{A}=(1,2), \mathrm{B}=(-1,4) \\ \mathrm{Slope} \mathrm{of} \overline{\mathrm{AB}}=\frac{4-2}{-1-1}=-1 \\ \Rightarrow \mathrm{m}=-1 \\ \mathrm{tan\theta }=\tan {135}^0\Rightarrow \mathrm{\theta }={135}^0 \\ \mathrm{If} \mathrm{the} \mathrm{axes} \mathrm{are} \mathrm{rotated} \mathrm{through} \mathrm{an} \mathrm{angle} \\ \mathrm{\theta }=\frac{\mathrm{\pi }}{12}={15}^0 \mathrm{then} inclination of line w.r.to new axes={135}^0-{15}^0={120}^0 \\ Now \mathrm{slope} \mathrm{of} \mathrm{line} \overline{\mathrm{AB} }\mathrm{w}.\mathrm{r}. \mathrm{to} \mathrm{new} \mathrm{axes} \mathrm{is} \mathrm{m}=\tan {120}^0=-\sqrt{3} \end{gathered}$$