If(alpha,alpha²) falls inside the angle made by the lines 2y=x, x>0 and y=3x,x>0 ,then set of values of alpha is

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$$\begin{gathered} \mathrm{the} \mathrm{given} \mathrm{lines} ,{\mathrm{l}}_1\to 2\mathrm{y}=\mathrm{x} \mathrm{and} {\mathrm{l}}_2\to \mathrm{y}=3\mathrm{x} \\ \Rightarrow \mathrm{y}=\frac{\mathrm{x}}{2} , \mathrm{m}1={\tan }^{-1}\left(\frac{1}{2}\right) \\ \Rightarrow \mathrm{y}=3\mathrm{x} , \mathrm{m}2={\tan }^{-1}\left(3\right) \\ \mathrm{since} \mathrm{the} \mathrm{point} \left(\mathrm{\alpha },{\mathrm{\alpha }}^2\right), \mathrm{lies} \mathrm{between} \mathrm{angle} \mathrm{subtended} \mathrm{by} \mathrm{lines} {\mathrm{l}}_1\mathrm{and} {\mathrm{l}}_2 \mathrm{at} \mathrm{the} \mathrm{origin} \\ \mathrm{the} \mathrm{angle} \mathrm{subtended} \mathrm{by} \mathrm{this} \mathrm{point} \mathrm{at} \mathrm{the} \mathrm{origin} \mathrm{will} \mathrm{lie} \mathrm{between} \mathrm{m}1 \mathrm{and} \mathrm{m}2 \\ \mathrm{hence}, \\ \mathrm{m}=\frac{\left({\mathrm{\alpha }}^2-0\right)}{\left(\mathrm{\alpha }-0\right)}=\mathrm{\alpha } \\ {\tan }^{-1}\left(\frac{1}{2}\right)<\mathrm{\alpha }<{\tan }^{-1}\left(3\right) \\ \mathrm{\alpha }\in \left({\tan }^{-1}\left(\frac{1}{2}\right),{\tan }^{-1}\left(3\right)\right) \end{gathered}$$

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