a
(a) According to Wein's law \(\lambda_mT\) = constant

==> \({\lambda _{{m_1}}}{T_1} = {\lambda _{{m_2}}}{T_2}\)

\(\Rightarrow {T_2} = \frac{{{\lambda _{{m_1}}}}}{{{\lambda _{{m_2}}}}}{T_1} = \frac{{{\lambda _0}}}{{3{\lambda _0}/4}} \times {T_1} = \frac{4}{3}{T_1}\)

Now \(P \propto {T^4} \Rightarrow \frac{{{P_2}}}{{{P_1}}} = {\left( {\frac{{{T_2}}}{{{T_1}}}} \right)^4}\)

==> \(\frac{{{P_2}}}{{{P_1}}} = {\left( {\frac{{4/3\;{T_1}}}{{{T_1}}}} \right)^4} = \frac{{256}}{{81}}\)