How is the temperature of stars determined?

The wavelength corresponding to the maximum energy emitted by a body at a certain temperature is ${\lambda _0}$. If the temperature of the body is increased such that the new maximum wavelength becomes $\frac{3{\lambda _0}}{4}$, then by what factor does the emissive power increase?

The Earth radiates in the infra-red region of the spectrum. The spectrum is correctly given by

The wavelength of maximum emitted energy of a body at $700 \ K$ is $4.08 \ \mu m$. If the temperature of the body is raised to $1400 \ K$,the wavelength of maximum emitted energy will be ........ $\mu m$.

$A$ hot source emits maximum radiation energy at a wavelength of $11 \times 10^{-5} \ cm$. According to Wien's displacement law,for what value of $n$ will a source at a temperature $n$ times the original temperature emit maximum energy at a wavelength of $5.5 \times 10^{-5} \ cm$?

Two metal spheres have radii $r$ and $2r$ and they emit thermal radiation with maximum intensities at wavelengths $\lambda$ and $2\lambda$ respectively. The respective ratio of the radiant energy emitted by them per second will be .........