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I'm not really sure about why d is false but for part a, I recall that lavelle briefly mentioned that since UV light has a lower wavelength, therefore a higher frequency and energy than visible or infrared lights, most metals will emit light when a UV light is shone on it. Most metals need this high of a frequency in order for the energy of the light to be greater than the work function of metal.
I assume that for part d, a higher frequency doesn't always imply that the emitted electrons have higher kinetic energies because the frequency has to be within a certain threshold (thus the wavelength has to also be within a certain threshold) in order for electrons to even be emitted.
For part d, higher frequency light has higher energy, so the energy of the photons in that light may be higher than the work function, giving the electrons emitted a higher energy. It is shown by the equation Kinetic Energy of electron=Energy of photon-Work Function. However, not all high frequencies of light will yield an equation where the energy of the photon is higher than the work function. It has to reach a certain threshold.
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