Use Einstein’s photoelectric equation

$K_{\max}=h\nu-\phi$

where:

• $K_{\max}$ = maximum kinetic energy of emitted photoelectrons
• $h$ = Planck’s constant
• $\nu$ = frequency of incident light
• $\phi$ = work function of the metal

Also,

$K_{\max}=\frac{1}{2}mv_{\max}^2$

Hence,

$\frac{1}{2}mv_{\max}^2=h\nu-\phi$ $v_{\max}^2=\frac{2h}{m}\nu-\frac{2\phi}{m}$

So $v_{\max}^2$ varies linearly with frequency $\nu$.

Therefore, (A) is correct.

Check statement (B)

Saturation current depends on the number of photoelectrons emitted per second, which depends on the intensity of incident light.

If the source is moved away from the metal surface, intensity decreases (approximately by inverse square law), so fewer photoelectrons are emitted.

Therefore saturation current decreases, not increases.

Hence, (B) is false.

Check statement (C)

Maximum kinetic energy depends on frequency, not on intensity/power of the source (provided frequency remains above threshold).

Decreasing the power of the LED reduces the number of emitted electrons, but not their maximum kinetic energy.

Hence, (C) is false.

Check statement (D)

Immediate emission of photoelectrons is actually explained by particle nature of light: a single photon transfers energy instantaneously to a single electron.

So the statement “can not be explained by particle nature” is wrong.

Hence, (D) is false.

Check statement (E)

Threshold wavelength (or threshold frequency) means emission occurs only if photon energy satisfies

$h\nu \ge \phi$

This is naturally explained by photon theory.

Classical wave theory cannot explain the existence of a threshold frequency/wavelength.

Hence, (E) is correct.

Evaluate options

Correct statements are:

• (A) correct
• (E) correct

Therefore, the correct option is:

$\boxed{\text{B: (A) and (E) only}}$