Correct option is (A)

$\begin{array}{rl}& {\mathrm{<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">IN</font>\; </font>}}_0<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">=</font>\; </font>\frac{\mathrm{<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">h</font>\; </font>}}{\mathrm{<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">It\; is\; </font>\; </font>}}f<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">-</font>\; </font>\frac{\mathrm{<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">h</font>\; </font>}}{\mathrm{<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">It\; is\; </font>\; </font>}}{\mathrm{<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">f</font>\; </font>}}_{\mathrm{<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">0</font>\; </font>}}\\ & \Rightarrow \text{<font style="vertical-align: inherit;"> <font style="vertical-align: inherit;"> Slope </font> </font>}<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">=</font>\; </font>\frac{\mathrm{<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">h</font>\; </font>}}{\mathrm{<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">It\; is\; </font>\; </font>}}\text{<font style="vertical-align: inherit;"> <font style="vertical-align: inherit;"> (S-I is correct) </font> </font>}\\ & \because {<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">(\; </font>\; </font>{\mathrm{<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">f\; </font>\; </font>}}_{\mathrm{<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">0\; </font>\; </font>}}<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">)\; </font>\; </font>}_{\mathrm{<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">1</font>\; </font>}}<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;"><</font>\; </font>{<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">(\; </font>\; </font>{\mathrm{<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">f\; </font>\; </font>}}_{\mathrm{<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">0\; </font>\; </font>}}<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">)\; </font>\; </font>}_{\mathrm{<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">2</font>\; </font>}}\\ & \therefore <font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">(</font>\; </font>\mathrm{S}<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">-</font>\; </font>\mathrm{<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">I</font>\; </font>}\mathrm{<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">I</font>\; </font>}\text{<font style="vertical-align: inherit;"> <font style="vertical-align: inherit;"> is incorrect) </font> </font>}\\ & <font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">\therefore </font>\; </font>\text{<font style="vertical-align: inherit;"> <font style="vertical-align: inherit;"> Option (3) is correct. </font> </font>}\end{array}$

Correct option is (A)

In order to determine the most appropriate answer to the question, let's analyze the given Assertion A and Reason R in detail.

Assertion A: Number of photons increases with increase in frequency of light.

This statement is not correct. The number of photons is determined by the intensity (or power) of the light and is given by the formula:

$N<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">=</font>\; </font>\frac{P}{\mathrm{<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">h</font>\; </font>}\mathrm{<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">f</font>\; </font>}}$

where $\mathrm{<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">N</font>\; </font>}$ is the number of photons per second, $\mathrm{<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">P</font>\; </font>}$ is the power (intensity) of the light, $\mathrm{<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">h</font>\; </font>}$ is Planck's constant, and $\mathrm{<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">f</font>\; </font>}$ is the frequency of the light. As the frequency increases, the energy per photon increases, but it does not necessarily mean that the number of photons increases unless the power of the light also increases proportionally.

Reason R: Maximum kinetic energy of emitted electrons increases with the frequency of incident radiation.

This statement is correct. According to the photoelectric effect, the maximum kinetic energy of emitted electrons is given by:

$K<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">.</font>\; </font>{\mathrm{<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">AND</font>\; </font>}}_{\text{<font style="vertical-align: inherit;"> <font style="vertical-align: inherit;">max</font> </font>}}<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">=</font>\; </font>\mathrm{<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">h</font>\; </font>}\mathrm{<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">f</font>\; </font>}<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">-</font>\; </font>\mathrm{<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">\varphi </font>\; </font>}$

where $\mathrm{<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">K</font>\; </font>}<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">.</font>\; </font>{\mathrm{<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">AND</font>\; </font>}}_{\text{<font style="vertical-align: inherit;"> <font style="vertical-align: inherit;">max</font> </font>}}$ is the maximum kinetic energy of the emitted electrons, $\mathrm{<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">h</font>\; </font>}$ is Planck's constant, $\mathrm{<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">f</font>\; </font>}$ is the frequency of the incident radiation, and $\mathrm{<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">\varphi </font>\; </font>}$ is the work function of the material. As the frequency of the incident light increases, the kinetic energy of the emitted electrons increases.

Now, let's match the statements with the options given:

Option A: $\mathbf{<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">A</font>\; </font>}$ is not correct but $\mathbf{<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">R</font>\; </font>}$ is correct.

This option is correct because Assertion A is incorrect while Reason R is correct.

Option B: $\mathbf{<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">A</font>\; </font>}$ is correct but $\mathbf{<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">R</font>\; </font>}$ is not correct.

This option is incorrect because Assertion A is not correct.

Option C: Both $\mathbf{<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">A</font>\; </font>}$ and $\mathbf{<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">R</font>\; </font>}$ are correct and $\mathbf{<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">R</font>\; </font>}$ is the correct explanation of $\mathbf{<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">A</font>\; </font>}$ .

This option is incorrect because Assertion A is not correct.

Option D: Both $\mathbf{<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">A</font>\; </font>}$ and $\mathbf{<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">R</font>\; </font>}$ are correct and $\mathbf{<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">R</font>\; </font>}$ is NOT the correct explanation of $\mathbf{<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">A</font>\; </font>}$ .

This option is incorrect because Assertion A is not correct.

Therefore, the most appropriate answer is:

Option A: $$\mathbf{A}$$ is not correct but $$\mathbf{R}$$ is correct. $\mathbf{<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">A</font>\; </font>}$ is not correct but $\mathbf{<font\; style="vertical-align:\; inherit;">\; <font\; style="vertical-align:\; inherit;">R</font>\; </font>}$ is correct.

Correct option is (C)

Stopping potential is independent on intensity but photocurrent increases non-linearly on increasing intensity.

Correct option is (C)

Since $\frac{\mathrm{f}}{2}<{\mathrm{f}}_0$

i.e. the incident frequency is less than threshold frequency. Hence there will be no emission of photoelectrons.

$\Rightarrow \text{ current }=0$

Correct option is (B)

$\begin{array}{rl}& \mathrm{E}=\varphi +{\mathrm{K}}_{max}\\ & \varphi =\frac{\mathrm{hc}}{{\lambda }_0}\\ & {\mathrm{K}}_{max}={\mathrm{eV}}_0\\ & 8\mathrm{e}=\frac{\mathrm{hc}}{\lambda }-\frac{\mathrm{hc}}{{\lambda }_0}\dots ..\text{ (i) }\\ & 2\mathrm{e}=\frac{\mathrm{hc}}{3\lambda }-\frac{\mathrm{hc}}{{\lambda }_0}\dots ...\text{ (ii) }\\ & \text{ on solving (i) \& (ii) }\\ & {\lambda }_0=9\lambda \end{array}$