This question has statement - 1 and statement - 2 of the four choice given after the statements choose the one that best describes the two statements
statement - 1 : A metallic surface is irradiated by a monochromatic light of frequency ` v gt v_(0)` (the threshold frequency). The maximum kinetic energy and the stopping potential are `K_(max) and V_(0)` respectively if the frequency incident on the surface is doubled , both the `K_(max) and V_(0)` are also doubled
statement - 2 : The maximum kinetic energy and the stopping potential of photoelectron emitted from a surface are linearly dependent on the frequency of incident light

Options

A. Statement -1 is true, statement -`2` is false.
B. Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-1
C. statement-1 is True, Statement-2 is True, Statement-2 is NOT a correct explanation for Statement -1
D. Statement -1 is False, Statement-2 is True

Explanation

To solve the given question, we need to analyze both statements carefully and determine their validity based on the principles of photoelectric effect.

### Step-by-Step Solution:

1. **Understanding Statement 1**:
- The statement claims that if a metallic surface is irradiated by monochromatic light of frequency \( v \) (where \( v > v_0 \), with \( v_0 \) being the threshold frequency), the maximum kinetic energy \( K_{max} \) and the stopping potential \( V_0 \) are both doubled when the frequency is doubled.
- The energy of a photon is given by \( E = h \nu \), where \( h \) is Planck's constant and \( \nu \) is the frequency of the light.
- The maximum kinetic energy of the emitted photoelectrons is given by the equation:
\[ K_{max} = h \nu - W \]
where \( W \) is the work function of the material.

2. **Doubling the Frequency**:
- If the frequency is doubled, then the new frequency is \( 2\nu \).
- The energy of the new photon becomes:
\[ E' = h(2\nu) = 2h\nu \]
- The new maximum kinetic energy becomes:
\[ K'_{max} = 2h\nu - W \]
- Comparing the original maximum kinetic energy \( K_{max} = h\nu - W \) with the new maximum kinetic energy \( K'_{max} \):
\[ K'_{max} = 2(h\nu) - W \neq 2K_{max} \]
- Therefore, \( K'_{max} \) is not equal to double \( K_{max} \) because of the work function \( W \). Hence, **Statement 1 is false**.

3. **Understanding Statement 2**:
- The statement claims that the maximum kinetic energy and the stopping potential of photoelectrons emitted from a surface are linearly dependent on the frequency of the incident light.
- From the equation \( K_{max} = h\nu - W \), we see that \( K_{max} \) is a linear function of \( \nu \) since it can be expressed as:
\[ K_{max} = h\nu - W \]
- The stopping potential \( V_0 \) is related to the maximum kinetic energy by the equation:
\[ K_{max} = eV_0 \]
- Thus, if \( K_{max} \) is linear in \( \nu \), then \( V_0 \) must also be linear in \( \nu \). Therefore, **Statement 2 is true**.

4. **Conclusion**:
- Since Statement 1 is false and Statement 2 is true, the correct option that describes the two statements is **D**: Statement 1 is false, Statement 2 is true.