Assertiom: The de-broglie wavelength of a neutrons when its kinetic energy is `k` is `lambda`. Its wavelength is `2lambda` when its kinetic energy is `4k`.
Reason : The de-broglie wavelength `lambda` is proportional to square root of the kinetic energy.

Options

A. If both, Assertion and Reason are true and the Reason is the correct explanation of the Assertion.
B. If both, Assertion and Reason are true but Reason is not a correct explanation of the Assertion.
C. If Assertion is true but the Reason is false.
D. If both Assertion and Reason are false.

Explanation

To solve the problem, we need to analyze the assertion and the reason given in the question step by step.

### Step 1: Understanding the de Broglie Wavelength
The de Broglie wavelength (\(\lambda\)) of a particle is given by the formula:
\[ \lambda = \frac{h}{p} \]
where \(h\) is Planck's constant and \(p\) is the momentum of the particle.

### Step 2: Relating Momentum to Kinetic Energy
The momentum \(p\) of a particle can be expressed in terms of its kinetic energy (\(K\)):
\[ K = \frac{1}{2}mv^2 \implies p = mv \]
From the kinetic energy formula, we can express \(v\) in terms of \(K\):
\[ v = \sqrt{\frac{2K}{m}} \implies p = m\sqrt{\frac{2K}{m}} = \sqrt{2mK} \]

### Step 3: Substituting Momentum into the de Broglie Wavelength Formula
Substituting the expression for momentum into the de Broglie wavelength formula:
\[ \lambda = \frac{h}{\sqrt{2mK}} \]
This shows that the de Broglie wavelength is inversely proportional to the square root of the kinetic energy:
\[ \lambda \propto \frac{1}{\sqrt{K}} \]

### Step 4: Analyzing the Assertion
The assertion states that when the kinetic energy is \(k\), the wavelength is \(\lambda\), and when the kinetic energy is \(4k\), the wavelength is \(2\lambda\).

Using our derived relationship:
1. For \(K = k\):
\[ \lambda_1 = \frac{h}{\sqrt{2mk}} \]
2. For \(K = 4k\):
\[ \lambda_2 = \frac{h}{\sqrt{2m(4k)}} = \frac{h}{\sqrt{8mk}} = \frac{h}{2\sqrt{2mk}} = \frac{\lambda_1}{2} \]

This means:
\[ \lambda_2 = \frac{\lambda_1}{2} \implies \lambda_2 = \frac{\lambda}{2} \]
Thus, the assertion that \(\lambda_2 = 2\lambda\) is incorrect.

### Step 5: Analyzing the Reason
The reason states that the de Broglie wavelength is proportional to the square root of the kinetic energy. However, we have established that:
\[ \lambda \propto \frac{1}{\sqrt{K}} \]
This means that the reason is also incorrect.

### Conclusion
Both the assertion and the reason are incorrect.

### Final Answer
- Assertion: Incorrect
- Reason: Incorrect