1. Use conservation of momentum

Initially, the particle of mass $4M$ is at rest. So initial momentum is $p_{\text{initial}}=0.$

After disintegration into particles of masses $M$ and $3M$, total momentum must still be zero: $\vec p_1 + \vec p_2 = 0.$

Hence, $\vec p_1 = -\vec p_2,$ so their magnitudes are equal: $p_1 = p_2.$

2. Apply de-Broglie wavelength formula

For any particle, $\lambda = \frac{h}{p}.$

Thus for the two particles, $\lambda_M = \frac{h}{p_1}, \qquad \lambda_{3M} = \frac{h}{p_2}.$

Since $p_1=p_2$, $\lambda_M = \lambda_{3M}.$

Therefore, $\lambda_M : \lambda_{3M} = 1:1.$

3. Check options

• A: $1:3$ ✗
• B: $3:1$ ✗
• C: $1:\sqrt{3}$ ✗
• D: $1:1$ ✓

So the correct option is D.