1. Given:

   • Frequency of electromagnetic wave: $\nu = 3.0\,\text{MHz}$
   • Wave goes from vacuum into a dielectric medium
   • Permittivity of medium: $\varepsilon_r = 4.0$

2. Key concepts:

   • When an electromagnetic wave passes from one medium to another, its frequency remains unchanged.
   • Speed in a dielectric medium is $v = \frac{c}{\sqrt{\varepsilon_r \mu_r}}$
   • For a non-magnetic dielectric, $\mu_r \approx 1$, so $v = \frac{c}{\sqrt{\varepsilon_r}} = \frac{c}{\sqrt{4}} = \frac{c}{2}$

3. Use the wave relation: $v = \nu \lambda$ Since frequency $\nu$ remains unchanged and speed becomes half, wavelength must also become half.

   In vacuum: $\lambda_0 = \frac{c}{\nu}$

   In medium: $\lambda = \frac{v}{\nu} = \frac{c/2}{\nu} = \frac{1}{2}\lambda_0$

4. Conclusion:

   • Frequency remains unchanged
   • Wavelength is halved

5. Check options:

   • A: wave length is halved and frequency remains unchanged ✅
   • B: incorrect
   • C: incorrect
   • D: incorrect

Therefore, the correct answer is A.