Why would amplitude modulation result in change of frequency? Question: (single choice correct; source: JEE Main 2015)


  A signal of $5 kHz$ frequency is amplitude modulated on a carrier wave of frequency $2 MHz$. The frequencies of the resultant signal is/are:
  
  
  2 MHz only
  2005 kHz, and 1995 kHz
  2005 kHz, 2000 kHz and 1995 kHz
  2000 kHz and 1995 kHz
  


My attempt:

I understand that amplitude modulation is the superimposition of an audio-frequency on a radio frequency carrier wave, in such a manner such that the frequency of the modulated wave is the same as that of the carrier wave, but its amplitude varies in accordance with the instantaneous amplitude of the modulating wave (message signal). 

Note the part I highlighted, because, based on that part, I believe that the answer to the quoted question should be (1). However, the official answer was given as (3), and I have no clue why. It might have been correct in frequency modulation, but I am not sure why it's true in amplitude modulation as well.

Note that this is high-school level question, please answer accordingly. Many thanks!
$$A[1+\mu\cos(2\pi 5000t)]\sin(2\pi 2000000t)= A\sin(2\pi 2000000t) + \dfrac{A\mu}{2}(\sin(2\pi[2000000-5000]t)+\sin(2\pi[2000000+5000]t))$$

If I didn't make an error with my trigonometric identities.

The three distinct frequencies are there: the carrier and the positive and negative frequency version of the tone around the carrier.

The above assumes an AM that is not suppressed carrier, since the problem statement didn't say "suppressed carrier".