A foreign DNA fragment was ligated at the BamH I restriction site...

The pBR322 cloning vector contains two antibiotic resistance genes: ampicillin resistance ($amp^R$) and tetracycline resistance ($tet^R$).

The BamH I restriction site is located within the tetracycline resistance gene ($tet^R$). When a foreign DNA fragment is ligated at the BamH I site, it disrupts the $tet^R$ gene, leading to its insertional inactivation. As a result, the recombinant plasmids lose their resistance to tetracycline but retain their resistance to ampicillin.

When these recombinant plasmids are introduced into host $E.\ coli$ cells, the resulting recombinant transformants will be resistant to ampicillin but sensitive to tetracycline.

In the given selection process:
- Plate 1 contains an ampicillin medium. The recombinant transformants will grow and form colonies here because they possess the intact $amp^R$ gene.
- Plate 2 contains a tetracycline medium. The recombinant transformants will not survive here because their $tet^R$ gene has been inactivated.

Therefore, the recombinant transformants will form colonies on Plate 1 but will not survive on Plate 2.

Answer: They will form colonies on Plate 1 but will not survive on Plate 2.