In which method is recombinant DNA directly injected into the nucleus…

The correct answer is Option (4) → Micro-injection

Recombinant DNA must be put into a host cell before it can work. DNA is a hydrophilic (water-loving) molecule, so it cannot pass through cell membranes on its own. Different methods are used for different hosts: bacteria, plant cells and animal cells. For an animal cell, the DNA is directly injected into the nucleus with a fine needle. This is micro-injection.

Option (1) Biolistics (gene gun) — Incorrect. In biolistics, cells are hit with tiny particles of gold or tungsten coated with DNA. The book gives this method as suitable for plants, not for putting DNA into the nucleus of an animal cell.

Option (2) Disarmed pathogen vectors — Incorrect. Disarmed pathogens, such as the Ti plasmid of Agrobacterium in plants or disarmed retroviruses in animal cells, carry the DNA into the cell by infecting it. The DNA is not injected straight into the nucleus.

Option (3) Heat shock in the presence of a divalent cation — Incorrect. Heat shock with a divalent cation such as calcium is used to make bacterial cells ‘competent’, so that they take up plasmids. It is a method for bacteria, not for animal cells.

Option (4) Micro-injection — Correct. In micro-injection, recombinant DNA is directly injected into the nucleus of an animal cell.

NCERT Extract:
Competent Host (for Transformation with Recombinant DNA) – “Since DNA is a hydrophilic molecule, it cannot pass through cell membranes. … In a method known as micro-injection, recombinant DNA is directly injected into the nucleus of an animal cell. In another method, suitable for plants, cells are bombarded with high velocity micro-particles of gold or tungsten coated with DNA in a method known as biolistics or gene gun. And the last method uses ‘disarmed pathogen’ vectors, which when allowed to infect the cell, transfer the recombinant DNA into the host.”