Human Health and Disease

Common cold is one of the most infectious human ailments; it is caused by rhino viruses, which infect the nose and respiratory passage.
Antibiotics act on bacteria by interfering with structures and processes such as the cell wall, ribosomes and metabolic enzymes.
Viruses have no cell wall, ribosomes or metabolism of their own; they multiply using the machinery of the host cell.
Therefore antibiotics have no effect on viruses.
Hence common cold is not cured by antibiotics because it is caused by a virus.

Pneumonia is caused by $Streptococcus\ pneumoniae$ (and $Haemophilus\ influenzae$); a healthy person gets it by inhaling the droplets or aerosols released by an infected person. This row is fully correct.
Elephantiasis (filariasis) is caused by $Wuchereria\ bancrofti$, but it is transmitted by the bite of the female $Culex$ mosquito, not by water and food.
Malaria is caused by $Plasmodium$, but it is transmitted by the bite of the female $Anopheles$ mosquito, not the male.
Typhoid is caused by $Salmonella\ typhi$, but it spreads through contaminated food and water, not inspired air.
Hence the correct match is pneumonia – $Streptococcus\ pneumoniae$ – droplet infection.

Antibodies are produced by B-lymphocytes (which form plasma cells), and T-lymphocytes (helper T-cells) help the B-cells to produce them. If both B- and T-lymphocytes are damaged, no antibodies will be produced against a pathogen. This statement is correct.
Hepatitis B vaccine is produced by recombinant DNA technology in yeast, which is a fungus, not a protozoan.
Snake antivenom contains preformed antibodies, so its injection is passive immunisation, not active.
Injection of dead or inactivated pathogens (vaccination) makes the body produce its own antibodies and memory cells, which is active immunity, not passive.
Hence the correct statement is the one about damage to B- and T-lymphocytes.