Q.With respect to Eichhornia
Statement A: It drains off oxygen from water and is seen growing in standing water.
Statement B: It is an indigenous species of our country.

Of the listed invasive species, Prosopis juliflora is the one that originated in South America and was later introduced into India, where it became invasive.

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India's environmental issues chapter lists several notorious biological invaders, each with a distinct geographic origin:

• Prosopis juliflora: a thorny leguminous tree/shrub native to the dry tropical regions of South America (and Central America/Mexico); it was deliberately introduced into India, including Tamil Nadu and Rajasthan, in the 1800s to green arid wastelands and provide fuelwood, but it has since become a highly aggressive invasive species that displaces native vegetation and reduces grazing land.
• Parthenium hysterophorus ('Congress grass'): native to Mexico and the tropical parts of Central America/the Caribbean; introduced accidentally into India (reportedly via contaminated wheat grain imports in the 1950s), it is now a widespread invasive weed causing allergies and reducing crop yields.
• Lantana camara: native to the tropical Americas (Central America and the Caribbean); introduced into India in the 19th century as an ornamental garden plant, it has since spread aggressively into forests and grasslands.
• Kappaphycus alvarezii: an invasive red seaweed native to the Philippines/Indo-Pacific region, introduced along the Tamil Nadu coast (Gulf of Mannar) for commercial seaweed (carrageenan) farming, where it now smothers native coral-reef and seagrass communities.

Since the question specifically asks for the species native to South America, Prosopis is the best-supported answer.

Clarias gariepinus, the African (walking) catfish, was introduced in India for aquaculture but has since turned invasive and threatens native catfish.

The African catfish Clarias gariepinus was introduced into Indian water bodies for aquaculture (fish farming) purposes because of its fast growth. However, being a hardy, opportunistic predator, it has since escaped into open water bodies and become an invasive species, out-competing and preying on native catfishes and other indigenous species, thereby illustrating how biological invasion by exotic species is a threat to biodiversity.

Water hyacinth (Eichhornia crassipes) creates the stagnant-water conditions that let disease-carrying mosquitoes breed.

Eichhornia crassipes is a notorious free-floating aquatic invasive species that rapidly multiplies to choke ponds, lakes, canals and irrigation channels with dense floating mats. These mats slow water movement, reduce light penetration and oxygen exchange, and create sheltered, stagnant micro-habitats among the roots and leaf bases that are ideal for mosquito larvae to develop undisturbed, including species of the genus Anopheles that transmit malaria. This is one of the major ecological and public-health harms attributed to the plant, alongside its blocking of waterways and displacement of native aquatic flora and fauna. Lantana camara, Prosopis juliflora and Parthenium hysterophorus, by contrast, are all terrestrial invasive weeds of wasteland, scrub and roadside habitats; they cause problems such as allelopathic suppression of native vegetation, thorny thicket formation, and contact allergies/respiratory irritation respectively, but none of them creates standing-water breeding sites for mosquitoes.