It contributes to significant rainfall along the Narmada and Tapi valleys - This is correct. The Arabian Sea branch of Southwest Monsoon does bring significant rainfall to the Narmada and Tapi river valleys as it moves inland from the western coast.
It causes heavy rainfall on the leeward side of the Western Ghats - This is incorrect. The leeward side (eastern side) of the Western Ghats actually receives much less rainfall due to the rain shadow effect. The windward side (western side) facing the Arabian Sea receives heavy rainfall as the moisture-laden winds are forced to rise over the mountains.
One of its branches causes scanty rainfall in western Rajasthan - This is correct. As the Arabian Sea branch moves northward, it splits, and one branch moves toward western Rajasthan, but by this time it has lost much of its moisture, resulting in scanty rainfall in this region.
It directly impacts the rainfall in the Tamil Nadu coastal regions - This is incorrect. Tamil Nadu coastal regions mainly receive rainfall from the retreating monsoon (Northeast Monsoon) during October-December, not directly from the Arabian Sea branch of the Southwest Monsoon.
