The correct answer is (a). The Katyuri dynasty was a Hindu dynasty that ruled parts of present-day Uttarakhand from the 7th to the 12th centuries AD. They were originally from the Katyur region of Garhwal, and their capital was at Nagarkot. The Katyuris were a powerful dynasty, and they extended their rule over much of the Himalayas. They were patrons of the arts and literature, and they built many temples and monasteries. The Katyuri dynasty declined in the 12th century, and they were eventually overthrown by the Chand dynasty.
Option (b) is incorrect because the Katyuri dynasty did not rule in parts of present-day Uttarakhand from 1500 BC to 500 AD. The Katyuri dynasty did not exist during this time period.
Option (c) is incorrect because the Katyuri dynasty did not rule in parts of present-day Uttarakhand from 1500 AD to 1800 AD. The Katyuri dynasty was overthrown in the 12th century, and they did not exist during this time period.
Option (d) is incorrect because the Katyuri dynasty did not rule in parts of present-day Uttarakhand from 2400 BC to 2000 BC. The Katyuri dynasty did not exist during this time period.