The question asks about the establishment date of the Bengal British India Society, which is essential information for Indian history and its governance system. The correct answer provided is "20 April, 1843". Let's explore the details:

1. The Bengal British India Society was an important organization formed in the early 19th century. It represented the intellectual voices advocating for economic and political reform under British rule in India.
2. According to historical records, the society was officially established on 20 April, 1843 in Calcutta (now Kolkata), which was then a major center for cultural and political activities in British India.
3. This society aimed to improve the condition of Indians under the British regime and had a significant role in influencing British policies towards India.

Now, let's examine why the other options are incorrect:

• 20 April, 1844: This date is one year after the actual establishment and hence incorrect.
• 21 April, 1843: This is one day after the correct establishment date, making it incorrect.
• 20 April, 1834: This date is significantly earlier than the actual establishment and thus incorrect.

In conclusion, the Bengal British India Society was established on 20 April, 1843, which aligns with historical records and the context provided by this question. Knowing such historical milestones helps in understanding the reformist and intellectual movements that shaped modern India.