Solution:

The correct answer is option 'B' which is Cash Book.

Explanation:
The Cash Book is a subsidiary book that records all cash transactions of a business. It is used to record both cash receipts and cash payments.

In this scenario, an allowance of Rs. 50 was offered for an early payment of cash amounting to Rs. 1,050. This means that the business is giving a discount of Rs. 50 to the customer for making an early payment.

Reason for selecting Cash Book:
The Cash Book is the appropriate book to record this transaction because it involves the receipt of cash.

Recording the transaction in the Cash Book:
The transaction can be recorded in the Cash Book as follows:

Date: The date of the transaction.
Particulars: The name of the customer or the description of the transaction, in this case, it could be "Allowance for early payment".
L.F.: Leave this column blank as it is not required for this transaction.
Amount: The amount of cash received, which is Rs. 1,050.

Explanation of the entry:
By recording this transaction in the Cash Book, it will reflect the inflow of cash of Rs. 1,050. The allowance of Rs. 50 is not recorded separately in the Cash Book as it is a reduction in the amount received. The total amount received, after deducting the allowance, is Rs. 1,000.

Advantages of using Cash Book:
1. It helps in maintaining a proper record of cash transactions.
2. It provides an easy way to reconcile the cash balance with the bank statement.
3. It helps in identifying any discrepancies or errors in cash transactions.
4. It serves as a source document for preparing financial statements.

Conclusion:
In conclusion, the transaction of offering an allowance of Rs. 50 for an early payment of cash amounting to Rs. 1,050 should be recorded in the Cash Book. The Cash Book is the appropriate book to record this transaction as it involves the receipt of cash.