Which one of the following options is correct?

Let us evaluate both statements:

1. Statement S1: *A hash index must be a dense index.*

   • In hash indexing, the hash function computes the address of a bucket directly from the search-key value. Because hashing does not maintain any sorted or sequential ordering of search keys, there is no way to locate a record using adjacent keys or sequential scanning if an index entry for a specific search key does not exist. Therefore, a hash index must contain an entry for every search-key value present in the indexed file, making it necessarily a dense index. Thus, S1 is true.

2. Statement S2: *A $B^+$ tree index can be a sparse index.*

   • A $B^+$ tree is an ordered/tree-based indexing structure. When the data file is ordered sequentially on the search key (as in a primary or clustering index), a $B^+$ tree can be constructed as a sparse index (e.g., having an index entry for only the first search key of each data block). Thus, S2 is true.

Since both statements S1 and S2 are true, option (A) is the correct choice.