Consider below statements – (I) Each non-leaf node in B⁺ tree has between…

In a B⁺ tree of order n, every non-leaf node must have between ⌈n/2⌉ and n children so the tree stays height-balanced. An index is dense when it stores one index entry for every distinct search-key value in the file, and sparse when it stores entries for only some of those values; a sparse index is possible only when the file is physically sorted on that key. A clustering index is the index built on the attribute that fixes this physical ordering — that attribute need not be the primary key — while a secondary index is built on an attribute that does not fix the physical order, so it can never skip entries and must always be dense.

1. Statement (I): a non-leaf B⁺-tree node's child count is bounded between ⌈n/2⌉ and n by the standard node-fanout rule, so statement (I) is true.

2. Statement (II): a clustering index's search key is whichever attribute fixes the file's physical order, and that attribute is not required to be the primary key, so statement (II) is false.

3. Statement (III): a secondary index cannot rely on the file being sorted by its own key, so it must carry an entry for every search-key value — matching the dense-index requirement — so statement (III) is true.

4. Statement (IV): carrying an entry for only some of the search-key values is the definition of a sparse index, not a dense one, so statement (IV) is false.

Cross-check: among the four offered pairs, only 'I and III' pairs two true statements — 'I and IV' and 'I and II' each pair a true statement with a false one, and 'II and III' pairs a false statement with a true one — so 'I and III' is the only internally consistent pair.

Therefore, statements (I) and (III) are true, and the correct option is 'I and III'.