An index in a DBMS is said to be dense if an index entry appears for every search-key value in the indexed file. Otherwise it is called a sparse index. Consider the following two statements.

S1: A hash index must be a dense index
S2: A $B^{+}$ tree index can be a sparse index

Which one of the following options is correct?

In a $B^{+}$- tree where each node can hold at most four key values, a root to leaf path consists of the following nodes:

$A=(49,77,83,-),B=(7,19,33,44),C=(20^{\ast },22^{\ast },25^{\ast },26^{\ast })$

The *-marked keys signify that these are data entries in a leaf.

Assume that a pointer between keys $k_1$ and $k_2$ points to a subtree containing keys in $[k_1,k_2)$, and that when a leaf is created, the smallest key in it is copied up into its parent.

A record with key value 23 is inserted into the $B^{+}$- tree.

The smallest key value in the parent of the leaf that contains $25^{\ast }$ is __________. (Answer in integer)

Consider the following $B^{+}$ tree with 5 nodes, in which a node can store at most 3 key values. The value 23 is now inserted in the $B^{+}$ tree. Which of the following options(s) is/are CORRECT?

Consider a database of fixed-length records, stored as an ordered file. The database has 25,000 records, with each record being 100 bytes, of which the primary key occupies 15 bytes. The data file is block-aligned in that each data record is fully contained within a block. The database is indexed by a primary index file, which is also stored as a block-aligned ordered file. The figure below depicts this indexing scheme.



Suppose the block size of the file system is 1024 bytes, and a pointer to a block occupies 5 bytes. The system uses binary search on the index file to search for a record with a given key. You may assume that a binary search on an index file of $b$ blocks takes $\lceil {\log }_{2}b\rceil$ block accesses in the worst case.
Given a key, the number of block accesses required to identify the block in the data file that may contain a record with the key, in the worst case, is __________.