Consider a database with a relation containing $1,500,000$ fixed length records.$$ The file system used by the database has a block size of $2048$ bytes, of which $2000$ bytes are available for the storage of records. The length of each record is $240$ bytes and the length of the attribute being used as a key is $12$ bytes. A block pointer takes up $6$ bytes, whereas a record pointer takes up $8$ bytes. The records are stored in a sequential $($sorted$)$ file, and records may not span be$-$ tween blocks. In order to allow for future expansion, space equivalent to one record is left free in each block.

$($i$)$ How many blocks are needed to store the data file for this relation? $[3$ marks$]$

$($ii$)$ How many blocks are needed to store a dense primary index for this relation?$[3$ marks$]$

$($iii$)$ How many blocks are needed to store a sparse primary index for this relation?$[3$ marks$]$

$($iv$)$ How many blocks are needed to store a secondary index for this relation, if keys are repeated?$[3$ marks$]$

$($v$)$ How many blocks are needed to store a secondary index for this relation, using indirection via buckets of pointers? You may assume that there are $100,000$ distinct values for the secondary key.$[3$ marks$]$