$($a$)$ 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 between blocks. In order to allow for future expansion, space equivalent to one record is left free in each block.

$($b$)$ The database in part $($a$)$ has been reconfigured to use a B$+$Tree sec$-$ ondary index over the relation. Assuming the same constraints as given in part $($a$),$ and replacing the free space requirement with an initial occupancy of $0\mathrm{.}6$:

$($i$)$ What is the maximum number of pointers in a non$-$leaf node of the index? $[3$ marks$]$

$($ii$)$ What is the minimum number of pointers in a leaf node of the index? $[3$ marks$]$

$($iii$)$ How many levels of index will initially be required? $[3$ marks$]$