Consider a file of 16384 records. Each record is 32 bytes long and its key field is of size 6 bytes. The file is ordered on a non-key field, and the file organization is unspanned. The file is stored in a file system with block size 1024 bytes, and the size of a block pointer is 10 bytes. If the secondary index is built on the key field of the file, and a multi-level index scheme is used to store the secondary index, the number of first-level and second-level blocks in the multi-level index are respectively

Given $${f_1},$$ $${f_3},$$ and $$f$$ in canonical sum of products form (in decimal) for the circuit. $$${f_1} = \sum {m\left( {4,5,6,7,8} \right)} $$$ $$${f_3} = \sum {m\left( {1,6,15} \right)} $$$ $$$f = \sum {m\left( {1,6,8,15} \right)} $$$
Then $${f_2}$$ is

If $$P, Q, R$$ are Boolean variables, then $$\left( {P + \overline Q } \right)$$ $$\left( {P.\overline Q + P.R} \right)\left( {\overline P .\overline R + \overline Q } \right)$$ Simplifies to

In the Karnaugh map shown below, $$X$$ denotes a don’t care term. What is the minimal form of the function represented by the Karnaugh map?