Consider a 32-bit system with 4 KB page size and page table entries of size 4 bytes each. Assume 1 KB = $2^{10}$ bytes. The OS uses a 2-level page table for memory management, with the page table containing an outer page directory and an inner page table. The OS allocates a page for the outer page directory upon process creation. The OS uses demand paging when allocating memory for the inner page table, i.e., a page of the inner page table is allocated only if it contains at least one valid page table entry.

An active process in this system accesses 2000 unique pages during its execution, and none of the pages are swapped out to disk. After it completes the page accesses, let $X$ denote the minimum and $Y$ denote the maximum number of pages across the two levels of the page table of the process.

The value of $X+Y$ is __________.

Consider a memory management system that uses a page size of 2 KB. Assume that both the physical and virtual addresses start from 0. Assume that the pages 0, 1, 2, and 3 are stored in the page frames 1, 3, 2, and 0, respectively. The physical address (in decimal format) corresponding to the virtual address 2500 (in decimal format) is ________

Consider the following two-dimensional array D in the C programming language, which is stored in row-major order:

int D[128] [128];

Demand paging is used for allocating memory and each physical page frame holds 512 elements of the array D. The Least Recently Used (LRU)( page-replacement policy is used by the operating system. A total of 30 physical page frames are allocated to a process which executes the following code snippet:

for (int i = 0; i < 128; i++)

$$\quad$$ for (int j = 0; j < 128; j++)

$$\quad\quad$$D[j] [i] *= 10;

The number of page faults generated during the execution of this code snippet is _____________.

Consider a computer system with 57-bit virtual addressing using multi-level tree-structured page tables with L levels for virtual to physical address translation. The page size is 4 KB (1 KB = 1024 B) and a page table entry at any of the levels occupies 8 bytes.

The value of L is ____________.