GATE CS 2026 Set 1 Q14 — Searching, Sorting, Hashing

We are given a hash table $P[0, 1, \ldots, 10]$ of size $m = 11$. The hash function is $h(x) = (x + 7) \bmod 11$, and collisions are resolved using linear probing.

Let's trace the insertions step-by-step:

1. Insert 1:
   $h(1) = (1 + 7) \bmod 11 = 8$.
   Position $8$ is empty. Insert $1$ at $P[8]$.
2. Insert 13:
   $h(13) = (13 + 7) \bmod 11 = 9$.
   Position $9$ is empty. Insert $13$ at $P[9]$.
3. Insert 22:
   $h(22) = (22 + 7) \bmod 11 = 7$.
   Position $7$ is empty. Insert $22$ at $P[7]$.
4. Insert 15:
   $h(15) = (15 + 7) \bmod 11 = 0$.
   Position $0$ is empty. Insert $15$ at $P[0]$.
5. Insert 11:
   $h(11) = (11 + 7) \bmod 11 = 7$.
   Position $7$ is occupied. Probing sequentially:
   - Index $8$ is occupied.
   - Index $9$ is occupied.
   - Index $10$ is empty. Insert $11$ at $P[10]$.
6. Insert 24:
   $h(24) = (24 + 7) \bmod 11 = 9$.
   Position $9$ is occupied. Probing sequentially:
   - Index $10$ is occupied.
   - Index $0$ is occupied.
   - Index $1$ is empty. Insert $24$ at $P[1]$.

After all insertions, the occupied positions in the hash table are:

• $P[0] = 15$
• $P[1] = 24$
• $P[7] = 22$
• $P[8] = 1$
• $P[9] = 13$
• $P[10] = 11$

The empty positions are $2, 3, 4, 5,$ and $6$. Among the given options, only position $2$ (Option C) is empty.