Shahid and Rohit start from the same point in opposite directions. After each 1 km, Shahid always turns left and Rohit always turns right. Which of the following statements is correct?

Correct Option (B)

Let the starting point be the origin (0,0). Assume Shahid initially moves along the positive x-axis (East) and Rohit along the negative x-axis (West).

• Shahid's path:
  • After 1 km: Position (1,0). Turns left (North).
  • After 2 km: Position (1,1). Turns left (West).
  • After 3 km: Position (0,1). Turns left (South).
• Rohit's path:
  • After 1 km: Position (-1,0). Turns right (North).
  • After 2 km: Position (-1,1). Turns right (East).
  • After 3 km: Position (0,1). Turns right (South).

After each has travelled 3 km, both Shahid and Rohit are at the position (0,1). Therefore, they meet at this point.

Incorrect Options:

• Option A: After both have travelled 2 km, the distance between them is 4 km.
  • After 2 km, Shahid is at (1,1) and Rohit is at (-1,1).
  • The distance between them is calculated as √[(1 - (-1))² + (1 - 1)²] = √[(2)² + (0)²] = √4 = 2 km.
  • Thus, the statement is incorrect.
• Option C: They meet for the first time after each has travelled 4 km.
  • As established, they meet after each has travelled 3 km.
  • After 4 km, Shahid is at (0,0) and Rohit is at (0,0). They meet again at the starting point, but this is not their first meeting.
  • Thus, the statement is incorrect.
• Option D: They go on without ever meeting again.
  • Their paths are cyclical. They meet at 3 km and again at 4 km.
  • Given the repetitive nature of their movements (turning left/right after every 1 km), they will continue to meet at regular intervals.
  • Thus, the statement is incorrect.