A train P starts from Delhi at 9 am, reaches its destination in UP at 1 pm.…

When two bodies start moving at the same time from opposite ends of a fixed distance and move towards each other, their relative speed equals the sum of their individual speeds, and the time taken for them to meet equals the total distance divided by that relative speed. If each body's own one-way travel time over that distance is known, its speed can be written as distance ÷ time, which lets the (unknown) distance cancel out algebraically.

1. Let the distance between Delhi and UP be x km.

2. Train P covers x km in 4 hours (9 am to 1 pm), so its speed = x/4 km/hr.

3. Train Q covers x km in 5 hours (9 am to 2 pm), so its speed = x/5 km/hr.

4. Since both trains start at 9 am from opposite ends and move towards each other, relative speed = x/4 + x/5 = 5x/20 + 4x/20 = 9x/20 km/hr.

5. Time to meet = total distance ÷ relative speed = x ÷ (9x/20) = 20/9 hours.

6. Convert 20/9 hours to hours and minutes: 20/9 = 2 whole hours + 2/9 hour remainder = 2 hours + (2/9 × 60) minutes ≈ 2 hours 13.3 minutes.

7. Adding 2 hours 13.3 minutes to the 9:00 am start gives approximately 11:13 am — the closest of the given options is 11.14 am.

Cross-check with the two-body meeting-time shortcut: if body A alone takes time t₁ to cover a distance and body B alone takes time t₂ to cover the same distance, and both start together from opposite ends, the time to meet is (t₁ × t₂)/(t₁ + t₂). Here that is (4 × 5)/(4 + 5) = 20/9 hours — the same result, confirming the answer.

So trains P and Q cross each other at approximately 11:13 am, matching the 11.14 am option among those given.