Directions: Read the table carefully and answer the following question. The table shows the data about distance (km) covered from one point to another point by the vehicle used between each point and speed of each vehicle (km/h). Note: (i) If one person wants to travel from P to S, then he/she can only use a vehicle available at the point P. (ii) ** means there is no direct connection between given points. | | P | Q | R | S | Transport Vehicle | Speed (km/h) | |---|---|---|---|---|---|---| | P | 0 | 360 | ** | 780 | Train | 60 | | Q | 540 | 0 | ** | 600 | Taxi | 50 | | R | 420 | 720 | 0 | ** | Bus | 40 | | S | 630 | ** | 900 | 0 | Bike | 45 | If Gaurav wants to travel from P to R, then find the minimum time required by him to cover this distance.

### Concept & Minimum Time Path To find the minimum time required, we must calculate the time taken for all possible routes from P to R and select the smallest value. $$ \text{Time} = \frac{\text{Distance}}{\text{Speed}} $$ ### Step-by-Step Solution 1. **Direct Route ($P \to R$):** The table shows "**", meaning no direct connection exists. 2. **Route via Q ($P \to Q \to R$):** - $P \to Q$: Distance = 360 km. Vehicle at P = Train (60 km/h). Time = $360 / 60 = 6$ hours. - $Q \to R$: The table shows "**" (no direct connection). This route is invalid. 3. **Route via S ($P \to S \to R$):** - $P \to S$: Distance = 780 km. Vehicle at P = Train (60 km/h). Time = $780 / 60 = 13$ hours. - $S \to R$: Distance = 900 km. Vehicle at S = Bike (45 km/h). Time = $900 / 45 = 20$ hours. - Total time for this route = $13 + 20 = 33$ hours. 4. Since the only valid logical path with provided direct links to R from reachable nodes is via S, 33 hours is the minimum time. ### Exam Strategy & Shortcut Quickly scan the destination column (R) for available incoming routes. Only S and P can potentially reach R, but P has no direct connection. Thus, any viable path must go through S. Calculating $P \to S \to R$ directly saves time mapping out impossible routes. ### Common Pitfall Applying the origin vehicle's speed to the entire journey. Note (i) specifies using the vehicle available at the starting point of the leg. For $S \to R$, you must use the Bike (45 km/h), not the Train. ### Final Answer Therefore, the correct answer is **33 hours**.