UPSC CDS 2 2019 Math Paper

Concept:Set up equations using time = distance/speed for each leg of the journey. Solve for the speeds of car and train, then find their ratio.Explanation:Let the speed of the car be $C$ km/h and the speed of the train be $T$ km/h.First journey: 120 km by train, remaining $600-120=480$ km by car. Total time = 11 hours.Equation: $\frac{120}{T} + \frac{480}{C} = 11$.Second journey: 200 km by train, remaining $600-200=400$ km by car. Time taken is 11 hours 40 minutes = $11 + \frac{40}{60} = \frac{35}{3}$ hours.Equation: $\frac{200}{T} + \frac{400}{C} = \frac{35}{3}$.Multiply the first equation by 5 and the second equation by 3, then subtract:$5 \cdot \left(\frac{120}{T} + \frac{480}{C}\right) - 3 \cdot \left(\frac{200}{T} + \frac{400}{C}\right) = 5 \cdot 11 - 3 \cdot \frac{35}{3}$$\Rightarrow \frac{600}{T} + \frac{2400}{C} - \frac{600}{T} - \frac{1200}{C} = 55 - 35$$\Rightarrow \frac{1200}{C} = 20$$\Rightarrow C = \frac{1200}{20} = 60$ km/h.Substitute $C=60$ into the first equation:$\frac{120}{T} + \frac{480}{60} = 11$$\Rightarrow \frac{120}{T} + 8 = 11$$\Rightarrow \frac{120}{T} = 3$$\Rightarrow T = \frac{120}{3} = 40$ km/h.Ratio of speed of car to that of train: $C : T = 60 : 40 = 3 : 2$.Answer:$3 : 2$ (Option A).