MAT Exam 2023 Paper

Concept:The late departure is compensated by increasing the speed so that the actual travel time is the scheduled time minus the delay of $0.5$ hour.Explanation:Let the usual speed be $x$ km/h.Scheduled time to cover $1500$ km is:$$\frac{1500}{x}$$As the plane left $30$ minutes late, available travel time becomes:$$\frac{1500}{x} - 0.5$$Increased speed is $x+250$ km/h, so the actual time taken is:$$\frac{1500}{x+250}$$Equating the two times:$$\frac{1500}{x+250} = \frac{1500}{x} - 0.5$$Solving this equation:$$1500x = 1500(x+250) - 0.5x(x+250)$$Simplifying gives:$$x^2 + 250x - 750000 = 0$$Using the quadratic formula:$$x = \frac{-250 \pm 1750}{2}$$Taking the positive value:$$x = 750 \text{ km/h}$$Answer:D. $750$ km/h