MAT Exam 2023 Paper

Concept:The first car’s speed increases in an arithmetic progression every hour. Its total distance is the sum of that progression. The second car moves at uniform speed, so its distance is simply speed multiplied by time.Explanation:Let the time after which both distances are equal be $t$ hours.For the car moving at constant speed:$$\text{Distance}_B = 60t.$$For the first car, the hourly speeds are:$25, 35, 45, \ldots$ for $t$ hours.This is an arithmetic progression with first term $a=25$ and common difference $d=10$.Distance covered by the first car is the sum of $t$ terms:$$S_t=\frac{t}{2}\left[2(25)+(t-1)(10)\right].$$Simplify:$$S_t=\frac{t}{2}(50+10t-10)=\frac{t}{2}(10t+40).$$$$S_t=5t^2+20t.$$Equate the two distances:$$5t^2+20t=60t.$$$$5t^2-40t=0.$$$$5t(t-8)=0.$$Since $t \neq 0$, we get:$$t=8 \text{ hours}.$$Answer:Both cars cover equal distance after $\mathbf{8}$ hours. Hence, the correct option is D.