UPSC CDS 1 2022 Math Paper

Given: Train $X$ takes 2 hours less than a train $Y$ Distance between two cities $192\ \mathrm{km}$ Average speeds differ by $16\ \mathrm{km}/\mathrm{hr}$ Formula Used: Speed $=$ Distance $/$ Time Calculation: According to the question $X$ is faster train since it takes less time than $y$ to cover a fixed distance Let the time taken by train $X$ be $x\ \mathrm{hr}$ So, the time is taken by $Y$ be $(x+2)\ \mathrm{hr}$ Now, According to the question $\Rightarrow\; \frac{192}{x} - \frac{192}{x+2} = 16$ $\Rightarrow\; \frac{12}{x} - \frac{12}{x+2} = 1$ From here, we can use the option to find directly the answer otherwise follow the given solution $\Rightarrow 12(x+2)-12x = x(x+2)$ $\Rightarrow 24 = x^2 + 2x$ $\Rightarrow x^2 + 2x - 24 = 0$ $\Rightarrow x^2 + 6x - 4x - 24 = 0$ $\Rightarrow x(x+6) - 4(x+6) = 0$ $\Rightarrow (x+6)(x-4) = 0$ $\Rightarrow x = -6 \; \text{or} \; x = 4$ We know that time can not be negative So, $x=4$ $\therefore$ The time taken by the faster train to cover the distance is $4\ \mathrm{hr}$ :