MAT Exam 2024 Paper

Concept:Time is calculated as $\frac{\text{distance}}{\text{speed}}$. Upstream speed is $(x - y)$ and downstream speed is $(x + y)$, where $x$ is the man's speed in still water and $y$ is the stream's speed.Explanation:Let $x$ be the speed of the man in still water and $y$ be the speed of the current.From the first condition:$$\frac{30}{x-y} + \frac{44}{x+y} = 10$$From the second condition:$$\frac{40}{x-y} + \frac{55}{x+y} = 13$$Let $p = \frac{1}{x-y}$ and $q = \frac{1}{x+y}$.Then the equations become:$$30p + 44q = 10 \quad \text{... (1)}$$$$40p + 55q = 13 \quad \text{... (2)}$$Multiply equation (1) by 4 and equation (2) by 3:$$120p + 176q = 40$$$$120p + 165q = 39$$Subtract the second equation from the first:$$11q = 1$$$$q = \frac{1}{11}$$Substitute $q = \frac{1}{11}$ into equation (1):$$30p + 44 \times \frac{1}{11} = 10$$$$30p + 4 = 10$$$$p = \frac{6}{30} = \frac{1}{5}$$Now:$$x - y = \frac{1}{p} = 5$$$$x + y = \frac{1}{q} = 11$$Add these two equations:$$(x - y) + (x + y) = 5 + 11$$$$2x = 16$$$$x = 8 \text{ km/hr}$$Substitute $x = 8$ into $x + y = 11$:$$8 + y = 11$$$$y = 3 \text{ km/hr}$$Answer:The rate of the current is $3$ km/hr.Correct option: A.