MAT Exam 2024 Paper

Concept:This is a successive dilution problem. Replacing a fixed volume of mixture with pure nitrogen reduces the oxygen concentration each time.Explanation:Total volume of the cylinder is $8$ litres. Initial oxygen volume is $16\%$ of $8$ litres, so initial oxygen fraction is $0.16$. After replacing $x$ litres of mixture with nitrogen twice, the final oxygen fraction becomes $9\%$, i.e. $0.09$. For two such operations, the relation is: $$\frac{\text{Final fraction}}{\text{Initial fraction}} = \left(1 - \frac{x}{\text{total volume}}\right)^2$$Substitute the values: $$\frac{9}{16} = \left(1 - \frac{x}{8}\right)^2$$Take square root on both sides: $$\sqrt{\frac{9}{16}} = 1 - \frac{x}{8}$$$$\frac{3}{4} = 1 - \frac{x}{8}$$$$\frac{x}{8} = 1 - \frac{3}{4} = \frac{1}{4}$$Therefore, $$x = 2 \text{ litres}$$Answer:The amount of mixture released each time is $2$ litres.Correct option: C. $2$ litres.