UPSC CDS II 2025 Mathematics Paper

Concept:The median formula for grouped data is used when the median lies in a known class interval. The formula is $Median = L + \left( \frac{N/2 - cf}{f} \right) \times h$, where $L$ is lower limit of median class, $N$ total frequency, $cf$ cumulative frequency before median class, $f$ frequency of median class, and $h$ class width.Explanation:Total number of students: $N = 55 + k$. So $N/2 = (55 + k)/2$. The median is 24, so it lies in the class 20–30. Lower limit $L = 20$, class width $h = 10$.Cumulative frequency before median class: $cf = 30$. Frequency of median class: $f = k$ (since cumulative frequency up to 20–30 is $30 + k$, so frequency is $(30 + k) - 30 = k$).Substitute into formula: $24 = 20 + \left( \frac{(55 + k)/2 - 30}{k} \right) \times 10$.Simplify: $4 = \left( \frac{55 + k - 60}{2k} \right) \times 10 = \frac{10(k - 5)}{2k} = \frac{5(k - 5)}{k}$.So $4k = 5(k - 5)$ => $4k = 5k - 25$ => $k = 25$.Answer:The value of $k$ is 25, which corresponds to option C.