UPSC CDS II 2025 Mathematics Paper

Concept:Use the median formula for the first group and the mean formula for the second group to set up two equations in u and v, then solve for v.Explanation:For the first group, median 20.8 lies in the class 20–25. Lower limit L = 20, class width h = 5, median class frequency f = 75.Cumulative frequency before median class: cf = u + v + 63 (since 11 + 52 = 63).Total frequency N = u + v + 160.Median formula: $Median = L + \frac{N/2 - cf}{f} \times h$.Substitute: $20.8 = 20 + \frac{(u+v+160)/2 - (u+v+63)}{75} \times 5$.Simplify stepwise: $0.8 = \frac{34 - (u+v)}{150} \times 5$, so $0.8 = \frac{34 - u - v}{30}$.Thus $34 - u - v = 24$, giving $u + v = 10$ ...(1).For the second group, midpoints x: 2.5, 7.5, 12.5, 17.5, 22.5, 27.5.Frequencies: 3u, 2v, 40, 50, 30, 28. Total frequency $\sum f = 3u + 2v + 148$.Sum of $f \times x$: $7.5u + 15v + 2820$.Mean formula: $Mean = \frac{\sum fx}{\sum f}$. Given mean = 17.3.So $\frac{7.5u + 15v + 2820}{3u + 2v + 148} = 17.3$.Cross-multiply: $7.5u + 15v + 2820 = 17.3(3u + 2v + 148)$.Expand: $7.5u + 15v + 2820 = 51.9u + 34.6v + 2560.4$.Rearrange: $44.4u + 19.6v = 259.6$. Divide by 2: $22.2u + 9.8v = 129.8$ ...(2).From (1), $v = 10 - u$. Substitute in (2): $22.2u + 9.8(10 - u) = 129.8$.Simplify: $22.2u + 98 - 9.8u = 129.8$, so $12.4u = 31.8$, giving $u \approx 2.56$, nearest integer u = 3.Then $v = 10 - 3 = 7$.Answer:The value of v is 7, option C.