UPSC NDA I 2025 Mathematics Paper

Concept:The most frequently occurring height is called the mode of grouped data. It is found using the formula for the modal class.Explanation:First, locate the modal class with the highest frequency. Class 164–166 has 24 students, the highest.Apply the mode formula: $Mode = L + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) \times h$.Here, $L = 164$ (lower boundary of modal class), $f_1 = 24$, $f_0 = 15$ (previous class frequency), $f_2 = 13$ (next class frequency), and $h = 2$ (class width).Substitute: $Mode = 164 + \left(\frac{24-15}{2 \times 24 - 15 - 13}\right) \times 2 = 164 + \left(\frac{9}{48-28}\right) \times 2 = 164 + \left(\frac{9}{20}\right) \times 2 = 164 + 0.9 = 164.9$.Thus, the most frequent height is approximately 164.9 cm.Answer:Option D: 164.9 cm