UPSC NDA 2 2021 Mathematics Solved Paper

Concept:Mode formula for grouped data: $\text{Mode} = L + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h$.Here, $L$ = lower limit of modal class, $h$ = class size, $f_1$ = frequency of modal class, $f_0$ = frequency of class before modal class, $f_2$ = frequency of class after modal class.Explanation:Identify the modal class as the class with the highest frequency.From the data, the highest frequency is 11, so modal class is $25 - 30$.Hence, $L = 25$, $h = 5$, $f_1 = 11$, $f_0 = 5$ (frequency of $20 - 25$), $f_2 = 6$ (frequency of $30 - 35$).Substitute into formula: $\text{Mode} = 25 + \left( \frac{11 - 5}{2 \times 11 - 5 - 6} \right) \times 5$.Simplify numerator: $11 - 5 = 6$.Denominator: $2 \times 11 - 5 - 6 = 22 - 11 = 11$.Fraction: $\frac{6}{11}$.Multiply by $5$: $\frac{6}{11} \times 5 = \frac{30}{11} \approx 2.7273$.Add to $25$: $25 + 2.7273 = 27.7273 \approx 27.73$.Answer:27.73 (Option B).