UPSC CDS 2 2019 Math Paper

Concept:The mode is the value that occurs most frequently. For grouped data, the modal class is the class with the highest frequency. We use the mode formula:$$\text{Mode} = l + \frac{f_1 - f_0}{(f_1 - f_0) + (f_1 - f_2)} \times h$$where $l$ = lower limit of modal class, $f_1$ = frequency of modal class, $f_0$ = frequency of class before modal class, $f_2$ = frequency of class after modal class, $h$ = class width.Explanation:From the table of Series-II, the frequencies are: 10-20: $4$, 20-30: $8$, 30-40: $4 = f_0$, 40-50: $58 = f_1$ (highest, so modal class is 40-50), 50-60: $26 = f_2$. Here $l = 40$, $h = 10$, $f_0 = 4$, $f_1 = 58$, $f_2 = 26$. Substitute into the formula: $$\text{Mode} = 40 + \frac{58 - 4}{(58 - 4) + (58 - 26)} \times 10$$$$= 40 + \frac{54}{54 + 32} \times 10$$$$= 40 + \frac{54}{86} \times 10$$$$= 40 + 6.279 \approx 46.28$$Rounding to the nearest whole number gives $46$.Answer:C. 46