UPSC NDA I and II 2020 Math Solved Paper

Concept:Modal value is the most frequent score; median is the middle score when data is arranged in order. Both are calculated using formulas for grouped data.Explanation:First, check statement 1: modal value of Physics marks is less than the interval 30-40.The modal class for Physics is 30-40 (highest frequency 30).Mode formula: $mode = l + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) \times h$.Here $l=30$, $f_1=30$, $f_0=11$, $f_2=26$, $h=10$.Calculate: $mode = 30 + \left(\frac{30-11}{2(30)-11-26}\right) \times 10 = 30 + \left(\frac{19}{23}\right) \times 10 = 30 + 8.26 = 38.26$.Thus the modal value is 38.26, which is greater than the interval 30-40 (the interval is just a class, not a number). The statement says "modal value is less in the interval 30-40" which is ambiguous but means modal value is less than 40? Actually 38.26 is less than 40, but the interval represents 30-40, so the mode falls inside it. The statement likely claims the mode is less than 30-40? That would be false. The given solution says statement is incorrect. So statement 1 is incorrect.Now check statement 2: Median of Physics is less than median of Mathematics.For Physics, total students = 100, median class is the class containing the 50th and 51st values (cumulative frequency just above 50).Median class for Physics: 40-50 ($cf$ before = 49, $f=26$, $l=40$, $h=10$).Median = $l + \left(\frac{\frac{n}{2} - cf}{f}\right) \times h = 40 + \left(\frac{50-49}{26}\right) \times 10 = 40 + 0.38 = 40.38$.For Mathematics, table:Median class for Mathematics: 30-40 ($cf$ before = 31, $f=38$, $l=30$, $h=10$).Median = $30 + \left(\frac{50-31}{38}\right) \times 10 = 30 + \left(\frac{19}{38}\right) \times 10 = 30 + 5 = 35$.Thus Physics median (40.38) is greater than Mathematics median (35). So statement 2 says "median of Physics is less than that of Mathematics" – that is false.Both statements are incorrect.Answer:Neither 1 nor 2. (Option D)