UPSC NDA 2 2021 Mathematics Solved Paper

Concept:The mean is the average of all numbers. The mode is the most frequent number. The median is the middle value when data is sorted.Explanation:We are given the set: $x$, $x$, $x+2$, $x+3$, $x+10$ where $x$ is a natural number. Take the smallest natural number $x = 1$ to check the statements. The numbers become: $1$, $1$, $3$, $4$, $11$. Mean = $\frac{1+1+3+4+11}{5} = \frac{20}{5} = 4$. Mode = $1$ (since $1$ appears twice). Therefore, Mean ($4$) > Mode ($1$). So statement 1 is correct. Now compute the median. Arrange the numbers: $1$, $1$, $3$, $4$, $11$. The middle value is $3$. We see that Median ($3$) < Mean ($4$). Alternatively, using the empirical formula: Mode $= 3 \times \text{Median} - 2 \times \text{Mean}$. Substitute: $1 = 3 \times \text{Median} - 2 \times 4$. Solve: $1 = 3 \times \text{Median} - 8 \implies 3 \times \text{Median} = 9 \implies \text{Median} = 3$. This confirms Median ($3$) < Mean ($4$), so statement 2 is false.Answer:Only statement 1 is correct. The correct option is A (1 only).