UPSC NDA 2 2019 Mathematics Solved Paper

Concept:If three numbers are in Arithmetic Progression (AP), then any linear transformation (adding, subtracting, multiplying, or dividing by a non‑zero constant) preserves the AP property.Explanation:Given: $a, b, c$ are in AP.This means $2b = a + c$.For statement 1: Multiply each term by $k \neq 0$.We get $ka, kb, kc$.Check: $2(kb) = 2kb$ and $ka + kc = k(a+c) = k(2b) = 2kb$.So $ka, kb, kc$ are in AP. Hence statement 1 is correct.For statement 2: Subtract each term from $k$.We get $k - a, k - b, k - c$.Check: $2(k - b) = 2k - 2b$ and $(k - a) + (k - c) = 2k - (a+c) = 2k - 2b$.So $k - a, k - b, k - c$ are in AP. Hence statement 2 is correct.For statement 3: Divide each term by $k \neq 0$.We get $\frac{a}{k}, \frac{b}{k}, \frac{c}{k}$.Check: $2\left(\frac{b}{k}\right) = \frac{2b}{k}$ and $\frac{a}{k} + \frac{c}{k} = \frac{a + c}{k} = \frac{2b}{k}$.So $\frac{a}{k}, \frac{b}{k}, \frac{c}{k}$ are in AP. Hence statement 3 is correct.Therefore, all three statements are correct.Answer:Statements 1, 2, and 3 are correct. The correct option is D.