UPSC NDA II 2024 Mathematics Paper

Concept:Use five terms of an arithmetic progression (AP) symmetric about the middle term: $a - 2d$, $a - d$, $a$, $a + d$, $a + 2d$.Explanation:The product of the five terms is given as $2,29,635$.So $(a - 2d)(a - d)(a)(a + d)(a + 2d) = 2,29,635$.This simplifies to $a(a^2 - d^2)(a^2 - 4d^2) = 2,29,635$ …(i).Also, the first three terms $a - 2d$, $a - d$, $a + 2d$ are in geometric progression (GP).For GP, $(a - d)^2 = (a - 2d)(a + 2d)$.Expanding gives $a^2 - 2ad + d^2 = a^2 - 4d^2$.Simplifying yields $5d^2 - 2ad = 0$ ⇒ $d(5d - 2a) = 0$.Since $d \neq 0$, we get $5d = 2a$ or $a = \frac{5d}{2}$ …(ii).Substitute $a = \frac{5d}{2}$ into equation (i):$\frac{5d}{2} \cdot \left( \frac{25d^2}{4} - d^2 \right) \cdot \left( \frac{25d^2}{4} - 4d^2 \right) = 2,29,635$Compute: $\frac{5d}{2} \cdot \frac{21d^2}{4} \cdot \frac{9d^2}{4} = 2,29,635$⇒ $\frac{5 \cdot 21 \cdot 9}{2 \cdot 4 \cdot 4} d^5 = 2,29,635$⇒ $\frac{945}{32} d^5 = 2,29,635$Solve: $d^5 = 2,29,635 \times \frac{32}{945} = 7776 = 6^5$.Thus $d = 6$.Answer:The common difference is $6$, which corresponds to option D.