UPSC CDS I 2019 Math Paper

Concept:Find Priya's present age from given conditions, then find the next age that satisfies the new condition (square in previous year, cube in next year).Explanation:Let Priya's present age be $T$ years.Four years ago, her age was $T-4$, which must be a perfect cube (not equal to $1$).Possible cubes: $8, 27, 64, \dots$. So $T = 12, 31, 69, \dots$.Four years later, her age $T+4$ must be a perfect square.Check $T=12$: $T+4 = 16 = 4^2$. Works.Check $T=31$: $T+4 = 35$ (not a square). So $T = 12$ is unique.Now, we need an age $A$ such that:• In the previous year ($A-1$), her age is a perfect square.• In the next year ($A+1$), her age is a perfect cube.For $A > 12$, the smallest square above $12$ is $16$, $25$, $36$, …The smallest cube above $12$ is $27$, $64$, …Test $A = 26$: previous year $25 = 5^2$, next year $27 = 3^3$. Condition satisfied.Wait time = $A - T = 26 - 12 = 14$ years.Answer:14 years