MAT Exam 2024 Paper

Concept:Raghu's age satisfies two nearby perfect-power conditions; find the age gap to the next perfect cube.Explanation:Let Raghu's current age be $x$ years.Last year his age was $x - 1$, which is a perfect square.Next year his age will be $x + 1$, which is a perfect cube.So the cube and the square differ by $2$: $(x + 1) - (x - 1) = 2$.The only nearby perfect powers fitting this are $25 = 5^2$ and $27 = 3^3$.Thus, $x - 1 = 25$ and $x + 1 = 27$.Hence, Raghu's current age is $x = 26$ years.After 27, the next perfect cube is $64 = 4^3$.So he must wait from age 26 to age 64.Required waiting time $= 64 - 26 = 38$ years.Answer:Raghu must wait 38 years. Option C.