JEE Main 2022 (Online) 29th July Morning ShiftQuestion 1
Let $p$ and $p+2$ be prime numbers and let $\Delta=\begin{vmatrix}p!&(p+1)!&(p+2)!\\(p+1)!&(p+2)!&(p+3)!\\(p+2)!&(p+3)!&(p+4)!\end{vmatrix}$. Then the sum of the maximum values of $\alpha$ and $\beta$, such that $p^\alpha$ and $(p+2)^\beta$ divide $\Delta$, is ________.
Determinant
Answer$4$
Solution
Row ops: $R_3\to R_3-R_2$, $R_2\to R_2-R_1$ and factoring factorials gives $\Delta=2\,(p!)^3(p+1)^2(p+2)$ (checked: $p=3$ gives $34560=2\cdot6^3\cdot16\cdot5$). For $p\ge3$ prime, $p+1,p+2$ are not divisible by $p$ and $p!$ has $p$ exactly once, so $\alpha=3$. $p+2$ is prime $>p$, so it does not divide $p!$ or $p+1$; it divides $\Delta$ exactly once, $\beta=1$. $\alpha+\beta=4$.