JEE Main 2022 · 29 Jul · Shift 1 · Mathematics Q27

← JEE Main 2022 · 29 Jul · Shift 1

MathematicsInteger+4 / -1Question 27 of 90

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 α and β, such that $p^\alpha$ and $(p+2)^\beta$ divide Δ, is _______.

Your answer

Answer and worked solution

Official answer: 4

Factor rows. Taking $p!$ , $(p+1)!$ , $(p+2)!$ out of the rows leaves rows $1,\ p+1+j,\ (p+1+j)(p+2+j)$ in shape; subtracting rows gives determinant 2. So $\Delta=2\,p!\,(p+1)!\,(p+2)!$ .

Powers. For an odd prime p (and $p+2<2p$ ), p divides each factorial once: $\alpha=3$ . $p+2$ appears only in $(p+2)!$ : $\beta=1$ . Sum 4.

Answer: 4

Topics:Determinants and properties Number systems

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JEE Main 2022 · 29 Jul · Shift 1, Mathematics question 27