JEE 2023 — Mathematics PYQ

Sabse pehle hum determinant $|A|$ ki value nikalenge. Matrix ke elements bade factorials hain, isliye hum rows se common factors bahar nikalenge:

• $R_1$ se $5!$ common lene par.

• $R_2$ se $6!$ common lene par.

• $R_3$ se $7!$ common lene par.

Ab row operations $R_2\to R_2-R_1$ aur $R_3\to R_3-R_2$ apply karne par:

Expanding along $C_1$:

Step 2: $|2A|$ ki value find karna

Property: $|kA|=k^n|A|$ (Yahan $n=3$ hai kyunki matrix $3\times 3$ hai)

Step 3: $|\text{adj}(\text{adj}(M))|$ ki property apply karna

Property: $|\text{adj}(\text{adj}(M))|=|M{|}^{(n-1{)}^2}$

Yahan humein $|\text{adj}(\text{adj}(2A))|$ nikalna hai, toh $M=2A$ aur $n=3$:

Step 4: Final Calculation

$|2A|=2^4$ ki value upar waale formula mein rakhne par:

Sahi Answer: (1) $2^{16}$