JEE Main 2024 — 9 April, Shift 1 — Question 27

$|3 \operatorname{adj}(2 \operatorname{adj}(|\mathrm{~A}| \mathrm{A}))|=\mid\operatorname{3adj}\left(2|\mathrm{~A}|^{2} \operatorname{adj}(\mathrm{~A}) \mid\right.$

$=\left.\left|3.2^{2}\right| \mathrm{A}\right|^{4} \operatorname{adj}\left(\left.\operatorname{adj}(\mathrm{~A})\left|=2^{6} 3^{3}\right| \mathrm{A}\right|^{12}|\mathrm{~A}|^{4}\right.$

$=2^{6} 3^{3}|\mathrm{~A}|^{16}=2^{-10} 3^{-13}$

$\Rightarrow|\mathrm{A}|^{16}=2^{-16} 3^{-16} \Rightarrow|\mathrm{~A}|=2^{-1} 3^{-1}$

Now $|3 \mathrm{adj}(2 \mathrm{~A})|=\left|3.2^{2} \operatorname{adj}(\mathrm{~A})\right|$

$=2^{6} 3^{3}|\mathrm{~A}|^{2}=2^{-\mathrm{m}} 3^{-\mathrm{n}}$

$\Rightarrow 2^{6} 3^{3} 2^{-2} 3^{-2}=2^{-\mathrm{m}} 3^{-\mathrm{n}}$

$\Rightarrow 2^{-\mathrm{m}} 3^{-\mathrm{n}}=2^{4} 3^{1}$

$\Rightarrow \mathrm{m}=-4, \mathrm{n}=-1$

$\Rightarrow|3 \mathrm{~m}+2 \mathrm{n}|=|-12-2|=14$