JEE Main 2025 — 8 April, Evening Shift — Question 26

$\left|\begin{array}{ccc}2 & 2+p & 2+p+q\\ 4 & 6+2 p & 8+3 p+2 q\\ 6 & 12+3 p & 20+6 p+3 q\end{array}\right|$

$$=\left|\begin{array}{ccc} 2 & 2 & 2+p+q\\ 4 & 6 & 8+3 p+2 q\\ 6 & 12 & 20+6 p+3 q \end{array}\right|+\underbrace{\left|\begin{array}{ccc} 2 & p & 2+p+q\\ 4 & 2 p & 8+3 p+2 q\\ 6 & 3 p & 20+6 p+3 q \end{array}\right|}_{=0}$$ $$=2 \times 2\left|\begin{array}{ccc} 1 & 1 & 2+p+q\\ 2 & 3 & 8+3 p+2 q\\ 3 & 6 & 20+6 p+3 q \end{array}\right|$$ $$C_{3} \rightarrow C_{3} \rightarrow p C_{2}$$ $$=4\left|\begin{array}{ccc} 1 & 1 & 2+q\\ 2 & 3 & 8+2 q\\ 3 & 6 & 20+3 q \end{array}\right|=4\left|\begin{array}{ccc} 1 & 1 & 2\\ 2 & 3 & 8\\ 3 & 6 & 20 \end{array}\right|+0$$ $$=4\left|\begin{array}{ccc} 1 & 1 & 1 \\2 & 3 & 4 \\3 & 6 & 10 \end{array}\right|$$ $$=8(1(6)-1(8)+1(3))$$ $$=8$$

$|\operatorname{adj}(\operatorname{adj}(3 A))|=(|3 A|)^{2^{2}}=|3 A|^{4}$

$=\left(3^{3}|A|\right)^{4}=3^{12} \cdot|A|^{4}$

$=3^{12} \cdot\left(2^{3}\right)^{4}$

$=3^{12} \cdot 2^{12}$