JEE Main 2026 — 21 January, Evening Shift — Question 14

Here $A^{n}=\left[\begin{array}{cc}2 n+1 & -4 n \\n & -2 n+1\end{array}\right]$

$\Rightarrow \mathrm{A}^{15}=\left[\begin{array}{ll}31 & -60 \\15 & -29\end{array}\right]$

$\Rightarrow \mathrm{A}^{15}+\mathrm{B}=\left[\begin{array}{ll}2 & -11 \\2 & -11\end{array}\right]$

Now $\left(A^{15}+B\right)\left[\begin{array}{l}x y\end{array}\right]=\left[\begin{array}{l}0 \\0\end{array}\right]$

$\Rightarrow\left[\begin{array}{cc}2 & -11\\ 2 & -11\end{array}\right]\left[\begin{array}{l}\mathrm{x} \\\mathrm{y}\end{array}\right]=\left[\begin{array}{l}0 \\0\end{array}\right]$

$\Rightarrow 2 \mathrm{x}-11 \mathrm{y}=0$

$x=11, y=2$