JEE Advanced · 2026 · 17 May 2026 · Paper 2 · Matrices

Let $\mathbb{R}$ denote the set of all real numbers and let $\mathrm{i}=\sqrt{-1}$. Consider the matrices
$\mathrm{S}=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$ and $\mathrm{T}=\left[\begin{array}{ll}1& 1\\ 0& 1\end{array}\right]$.
Let $\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}$ be real numbers such that
$\mathrm{ST}=\left[\begin{array}{cc}\mathrm{a}& \mathrm{b}\\ \mathrm{c}& \mathrm{d}\end{array}\right]$.
Let
$\mathrm{H}=\{\mathrm{x}+\mathrm{iy}:\mathrm{x},\mathrm{y}\in \mathbb{R}$ and $\mathrm{y}>0\}$.
Then which of the following statements is (are) TRUE?
(A) $\frac{\mathrm{b}+\mathrm{ia}}{\mathrm{d}+\mathrm{ic}}=\mathrm{i}$
(B) If $\omega =\frac{-1+\mathrm{i}\sqrt{3}}{2}$, then $\frac{\mathrm{a}\omega +\mathrm{b}}{\mathrm{c}\omega +\mathrm{d}}=\omega$
(C) If m is an integer greater than 2 such that $(\mathrm{ST}{)}^2=(\mathrm{ST}{)}^{\mathrm{m}}$, then m is an integer multiple of 8
(D) If $\mathrm{z}\in \mathrm{H}$, then $\frac{\mathrm{az}+\mathrm{b}}{\mathrm{cz}+\mathrm{d}}\in \mathrm{H}$