JEE Advanced 2026 · Paper 2, Mathematics question 9

$$ST=\begin{bmatrix}0&-1\\1&0\end{bmatrix}\begin{bmatrix}1&1\\0&1\end{bmatrix}=\begin{bmatrix}0&-1\\1&1\end{bmatrix},$$
so $a=0,\ b=-1,\ c=1,\ d=1$.

• (A) $\dfrac{b+ia}{d+ic}=\dfrac{-1}{1+i}=\dfrac{-1+i}{2}\ne i$. False.
• (B) $\dfrac{-1}{\omega+1}=\dfrac{-1}{-\omega^2}=\omega^{-2}=\omega$. True.
• (C) $M=ST$ has characteristic polynomial $\lambda^2-\lambda+1$, so its eigenvalues are $e^{\pm i\pi/3}$ and $M^6=I$ (with no smaller power equal to $I$). $M^2=M^m$ iff $6\mid m-2$; e.g. $m=14$ works and is not a multiple of 8. False.
• (D) $z\mapsto\dfrac{-1}{z+1}$ is a Möbius map with real coefficients and $ad-bc=1>0$, so $\operatorname{Im}\dfrac{az+b}{cz+d}=\dfrac{(ad-bc)\operatorname{Im}z}{|cz+d|^2}>0$. True.