JEE 2024 — Mathematics PYQ

Given $f(x)=\left[\begin{array}{ccc}\cos x& amp;-\sin x& amp;0\\ \sin x& amp;\cos x& amp;0\\ 0& amp;0& amp;1\end{array}\right]$

$f(x)=\left[\begin{array}{ccc}\cos (-x)& amp;-\sin (-x)& amp;0\\ \sin (x)& amp;\cos (-x)& amp;0\\ 0& amp;0& amp;1\end{array}\right]$

$f(x)f(-x)=\left[\begin{array}{ccc}\cos x& amp;\sin x& amp;0\\ -\sin x& amp;\cos x& amp;0\\ 0& amp;0& amp;1\end{array}\right]\left[\begin{array}{ccc}\cos x& amp;-\sin x& amp;0\\ \sin x& amp;\cos x& amp;0\\ 0& amp;0& amp;1\end{array}\right]=\left[\begin{array}{ccc}{\cos }^{2}x+{\sin }^{2}x+0& amp;\sin x\cos x-\frac{\sin x\cos x}{\sin x\cos x+0}& amp;0+0+1\\ \sin x\cos x-\frac{{\sin }^{2}x+{\cos }^{2}x+0}{0}& amp;{\sin }^{2}x+{\cos }^{2}x+0& amp;0+0+0\\ 0+0+0& amp;0+0+0& amp;0+0+1\end{array}\right]$

$=\left[\begin{array}{ccc}1& amp;0& amp;0\\ 0& amp;1& amp;0\\ 0& amp;0& amp;1\end{array}\right]=I$

Statement - I is true

$f(x)f(y)=\left[\begin{array}{ccc}\cos x& amp;-\sin x& amp;0\\ \sin x& amp;\cos x& amp;0\\ 0& amp;0& amp;1\end{array}\right]\left[\begin{array}{ccc}\cos y& amp;-\sin y& amp;0\\ \sin y& amp;\cos y& amp;0\\ 0& amp;0& amp;1\end{array}\right]=\left[\begin{array}{ccc}\cos x\cos y-\sin x\sin y& amp;-\cos x\sin y& amp;0\\ \sin x\sin y& amp;-\sin x\sin y& amp;0\\ 0& amp;0& amp;1\end{array}\right]$

$\left[\begin{array}{ccc}\cos x\cos y-\sin x\sin y& amp;-\cos x\sin y& amp;0\\ \sin x\sin y& amp;-\sin x\sin y& amp;0\\ 0& amp;0& amp;1\end{array}\right]=\left[\begin{array}{ccc}\cos (x+y)& amp;-\sin (x+y)& amp;0\\ \sin (x+y)& amp;\cos (x+y)& amp;0\\ 0& amp;0& amp;1\end{array}\right]=f(x+y)$