UPSC NDA I 2025 Mathematics Paper

Concept:The matrix $f(\theta)$ is a rotation matrix. Squaring it gives a rotation by twice the angle. For $\theta = \pi$, a $180^\circ$ rotation, its square is a $360^\circ$ rotation, which is the identity matrix.Explanation:First, substitute $\theta = \pi$: $\cos\pi = -1$, $\sin\pi = 0$. So$$f(\pi) = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}.$$Now compute $(f(\pi))^2$ by multiplying the matrix by itself:$$\begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \times \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}.$$Using matrix multiplication:Top-left: $(-1)(-1)+(0)(0)=1$Top-right: $(-1)(0)+(0)(-1)=0$Bottom-left: $(0)(-1)+(-1)(0)=0$Bottom-right: $(0)(0)+(-1)(-1)=1$Thus, $(f(\pi))^2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, the identity matrix.Answer:Option D: $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$.