MCQ
If $f(\theta)=\left[\begin{array}{rr}\cos \theta & -\sin \theta \\ \sin \theta & -\cos \theta\end{array}\right]$, then $f\left(\frac{\pi}{6}\right)=$ __________ .
- ✓$-\frac{1}{2}$
- B$\frac{1}{2}$
- C$\frac{\sqrt{3}}{2}$
- D$-\frac{\sqrt{3}}{2}$
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
$P_6=\left[\begin{array}{lll}0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0\end{array}\right]$ and $X=\sum_{k=1}^6 P_k\left[\begin{array}{lll}2 & 1 & 3 \\ 1 & 0 & 2 \\ 3 & 2 & 1\end{array}\right] P_k^{\top}$
where $P _{ K }^{ T }$ denotes the transpose of the matrix $P _{ K }$. Then which of the following options is/are correct?
$(1)$ $X -30 I$ is an invertible matrix
$(2)$ The sum of diagonal entries of $X$ is 18
$(3)$ If $X \left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]=\alpha\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]$, then $\alpha=30$
$(4)$ $X$ is a symmetric matrix
| $\text{X}$ | $2$ | $3$ | $4$ | $5$ |
| $\text{P}(\text{X})$ | $\frac{5}{\text{k}}$ | $\frac{7}{\text{k}}$ | $\frac{9}{\text{k}}$ | $\frac{11}{\text{k}}$ |