Match the statements/expressions given in Column I with the values...

(a) $2 \sin ^2 \theta+\sin ^2 2 \theta=2$ $\Rightarrow \quad \sin ^2 2 \theta=2 \cos ^2 \theta$ $\Rightarrow \quad 4 \sin ^2 \theta \cos ^2 \theta=2 \cos ^2 \theta$ $\Rightarrow \quad \cos ^2 \theta=0$ or $\sin ^2 \theta=\frac{1}{2}$ $\Rightarrow \quad \cos \theta=0$ or $\sin \theta=\pm \frac{1}{\sqrt{2}}$ $\Rightarrow \quad \theta=\pm \frac{\pi}{4}$ or $\frac{\pi}{2}$ (b) $f(x)=$\left[\frac{6 x}{\pi}\right] \cos $\left[\frac{3 x}{\pi}\right]$ Possible points of discontinuity of $\left[\frac{6 x}{\pi}\right]$ are $$\begin{gathered} \frac{6 x}{\pi}=n, n \in I \\ \Rightarrow \quad x=\frac{n \pi}{6} \Rightarrow x=\frac{\pi}{6}, \frac{\pi}{3}, \frac{\pi}{2}, \pi \\ \lim _{x \rightarrow \pi^{-} / 6} f(x)=0 \cos 0=0 \\ \lim _{x \rightarrow \pi^{+} / 6} f(x)=1 \cos 0=1 \end{gathered}$$ $\therefore$ Discontinuous at $x=\frac{\pi}{6}$. Similarly, discontinuous at $x=\frac{\pi}{3}, \frac{\pi}{2}, \pi$. (c) Here, V=\left\|$\begin{array}{lll}1 & 1 & 0 \\ 1 & 2 & 0 \\ 1 & 1 & \pi\end{array}$\right\|=\pi cubic unit (d) Given, $\mathbf{a}+\mathbf{b}+\sqrt{3} \mathbf{c}=\mathbf{0}$ \begin{array}{ll} \Rightarrow & \mathbf{a}+\mathbf{b}=-\sqrt{3} \mathbf{c} \\ \Rightarrow & |\mathbf{a}+\mathbf{b}|^2=\mid \sqrt{3} \mathbf{c}^2 \end{array}\begin{array}{lrl} \Rightarrow & a^2+b^2+2 \mathbf{a} \cdot \mathbf{b}=3 c^2 \\ \Rightarrow & 2+2 \cos \theta=3 \\ \Rightarrow & \cos \theta=\frac{1}{2} \Rightarrow \theta=\frac{\pi}{3} \end{array}