If x^ 2 = | cases sinθ & cosθ & 0 - cosθ & sinθ & 1 sinθ & cosθ & 2...

Given,& x^{2} = \left|\begin{cases}sin\theta & cos\theta & 0 \\ - cos\theta & sin\theta & 1 \\ sin\theta & cos\theta & 2\end{cases}\right|

On expanding \left|$\begin{cases}sin\theta & cos\theta & 0 \\ - cos\theta & sin\theta & 1 \\ sin\theta & cos\theta & 2\end{cases}$\right|$along${C}_{3}$, we get</p><p>$x^{2} = 0 - 1\left|$\begin{cases}sin\theta & cos\theta \\ sin\theta & cos\theta\end{cases}$\right| + 2\left|$\begin{cases}sin\theta & cos\theta \\ - cos\theta & sin\theta\end{cases}$\right| = - 1 \left($sin\theta cos$\theta -$sin\theta cos$\theta\right) +2\left(\sin^{2}\theta + \cos^{2}\theta\right)

$= - 1$\times0 + 2 \times1$</p><p>$⇒ x^{2} = 2 ⇒ x = $\pm \sqrt{2}</p><p>If$ x = \sqrt{2}, then

4x^{2} + xsin\dfrac{3\pi}{2} +5 = 4 \times \left(\sqrt{2}\right)$2 -$$\sqrt{2} +$$5$

$= 8 -$$\sqrt{2} +$$5 =$$\left(13 - \sqrt{2}\right)$

If $x = -$$\sqrt{2},$ then

4x^{2} + xsin\dfrac{3\pi}{2} +5 = 4 \times \left(- \sqrt{2}\right)2 + \sqrt{2} +5$</p><p>$= 8 + \sqrt{2} +5 = \left(13 + \sqrt{2}\right)$</p>$