A value of theta in left(0, frac{pi}{2}right) | vedclass

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Question

(D) Given the determinant equation:$\Delta = \left|\begin{array}{ccc}1+\sin ^2 	heta & \cos ^2 	heta & 4 \sin 4 	heta \ \sin ^2 	heta & 1+\cos ^2

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$\frac{7 \pi}{24}$

Vedclass · Answer

(D) Given the determinant equation:$\Delta = \left|\begin{array}{ccc}1+\sin ^2 	heta & \cos ^2 	heta & 4 \sin 4 	heta \ \sin ^2 	heta & 1+\cos ^2 	heta & 4 \sin 4 	heta \ \sin ^2 	heta & \cos ^2 	heta & 1+4 \sin 4 	heta\end{array}ight| = 0$Applying the column operation $C_1 ightarrow C_1 + C_2 + C_3$:$\Delta = \left|\begin{array}{ccc}1+\sin^2 	heta + \cos^2 	heta + 4\sin 4	heta & \cos^2 	heta & 4\sin 4	heta \ \sin^2 	heta + 1 + \cos^2 	heta + 4\sin 4	heta & 1+\cos^2 	heta & 4\sin 4	heta \ \sin^2 	heta + \cos^2 	heta + 1 + 4\sin 4	heta & \cos^2 	heta & 1+4\sin 4	heta\end{array}ight| = 0$Since $\sin^2 	heta + \cos^2 	heta = 1$,the first column becomes $2 + 4\sin 4	heta$:$\Delta = (2 + 4\sin 4	heta) \left|\begin{array}{ccc}1 & \cos^2 	heta & 4\sin 4	heta \ 1 & 1+\cos^2 	heta & 4\sin 4	heta \ 1 & \cos^2 	heta & 1+4\sin 4	heta\end{array}ight| = 0$Applying row operations $R_2 ightarrow R_2 - R_1$ and $R_3 ightarrow R_3 - R_1$:$\Delta = (2 + 4\sin 4	heta) \left|\begin{array}{ccc}1 & \cos^2 	heta & 4\sin 4	heta \ 0 & 1 & 0 \ 0 & 0 & 1\end{array}ight| = 0$Expanding the determinant,we get $(2 + 4\sin 4	heta)(1) = 0$,which implies $2 + 4\sin 4	heta = 0$.Therefore,$\sin 4	heta = -\frac{1}{2}$.Since $	heta \in (0, \frac{\pi}{2})$,$4	heta \in (0, 2\pi)$. The values for $4	heta$ where $\sin 4	heta = -\frac{1}{2}$ are $\frac{7\pi}{6}$ and $\frac{11\pi}{6}$.For $4	heta = \frac{7\pi}{6}$,$	heta = \frac{7\pi}{24}$.For $4	heta = \frac{11\pi}{6}$,$	heta = \frac{11\pi}{24}$.Comparing with the options,$\frac{7\pi}{24}$ is the correct value.

$A$ value of $\theta$ in $\left(0, \frac{\pi}{2}\right)$ satisfying $\left|\begin{array}{ccc}1+\sin ^2 \theta & \cos ^2 \theta & 4 \sin 4 \theta \\ \sin ^2 \theta & 1+\cos ^2 \theta & 4 \sin 4 \theta \\ \sin ^2 \theta & \cos ^2 \theta & 1+4 \sin 4 \theta\end{array}\right|=0$ is

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