The solutions of the equation

Question:

The solutions of the equation

$\left|\begin{array}{ccc}1+\sin ^{2} x & \sin ^{2} x & \sin ^{2} x \\ \cos ^{2} x & 1+\cos ^{2} x & \cos ^{2} x \\ 4 \sin 2 x & 4 \sin 2 x & 1+4 \sin 2 x\end{array}\right|=0,(0

 

 

  1. $\frac{\pi}{12}, \frac{\pi}{6}$

  2. $\frac{\pi}{6}, \frac{5 \pi}{6}$

  3. $\frac{5 \pi}{12}, \frac{7 \pi}{12}$

  4. $\frac{7 \pi}{12}, \frac{11 \pi}{12}$


Correct Option: , 4

Solution:

$\left|\begin{array}{ccc}1+\sin ^{2} x & \sin ^{2} x & \sin ^{2} x \\ \cos ^{2} x & 1+\cos ^{2} x & \cos ^{2} x \\ 4 \sin 2 x & 4 \sin 2 x & 1+4 \sin 2 x\end{array}\right|=0$

use $\mathrm{R}_{1} \rightarrow \mathrm{R}_{1}+\mathrm{R}_{2}+\mathrm{R}_{3}$

$\Rightarrow(2+4 \sin 2 x)\left|\begin{array}{ccc}1 & 1 & 1 \\ \cos ^{2} x & 1+\cos ^{2} x & \cos ^{2} x \\ 4 \sin 2 x & 4 \sin 2 x & 1+4 \sin 2 x\end{array}\right|=0$

$\Rightarrow \sin 2 x=-\frac{1}{2}$

$\Rightarrow 2 x=\pi+\frac{\pi}{6}, 2 \pi-\frac{\pi}{6}$

$x=\frac{\pi}{2}+\frac{\pi}{12}, \pi-\frac{\pi}{12}$

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