Question

If the system of linear equations $\left(\cos \theta \right)x+\left(\sin \theta \right)y+\cos \theta =0,$ $\left(\sin \theta \right)x+\left(\cos \theta \right)y+\sin \theta =0$ and $\left(\cos \theta \right)x+\left(\sin \theta \right)y-\cos \theta =0$ is consistent, then the possible value(s) of $\theta \in [0,2\pi ]$ is/are

• A. $\frac{\pi }{2}$
• B. $\frac{3\pi }{4}$
• C. $\frac{3\pi }{2}$
• D. $\frac{7\pi }{4}$

Answer 1

Answer: (A), (C)

$\frac{3\pi }{2}$

If the system of linear equations is consistent, then we have
$\Delta =0$
where $\Delta =\left|\begin{array}{ccc}\cos \theta & \sin \theta & \cos \theta \\ \sin \theta & \cos \theta & \sin \theta \\ \cos \theta & \sin \theta & -\cos \theta \end{array}\right|$

$\Rightarrow \left|\begin{array}{ccc}\cos \theta & \sin \theta & \cos \theta \\ \sin \theta & \cos \theta & \sin \theta \\ \cos \theta & \sin \theta & -\cos \theta \end{array}\right|=0$

$\Rightarrow \cos \theta [-{\cos }^2\theta -{\sin }^2\theta ]-\sin \theta [-\sin \theta \cos \theta -\sin \theta \cos \theta ]+\cos \theta [{\sin }^2\theta -{\cos }^2\theta ]=0$

$\Rightarrow \cos \theta (2{\sin }^2\theta -1)-\cos \theta \cos 2\theta =0$
$\Rightarrow \cos \theta \cos 2\theta =0$
$\Rightarrow \theta =\frac{\pi }{4},\frac{\pi }{2},\frac{3\pi }{4},\frac{5\pi }{4},\frac{3\pi }{2},\frac{7\pi }{4}$
But $\theta =\frac{\pi }{4},\frac{3\pi }{4},\frac{5\pi }{4},\frac{7\pi }{4}$ (rejected)
because lines are parallel, so system is inconsistent.