Source: MAT $2008$
Q. The following simultaneous equations are solvable for how many values of $\theta$ in the range $0 \le \theta \lt 2 \pi$? $$(\cos \theta)x - (sin\theta)y = 2$$ $$(\sin\theta)x + (\cos\theta)y = 1$$
Eliminating $y$ from the equations, we get $$x = 2\cos\theta + \sin\theta$$ Finding $y$ in terms of $x$, we get $$y = \frac{(\cos\theta)x - 2}{sin\theta}, \; \sin\theta \neq 0$$ Since $\sin\theta \neq 0, \, 0 \, \text{and} \, \pi$ cannot be values of $\theta$. Hence, all except $2$ values of $\theta$ in the specified range render the simultaneous equations solvable.
This is an incorrect conclusion, since all values of $\theta$ will work in the specified range. Where did I make a mistake?