isi-entrance 2019 Q10

isi-entrance · India · UGA Simultaneous equations
For a real number $\theta$, consider the following simultaneous equations:
$$\begin{aligned} & \cos ( \theta ) x - \sin ( \theta ) y = 1 \\ & \sin ( \theta ) x + \cos ( \theta ) y = 2 \end{aligned}$$
The number of solutions of these equations in $x$ and $y$ is
(A) 0
(B) 1
(C) infinite for some values of $\theta$
(D) finite only when $\theta = \frac { m \pi } { n }$ for integers $m$, and $n \neq 0$.
For a real number $\theta$, consider the following simultaneous equations:

$$\begin{aligned}
& \cos ( \theta ) x - \sin ( \theta ) y = 1 \\
& \sin ( \theta ) x + \cos ( \theta ) y = 2
\end{aligned}$$

The number of solutions of these equations in $x$ and $y$ is\\
(A) 0\\
(B) 1\\
(C) infinite for some values of $\theta$\\
(D) finite only when $\theta = \frac { m \pi } { n }$ for integers $m$, and $n \neq 0$.