The simultaneous equations in x and y:

cos(θ)x - sin(θ)y = 2

sin(θ)x + cos(θ)y = 1

are solvable

(a) for all values of θ in the range 0 ≤ θ ≤ 2π;

(b) except for one value of θ in the range 0 ≤ θ < 2π;

(c) except for two values of θ in the range 0 ≤ θ < 2π;

(d) except for three values of θ in the range 0 ≤ θ < 2π;

Solution:

The equations are not solvable if the slopes of the two equations are equal.

The slope of the first line is [cos(θ)/sin(θ)] and the slope of the second line is [-sin(θ)/cos(θ)].

cos(θ)/sin(θ) = -sin(θ)/cos(θ)

There are no θ that satisfy this equation, so the equations are solvable for all values of θ in the range 0 ≤ θ ≤ 2π.