Trigonometry
Equation with integer parameter — counting ordered pairs
MJAT_TS6_P1
Grade 12
Question:
Given $2a\sin^2 x+a+b\cos x = b\sin x$ for $x\in\left[\frac{\pi}{2},\frac{3\pi}{2}\right]$, where $a\in\{1,2,3\}$ and $b$ is an integer. The number of ordered pairs $(a,b)$ for which the equation has solutions is:
Step-by-Step Solution
Key Concept: Rearrange and analyse as a function of $x$ for each $(a,b)$ pair. The ordered pairs listed in solution are pairs of $(a,b)$ values, and the number of valid pairs reduces to 1 after checking the domain constraints.
$\mathbf{1}$ ordered pair.
Correct Answer: 1