Trigonometry
Range of Expression
MMTS_Full_Test_16
Grade 12
Question:
If $\theta\in\mathbb{R}$, then the range of $f(\theta)=\begin{vmatrix}1&\cos\theta&1\\-\cos\theta&1&\cos\theta\\-1&-\cos\theta&1\end{vmatrix}$ is
$(2,4]$
$[2,4]$
$(0,4]$
$(0,\infty)$
Step-by-Step Solution
Key Concept: Expand the $3\times 3$ determinant
Expanding: $f(\theta)=1(1+\cos^2\theta)-\cos\theta(-\cos\theta+\cos\theta)+1(\cos^2\theta+1)=2+2\cos^2\theta=2+2\cos^2\theta\in[2,4]$.
Correct Answer: 3