AP DSC School Assistant Math Model Test 2

Concept:The determinant of a matrix is evaluated by expansion. Here the matrix contains trigonometric terms, so the determinant becomes a function of $\sin\theta$. The range of $\sin^2\theta$ determines the possible values of the determinant.Explanation:Expand the given determinant along the first row:$\Delta = 1 \cdot (1 \cdot 1 - \sin\theta \cdot (-\sin\theta)) - \sin\theta \cdot ((-\sin\theta) \cdot 1 - \sin\theta \cdot (-1)) + 1 \cdot ((-\sin\theta) \cdot (-\sin\theta) - 1 \cdot (-1))$Simplify each minor:$= 1(1 + \sin^2\theta) - \sin\theta(-\sin\theta + \sin\theta) + 1(\sin^2\theta + 1)$The second term is $-\sin\theta \times 0 = 0$, so:$\Delta = (1 + \sin^2\theta) + (\sin^2\theta + 1) = 2 + 2\sin^2\theta$Since $0 \leq \sin^2\theta \leq 1$ for all real $\theta$, we have:When $\sin^2\theta = 0$, $\Delta = 2$; when $\sin^2\theta = 1$, $\Delta = 4$.Thus $\Delta$ takes every value between 2 and 4 inclusive, i.e., the interval $[2, 4]$.Answer:The determinant $\Delta$ lies in the interval $[2, 4]$, which corresponds to option B.