Applications of Derivatives
Determinant as Function — Max/Min via Trig
nta_pyq_2023_jan
Grade 12
Question:
Let $f(x)=\begin{vmatrix}1+\sin^2x & \cos^2x & \sin2x\\\sin^2x & 1+\cos^2x & \sin2x\\\sin^2x & \cos^2x & 1+\sin2x\end{vmatrix}$, $x\in\left[\dfrac{\pi}{6},\dfrac{\pi}{3}\right]$. If $\alpha$ and $\beta$ are the maximum and minimum values of f, then:
$\beta^2-2\sqrt{\alpha}=\dfrac{19}{4}$
$\beta^2+2\sqrt{\alpha}=\dfrac{19}{4}$
$\alpha^2-\beta^2=4\sqrt{3}$
$\alpha^2+\beta^2=\dfrac{9}{2}$
Step-by-Step Solution
Key Concept: Apply $C_1\to C_1+C_2+C_3$, then row operations. $f(x)=(2+\sin2x)\cdot1=2+\sin2x$.
$f(x)=2+\sin2x$, $\alpha=3$, $\beta=2+\frac{\sqrt{3}}{2}$. $\beta^2-2\sqrt{\alpha}=\frac{19}{4}$.
Correct Answer: 1