Matrices & Determinants
Matrices And Determinants
nta_abhyas_2025
Grade 12
Question:
If the maximum and minimum values of the determinant $\begin{vmatrix} 1 + \sin^2 x & \cos x & \sin 2x \\ \sin^2 x & 1 + \cos^2 x & \sin 2x \\ \sin^2 x & \cos^2 x & 1 + \sin 2x \end{vmatrix}$ are $c$ and $d$ respectively, then $c + 2d$ is equal to
Step-by-Step Solution
Key Concept: Row and column operations preserve determinant properties; systematic simplification reveals underlying structure.
Apply $C_1 \to C_1 + C_2$ to get $\begin{vmatrix} 2 & \cos^2 x & \sin 2x \\ 2 & 1 + \cos^2 x & \sin 2x \\ 1 & \cos^2 x & 1 + \sin 2x \end{vmatrix}$. Then apply $R_2 \to R_2 - R_1$ and $R_3 \to R_3 - R_1$ to obtain $\begin{vmatrix} 2 & \cos^2 x & \sin 2x \\ 0 & 1 & 0 \\ -1 & 0 & 1 \end{vmatrix}$. Expanding along $R_2$: $\Delta = -1(2 \cdot 1 - \sin 2x \cdot (-1)) = -(2 + \sin 2x)$. Since the maximum value of $\sin 2x$ is 1 and minimum is -1, $\Delta = 0 + (2 + \sin 2x) = 0 - 2 + \sin 2x$. Reconalculating: $\Delta = 2\cos^2 x - \sin 2x = 2 - 2\sin 2x$.
Correct Answer: 2