Matrices & Determinants
Matrices and Determinants
star_batch_jee_advanced_2025
Grade 12
Question:
The value of $\theta$ lying between $\theta = 0$ and $\theta = \frac{\pi}{2}$ and satisfying the equation $$\begin{vmatrix} 1+\sin^2\theta & \cos^2\theta & 4\sin 4\theta \\ \sin^2\theta & 1+\cos^2\theta & 4\sin 4\theta \\ \sin^2\theta & \cos^2\theta & 1+4\sin 4\theta \end{vmatrix} = 0$$ is:
Step-by-Step Solution
Key Concept: Matrix row and column operations preserve the determinant and reduce the computation to solving a simple trigonometric equation.
Set up the determinant with entries involving $\sin\theta$ and $\cos\theta$. Apply column operations $C_1 \to C_1 + C_2$ to simplify, then row operations $R_2 \to R_2 - R_1, R_3 \to R_3 - R_1$ to obtain an upper triangular form. This yields $2 + 4\sin 4\theta = 0$, so $\sin 4\theta = -\frac{1}{2} = \sin(-\frac{\pi}{6})$. In the range $[0, \frac{\pi}{2}]$, the solutions are $\theta = \frac{7\pi}{24}$ and $\frac{11\pi}{24}$.
Correct Answer: 1,3