JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

Let $x, y, z > 1$ and $A = \begin{bmatrix} 1 & \log_x y & \log_x z \\ \log_y x & 2 & \log_y z \\ \log_z x & \log_z y & 3 \end{bmatrix}$ . Then $|\text{adj}(\text{adj } A^2)|$ is equal to:

Choose the correct answer:

Explanation

Solution:

Determinant $|A|$ nikaalne ke liye $\log$ property use karein:

|A| = \begin{vmatrix} 1 & \frac{\ln y}{\ln x} & \frac{\ln z}{\ln x} \\ \frac{\ln x}{\ln y} & 2 & \frac{\ln z}{\ln y} \\ \frac{\ln x}{\ln z} & \frac{\ln y}{\ln z} & 3 \end{vmatrix} = \frac{1}{\ln x \ln y \ln z} \begin{vmatrix} \ln x & \ln y & \ln z \\ \ln x & 2\ln y & \ln z \\ \ln x & \ln y & 3\ln z \end{vmatrix}

|A| = (1)(6-1) - (1)(3-1) + (1)(1-2) = 5 - 2 - 1 = 2

Ab $|A^2| = |A|^2 = 2^2 = 4$ .

Formula: $|\text{adj}(\text{adj } M)| = |M|^{(n-1)^2}$ , yahan $n=3$ :

|\text{adj}(\text{adj } A^2)| = |A^2|^{(3-1)^2} = 4^4 = (2^2)^4 = 2^8

Correct Option: (1)