\[ D = \begin{vmatrix} -\frac{bc}{a^2} & \frac{c}{a} & \frac{b}{a} \\ \frac{c}{b} & -\frac{ac}{b^2} & \frac{a}{b} \\ \frac{b}{c} & \frac{a}{c} & -\frac{ab}{c^2} \end{vmatrix} \]
\( \frac{a^2 + b^2 + c^2}{a^2 b^2 c^2} \)
We need to evaluate the determinant:
\[ D = \begin{vmatrix} -\frac{bc}{a^2} & \frac{c}{a} & \frac{b}{a} \\ \frac{c}{b} & -\frac{ac}{b^2} & \frac{a}{b} \\ \frac{b}{c} & \frac{a}{c} & -\frac{ab}{c^2} \end{vmatrix}. \]
Step 1: Expand the Determinant Along the First Row
Using the determinant expansion along the first row:
\[ D = -\frac{bc}{a^2} \begin{vmatrix} - \frac{ac}{b^2} & \frac{a}{b} \\ \frac{a}{c} & -\frac{ab}{c^2} \end{vmatrix} - \frac{c}{a} \begin{vmatrix} \frac{c}{b} & \frac{a}{b} \\ \frac{b}{c} & -\frac{ab}{c^2} \end{vmatrix} + \frac{b}{a} \begin{vmatrix} \frac{c}{b} & -\frac{ac}{b^2} \\ \frac{b}{c} & \frac{a}{c} \end{vmatrix}. \]
Step 2: Compute the 2×2 Determinants
Each determinant is evaluated as follows:
\[ \begin{vmatrix} - \frac{ac}{b^2} & \frac{a}{b} \\ \frac{a}{c} & -\frac{ab}{c^2} \end{vmatrix} = \left(-\frac{ac}{b^2} \times -\frac{ab}{c^2} \right) - \left( \frac{a}{b} \times \frac{a}{c} \right) = \frac{a^2 b}{b^2 c^2} - \frac{a^2}{bc}. \]
\[ \begin{vmatrix} \frac{c}{b} & \frac{a}{b} \\ \frac{b}{c} & -\frac{ab}{c^2} \end{vmatrix} = \left(\frac{c}{b} \times -\frac{ab}{c^2} \right) - \left( \frac{a}{b} \times \frac{b}{c} \right) = -\frac{ac}{b c^2} - \frac{a}{c b}. \]
\[ \begin{vmatrix} \frac{c}{b} & -\frac{ac}{b^2} \\ \frac{b}{c} & \frac{a}{c} \end{vmatrix} = \left(\frac{c}{b} \times \frac{a}{c} \right) - \left( -\frac{ac}{b^2} \times \frac{b}{c} \right) = \frac{a}{b} + \frac{a}{b}. \]
Step 3: Compute the Final Value
After simplifications, we obtain:
\[ D = 4. \]
Let \( A = [a_{ij}] \) be a \( 3 \times 3 \) matrix with positive integers as its elements. The elements of \( A \) are such that the sum of all the elements of each row is equal to 6, and \( a_{22} = 2 \).
\[ \textbf{If } | \text{Adj} \ A | = x \text{ and } | \text{Adj} \ B | = y, \text{ then } \left( | \text{Adj}(AB) | \right)^{-1} \text{ is } \]
The roots of the equation \( x^3 - 3x^2 + 3x + 7 = 0 \) are \( \alpha, \beta, \gamma \) and \( w, w^2 \) are complex cube roots of unity. If the terms containing \( x^2 \) and \( x \) are missing in the transformed equation when each one of these roots is decreased by \( h \), then
If \( a \neq b \neq c \), then
\[ \Delta_1 = \begin{vmatrix} 1 & a^2 & bc \\ 1 & b^2 & ca \\ 1 & c^2 & ab \end{vmatrix}, \quad \Delta_2 = \begin{vmatrix} 1 & 1 & 1 \\ a^2 & b^2 & c^2 \\ a^3 & b^3 & c^3 \end{vmatrix} \]and
\[ \frac{\Delta_1}{\Delta_2} = \frac{6}{11} \]then what is \( 11(a + b + c) \)?
\[ \text{The domain of the real-valued function } f(x) = \sin^{-1} \left( \log_2 \left( \frac{x^2}{2} \right) \right) \text{ is} \]