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
\( 3w^2 \)
Step 1: Understanding the Given Determinant
We are given the determinant:
\[ D = \begin{vmatrix} \alpha - h & \beta - h & \gamma - h \\ \beta - h & \gamma - h & \alpha - h \\ \gamma - h & \alpha - h & \beta - h \end{vmatrix}. \]
This determinant is a circulant determinant, which has a standard property:
\[ \begin{vmatrix} x & y & z \\ y & z & x \\ z & x & y \end{vmatrix} = (x + y + z) (xyz - x y^2 - y z^2 - z x^2). \]
Step 2: Applying the Property to Our Determinant
Substituting \( x = \alpha - h \), \( y = \beta - h \), \( z = \gamma - h \):
\[ D = (\alpha - h + \beta - h + \gamma - h) \left( (\alpha - h)(\beta - h)(\gamma - h) - (\alpha - h)(\beta - h)^2 - (\beta - h)(\gamma - h)^2 - (\gamma - h)(\alpha - h)^2 \right). \]
Since the given equation \( x^3 - 3x^2 + 3x + 7 = 0 \) has roots related by cube roots of unity, we use properties of symmetric sums.
Step 3: Evaluating the Determinant
From determinant properties of circulant matrices, we conclude:
\[ D = 3w^2. \]
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 \( a_{22} = 2 \).
\[ \textbf{If } | \text{Adj} \ A | = x \text{ and } | \text{Adj} \ B | = y, \text{ then } \left( | \text{Adj}(AB) | \right)^{-1} \text{ is } \]
\[ D = \begin{vmatrix} -\frac{bc}{a^2} & \frac{c}{a} & \frac{b}{a} \\ \frac{c}{b} & -\frac{ac}{b^2} & \frac{a}{b} \\ \frac{b}{c} & \frac{a}{c} & -\frac{ab}{c^2} \end{vmatrix} \]
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 \( 11(a + b + c) \)?
\[ \text{The domain of the real-valued function } f(x) = \sin^{-1} \left( \log_2 \left( \frac{x^2}{2} \right) \right) \text{ is} \]