Question

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 

• A. \( \frac{3}{w^2} \)
• B. \( 3w \)
• C. \( 0 \)
• D. \( 3w^2 \)

Hint:

Circulant determinants have specific properties that simplify computations. Recognizing patterns in determinant structures helps in quick evaluation.

Answer 1

The Correct Option is D

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:

\[ D = 3w^2. \]