Question

If \( \alpha, \beta, \gamma \) are the roots of the determinant equation: \[ \begin{vmatrix} 1-x & -2 & 1
-2 & 4-x & -2
1 & -2 & 1-x \end{vmatrix} = 0 \] then \( \alpha \beta + \beta \gamma + \gamma \alpha \) is:

• A. \( 6 \)
• B. \( 8 \)
• C. \( 0 \)
• D. \( -4 \)

Hint:

For determinant equations, use characteristic equations and Vieta’s formulas to find sum and product of roots.

Answer 1

The Correct Option is C

Step 1: Characteristic Equation Expanding the determinant, we obtain a cubic equation in \( x \). Step 2: Use Sum and Product of Roots From the properties of determinants: \[ \alpha + \beta + \gamma = 4, \quad \alpha \beta + \beta \gamma + \gamma \alpha = 0 \] Thus, the correct answer is 0.