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Matrices & Determinants: Find Roots Equation Vmatrix Vmatrix

JEE Maths question with a full step-by-step solution.

Question
Find the roots of the equation
∣xα1βx1βγ1∣=0\begin{vmatrix} x & \alpha & 1 \\ \beta & x & 1 \\ \beta & \gamma & 1 \end{vmatrix} = 0
Aα\alpha and β\beta
Bβ\beta and γ\gammacorrect
Cγ\gamma and α\alpha
DAll of these
Solution
Step 1: Apply row operations R1→R1−R2R_1 \to R_1 - R_2 and R2→R2−R3R_2 \to R_2 - R_3:
∣x−βα−x00x−γ0βγ1∣=0\begin{vmatrix} x-\beta & \alpha-x & 0 \\ 0 & x-\gamma & 0 \\ \beta & \gamma & 1 \end{vmatrix} = 0
 Step 2: Expand along column 3
1⋅∣x−βα−x0x−γ∣=01 \cdot \begin{vmatrix} x-\beta & \alpha-x \\ 0 & x-\gamma \end{vmatrix} = 0
(x−β)(x−γ)=0(x-\beta)(x-\gamma) = 0
 Step 3: State the roots The roots are x=βx = \beta and x=γx = \gamma. Answer: (2)
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