Question

If \[ \begin{vmatrix} 1 & 1 & 1 \\ a & b & c \\ a^2 & b^2 & c^2 \end{vmatrix}=0 \] then:

• A. $a=b=c$
• B. At least two of $a,b,c$ are equal
• C. $a+b+c=0$
• D. $abc=0$

Hint:

Always remember the Vandermonde determinant: \[ \begin{vmatrix} 1 & 1 & 1 a & b & c a^2 & b^2 & c^2 \end{vmatrix} =(b-a)(c-a)(c-b) \] It appears very frequently in determinant problems.

Answer 1

The Correct Option is B

Concept: The given determinant is a Vandermonde determinant. The standard form is: \[ \begin{vmatrix} 1 & 1 & 1 \\ a & b & c \\ a^2 & b^2 & c^2 \end{vmatrix} =(b-a)(c-a)(c-b) \] A product becomes zero when at least one factor is zero.

Step 1: Write the determinant formula.
Using the Vandermonde determinant property: \[ \Delta=(b-a)(c-a)(c-b) \] Given: \[ \Delta=0 \] Therefore, \[ (b-a)(c-a)(c-b)=0 \]

Step 2: Interpret the condition.
A product is zero if at least one factor is zero. Thus: \[ b-a=0 \] or \[ c-a=0 \] or \[ c-b=0 \] This means: \[ a=b \] or \[ a=c \] or \[ b=c \] Hence, at least two of the numbers are equal. Therefore, \[ \boxed{\text{At least two of }a,b,c\text{ are equal}} \]