If \(\begin{vmatrix} -1 & a & a^2\\  -1 & b & b^2\\ -1 & c& c^2\\ \end{vmatrix}\) = λ and a - b = 1, b - c = 2 and c - a = -3, then the value if λ is

6

The correct answer is Option (1) → 6

We are given the determinant:

$\lambda = \begin{vmatrix} -1 & a & a^2 \\ -1 & b & b^2 \\ -1 & c & c^2 \end{vmatrix}$

$\lambda = (-1) \begin{vmatrix} 1 & a & a^2 \\ 1 & b & b^2 \\ 1 & c & c^2 \end{vmatrix}$

Now, this is a standard form of a determinant whose value is: (b−a)(c−a)(c−b)

Given the algebraic values:

• $b - a = -1$
• $c - b = -2$
• $c - a = -3$

So,

(b−a)(c−a)(c−b)=(−1)(−3)(−2)  = -6

$\lambda$ = −(−6)= 6 

Thus, the value of the determinant is $6$.

*****