Matrices & Determinants
Matrices and Determinants
star_batch_jee_advanced_2025
Grade 12
Question:
For $a, b, c, x, y, z \in \mathbb{R}$, if $\Delta_1 = \begin{vmatrix} (a-x)^2 & (b-x)^2 & (c-x)^2 \\ (a-y)^2 & (b-y)^2 & (c-y)^2 \\ (a-z)^2 & (b-z)^2 & (c-z)^2 \end{vmatrix}$ and $\Delta_2 = \begin{vmatrix} (1+ax)^2 & (1+bx)^2 & (1+cx)^2 \\ (1+ay)^2 & (1+by)^2 & (1+cy)^2 \\ (1+az)^2 & (1+bz)^2 & (1+cz)^2 \end{vmatrix}$ then $|\Delta_1/\Delta_2| = $
Step-by-Step Solution
Key Concept: Express a determinant as a product of two simpler determinants and use column operations to reveal factorization patterns.
The determinant is written as a product of two determinants: one with columns $x^2-2x$, $y^2-2y$, $z^2-2z$ and another with columns $1, a, a^2$. By interchanging columns and factoring, the first determinant becomes $(1+ax)^2(1+by)^2(1+cz)^2$, which factorizes as the product of linear terms and can be compared with $\Delta_2$ shown on the right.
Correct Answer: 1