Question

The determinant $\Delta =$ $\left|\begin{array}{ccc}{a}^2+x& ab& ac\\ ab& b^2+x& bc\\ ac& bc& c^2+x\end{array}\right|$ is divisible by

• A. $x$
• B. $x^2$
• C. $x^3$
• D. None of these

Answer 1

Answer: (A), (B)

The correct options are
A $x$
B $x^2$

$△=\left|\begin{array}{ccc}{a}^2+x& ab& ac\\ ab& b^2+x& bc\\ ac& bc& c^2+x\end{array}\right|$

$\left(a^2+x\right)\left(b^2{c}^2+b^{2}x+c^{2}x+x^2-b{x}^2\right)-ab\left(ab{c}^2+a{b}^2-a{b}^2\right)+ac(a{b}^{2}c-ac{b}^2-acx)$

$a^2{x}^2+b^2{x}^2+c^2{x}^2+x^3$

$\left(a^2+b^2+c^2\right)x^2+x^3$

$x^2\left(a^2+b^2+c^2+x\right)$

$\therefore$ Determinant is divisible by $x^2$

also divisible by $x$.