Question

If $a\ne b\ne c$, the value of $x$ which satisfies the question
$\left|\begin{array}{ccc}0& x-a& x-b\\ x+a& 0& x-c\\ x+b& x+c& 0\end{array}\right|=0$ is

• A. $x=0$
• B. $x=a$
• C. $x=b$
• D. $x=c$

Answer 1

The correct option is A $x=0$

$a\ne b\ne c$

$\left|\begin{array}{ccc}0& x-a& x-b\\ x+a& 0& x-c\\ x+b& x+c& 0\end{array}\right|=0$

$\Rightarrow (x-a)(x+b)(x-c)+(x-b)(x+a)(x+c)=0$ ___ (1)

since $a\ne b\ne c$

putting $x=a$ or $x=b$ or $x=c$

The given determinant can never be zero.

putting $x=0$ in (1), gives

$(0-a)(0+b)(0-c)+(0-b)(0+a)(0+c)$

$=abc-abc$

$=0$.

$\therefore x=0$