Question

Say true or false:

If a, b, c are sides of scalene triangle, then the value of $\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|$ is positive.

• A. True
• B. False

Answer 1

The correct option is B False

$a,b,c$ are sides of scalene triangle.
i.e $a+b+c>0$
$\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|$
$\to (a+b+c)\left|\begin{array}{ccc}1& 1& 1\\ b& c& a\\ c& a& b\end{array}\right|$
$=-\frac{1}{2}(a+b+c)\left(\right(a-b{)}^2+(b-c{)}^2+(c-a{)}^2)<0$
$\therefore \left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|$ is not positive.
Hence, option B.