Question

If a. b, c are negative and different real numbers then $\Delta =\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|$ is

• A. $<0$
• B. 0
• C. $>0$
• D. $\le 0$

Answer 1

The correct option is C $>0$

$\Delta =\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|$

$\Delta =(a+b+c)\left|\begin{array}{ccc}1& b& c\\ 1& c& a\\ 1& a& b\end{array}\right|$

$=-(a+b+c)[a^2+b^2+c^2-ab-bc-ca]\Delta =-\left(\frac{a+b+c}{2}\right)\left[\right(a-b{)}^2+(b-c{)}^2+(c-a{)}^2]$

$\Delta >0$ as $a\ne b\ne c,a,b,c<0$

$\therefore a+b+c<0$

$\therefore \frac{-(a+b+c)}{2}>0$