Question

Assertion :If a+b+c=0 and $a^2+b^2+c^2\ne bc+ca+ab$, then the system of homogenous equations $ax+by+cz=0$
$bx+cy+az=0$
$cx+ay+bz=0$ has infinite number of solutions. Reason: If |A|=0, the system of equations AX=B has infinite number of solutions.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is C Assertion is correct but Reason is incorrect

Reason is not always true. For instance, the system of equations
$x+2y+3z=12x+3y+4z=23x+4y+5z=4$
has no solution but $\left|A\right|=0$
For assertion
$△=\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|$
Applying $C_1\to C_1+C_2+C_3$, we obtain
$△=(a+b+c)\left|\begin{array}{ccc}1& b& c\\ 1& c& a\\ 1& a& b\end{array}\right|=0$
Also, note that $x=y=z$ satisfies each of the three equation. Thus, the system of equation has infinite number of solution.