Question

Assertion :If a, b, c are distinct real number and x, y, z are not all zero given that $ax+by+cz=0,bx+cy+az=0,cx+ay+bz=0$, then $a+b+c\ne 0$ Reason: If a, b, c are distinct positive real number then $a^2+b^2+c^2\ne ab+bc+ca$.

• 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. Assertion is incorrect but Reason is correct

Answer 1

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

$D=\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|=-(a^3+b^3+c^3-3abc)=-(a+b+c)$
$(a^2+b^2+c^2-ab-bc-ca).$ Now the other bracket cannot vanish in the light of the correct reason $R.$
$\Rightarrow$ The system can have non-zero solutions $(x,y,z)$
are not all zero
If$D=0$
$\Rightarrow a+b+c=0$
$\Rightarrow$ Assertion A is false