question_answer

For the system of equations

\[ax+by+cz=q-r,\]

\[bx+cy+az=r-p,\]

\[cx+ay+bz=p-q,\]

Which of the following is FALSE?

A) Consistent if \[p=q=r\]                                

B) Consistent if \[a=b=c\]and \[p,q,r\] are distinct

C) Consistent if \[a,\text{ }b,\text{ }c\]are distinct and \[a+b+c\ne 0\]        

D) Consistent if \[a=b=c\]

Correct Answer: B

Solution :

[b] \[=-\frac{1}{2}(a+b+c)\,({{(a-b)}^{2}}+{{(b-c)}^{2}}+{{(c-a)}^{2}})\] \[=-\frac{1}{2}\,(a+b+c)({{(a-b)}^{2}}+{{(b-c)}^{2}}+{{(c-a)}^{2}})\] Now, when \[p=q=r,\]then system is homogenous and hence, consistent. If \[a=b=c\]and p, q, r are distinct then system represents three parallel planes and hence, inconsistent.