For the system of equations |
\[ax+by+cz=q-r,\] |
\[bx+cy+az=r-p,\] |
\[cx+ay+bz=p-q,\] |
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.You need to login to perform this action.
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