Question

The system of equations
$ax+by+cz=q-r$
$bx+cy+az=r-p$
$cx+ay+bz=p-q$
is

• A. consistent if $p=q=r$
• B. inconsistent if $a=b=c$, and $p,q,r$ are distinct
• C. consistent if $a,b,c$ are distinct and $a+b+c\ne 0$
• D. none of these

Answer 1

Answer: (A), (C)

The correct options are
A consistent if $p=q=r$
C consistent if $a,b,c$ are distinct and $a+b+c\ne 0$

If $p=q=r$, then $x=0,y=0,z=0$ is a solution of the system of equation.
If $a,b,c$ are distinct, then determinant of the coefficient matrix is
$\frac{1}{2}(a+b+c)\left[\right(b-c{)}^2+(c-a{)}^2+(a-b{)}^2]\ne 0$
and hence the system of equations has a unique solution.
hence, options A and C.