Question

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?
1. consistent if p = q = r
2. consistent if a = b = c and p,q,r are distinct
3. consistent if a,b,c are distinct and a + b + c ¹ 0
4. consistent if a = b = c

Answer 1

Correct Answer - Option 2 : consistent if a = b = c and p,q,r are distinct

\(\Delta = \left| {\begin{array}{*{20}{c}} a&b&c\\ b&c&a\\ c&a&b \end{array}} \right|\) = 3abc – a³ – b³ – c³

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