Question

Consider the following system of equations :

x + 2y - 3z = a

2x + 6y - 11z = b

x - 2y + 7z = c,

where a, b and c are real constants. Then the system of equations :

(1) has a unique solution when 5a = 2b + c

(2) has infinite number of solutions when 5a = 2b + c

(3) has no solution for all a, b and c

(4) has a unique solution for all a, b and c

Answer 1

Correct option is (2) has infinite number of solutions when 5a = 2b + c

P₁ : x + 2y - 3z = a

P₂ : 2x + 6y - 11z = b

P₃ : x - 2y + 7z = c

Clearly

5P₁ = 2P₂ + P₃   if 5a = 2b + c

⇒ All the planes sharing a line of intersection

⇒ infinite solutions

Answer 2

P1 : x + 2y - 3z = a

P2 : 2x + 6y - 11z = b

P3 : x - 2y + 7z = c

Clearly

5P1 = 2P2 + P3  if 5a = 2b + c

 ⇒ All the planes sharing a line of intersection

 ⇒ infinite solutions

Comments

Why cheating?

Answer 3

Answer by Mukesh

Let

P1 : x + 2y - 3z = a
P2 : 2x + 6y - 11z = b
P3 : x - 2y + 7z = c
Clearly
5P1 = 2P2 + P3  if 5a = 2b + c
 

⇒ All the planes sharing a line of intersection
⇒ infinite solutions