Consider three planes P1 : x - y + z = 1 P2 : x + y - z = -1 and P3 : x - 3y + 3z = 2

Question

For the following questions, choose the correct answer from the codes (a), (b), (c) and (d) defined as follows.

(a) Statement I is true, Statement II is also true; Statement

II is the correct explanation of Statement I.

(b) Statement I is true, Statement II is also true; Statement

II is not the correct explanation of Statement I.

(c) Statement I is true; Statement II is false.

(d) Statement I is false; Statement II is true.

Consider three planes

P₁ : x - y + z = 1

P₂ : x + y - z = -1

and P₃ : x - 3y + 3z = 2

Let L₁, L₂, L₃ be the lines of intersection of the planes P₂ and P_3,P₃ and P₁, P₁ and P₂, respectively.

Statement I : At least two of the lines L₁, L₂ and L₃ are non-parallel.

Statement II : The three planes do not have a common point.

Answer 1

Correct option: (d) Statement I is false; Statement II is true.

Explanation:

The given equations are

 x - y + z = 1

x + y - z = -1

and x - 3y + 3z = 2

The system of equations can be put in matrix form as AX = B

=> The three planes do not have a common point.

=> Statement II is true.

Since, planes P₁, P₂, P₃ are pairwise intersection, then their lines of intersection are parallel.

Statement I is false.