Let the vectors
(2 + a + b)î + (a + 2b + c)ĵ - (b + c)k̂,(1 + b)î + 2bĵ - bk̂ and (2 + b)î + 2bĵ + (1 - b)k̂, a,b,c, ∈ R
be co-planar. Then which of the following is true?

Solution

Maths sir · 4 steps

Coplanar means the determinant is zero, and two row operations make most of the letters cancel.

1. Coplanar condition. Three vectors are coplanar exactly when their scalar triple product is zero. So the 3 × 3 determinant of their components must vanish.
2. Simplify the rows. Do R₁ → R₁ - R₂ and R₃ → R₃ - R₂. Row 1 becomes (1+a, a+c, -c) and row 3 becomes (1, 0, 1). Then C₁ → C₁ - C₃ turns row 3 into (0, 0, 1). These operations do not change the determinant.

   |1+a+c, a+c, -c; 1+2b, 2b, -b; 0, 0, 1| = (1+a+c)(2b) - (a+c)(1+2b)

3. Expand. Expanding along row 3 gives the expression above. The 2b(a+c) terms cancel, leaving 2b - (a+c).
4. So. Setting 2b - (a+c) = 0 gives 2b = a + c, which is option (a).

The idea to keep. For coplanarity, set the determinant to zero, but first subtract rows that share terms. Aim for a row with two zeros, then expand along it.

Answer from the NTA final answer key. Solution by PrepWiser.