Let the position vectors of the points A, B, C and D be 5 î+5 ĵ+2 λ k̂, î+2 ĵ+3 k̂,-2 î+λ ĵ+4 k̂ and -î+5 ĵ+6 k̂. Let the set S={λ ∈ ℝ : the points A, B, C and D are coplanar }.
Then Σλ ∈ S(λ+2)² is equal to :

Solution

Maths sir · 3 steps

Build three edges from B and set their box product to zero, because coplanar means zero volume.

1. Edges from B. Take B as the base point, since it has no λ in it. Then BA⃗ = 4î+3ĵ+(2λ-3)k̂, BC⃗ = -3î+(λ-2)ĵ+k̂ and BD⃗ = -2î+3ĵ+3k̂.

   |4, 3, 2λ-3; -3, λ-2, 1; -2, 3, 3| = 0

2. Expand the determinant. Along the first row: 4(3λ-9) - 3(-7) + (2λ-3)(2λ-13) = 4λ²-20λ+24. Setting this to zero gives λ²-5λ+6=0, so λ = 2 or λ = 3.
3. So. S = {2, 3}, and Σ(λ+2)² = 4² + 5² = 16 + 25 = 41, which is option (d).

The idea to keep. For four points, make three vectors from one common point and set their scalar triple product to zero. Pick the base point without the parameter to keep the algebra light.

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