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JEE Main20236 Apr 2023Morning ShiftMathematicsVector AlgebraActual

Let the position vectors of the points A , B , C and D be 5 i ^ + 5 j ^ + 2 λ k ^ , i ^ + 2 j ^ + 3 k ^ , - 2 i ^ + λ j ^ + 4 k ^ and - i ^ + 5 j ^ + 6 k ^ . Let the set S = λ ∈ ℝ : the points A , B , C and D are coplanar . The ∑ λ ∈ S ( λ + 2 ) 2 is equal to

Options

  1. A25
  2. B37 2
  3. C14
  4. D41

Correct answer

D. 41

Step-by-step solution

Given that: A ( 5 , 5 , 2 λ ) , B 1 , 2 , 3 , C - 2 , λ , 4 and D - 1 , 5 , 6 . Now, A B → ≡ ( - 4 , - 3 , 3 - 2 λ ) A C → ≡ ( - 7 , λ - 5 , 4 - 2 λ ) A D → ≡ ( - 6 , 0 , 6 - 2 λ ) Given that A , B , C   and   D are coplanar. ∴ [ A B →   A C →   A D → ] = 0 ⇒ - 4 - 3 3 - 2 λ - 7 λ - 5 4 - 2 λ - 6 0 6 - 2 λ = 0 Expand the determinant along the first row. ⇒ - 4 λ - 5 6 - 2

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