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JEE Advanced2023MathematicsThree Dimensional GeometryActual

Let Q be the cube with the set of vertices x 1 , x 2 , x 3 ∈ ℝ 3 : x 1 , x 2 , x 3 ∈ 0 , 1 . Let F be the set of all twelve lines containing the diagonals of the six faces of the cube Q . Let S be the set of all four lines containing the main diagonals of the cube Q ; for instance, the line passing through the vertices 0 , 0 , 0 and 1 , 1 , 1 is in S . For lines ℓ 1 and ℓ 2 , let d &#846

Options

  1. A1 6
  2. B1 8
  3. C1 3
  4. D1 12

Correct answer

A. 1 6

Step-by-step solution

Plotting the diagram of cube we get, Now equation of O D line will be, r → = 0 → + λ i ^ + j ^ And equation of diagonal B E will be, r → 1 = j ^ + μ i ^ - j ^ + k ^ Now finding the shortest distance between line O D   &   B E we get, S . D = j ^ - 0 · i ^ + j ^ × i ^ - j ^ + k ^ 1 2 + 1 2 + 0 2 · 1 2 + 1 2 + 1 2 ⇒ S . D = j ^ · i ^ - j ^ - 2 k ^ 6 = 1 6 Now in other case shortest distance will be zero.

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