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 line

Question

Let Q  be the cube with the set of vertices x 1 , &#160; x 2 , &#160; x 3 &#8712; &#8477; 3 &#160; : &#160; x 1 , &#160; x 2 , &#160; x 3 &#8712; 0 , &#160; 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 , &#160; 0 , &#160; 0  and 1 , &#160; 1 , &#160; 1 is in S . For lines &#8467; 1  and &#8467; 2 , let d &#8467; 1 , &#160; &#8467; 2 denote the shortest distance between them. Then the maximum value of d &#8467; 1 , &#160; &#8467; 2 , as &#8467; 1  varies over F and &#8467; 2  varies over S , is

Quantrex Academy · Accepted Answer

Correct answer: A. 1 6

Plotting the diagram of cube we get, Now equation of O D line will be, r &#8594; = 0 &#8594; + &#955; i ^ + j ^ And equation of diagonal B E  will be, r &#8594; 1 = j ^ + &#956; i ^ - j ^ + k ^ Now finding the shortest distance between line  O D &#160; &#38; &#160; B E  we get, S . D = j ^ - 0 &#183; i ^ + j ^ &#215; i ^ - j ^ + k ^ 1 2 + 1 2 + 0 2 &#183; 1 2 + 1 2 + 1 2 &#8658; S . D = j ^ &#183; i ^ - j ^ - 2 k ^ 6 = 1 6 Now in other case shortest distance will be zero.

JEE Advanced Mathematics Question (2023) — Solution

Question

Options

Answer

Step-by-step solution