Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Advanced 2023Paper 1
Exam
JEE Advanced
Year
2023
Paper
Paper 1
Subject
Mathematics
Chapter
Three Dimensional Geometry

JEE Advanced 2023 · Paper 1 · Mathematics · Three Dimensional Geometry

Let Q be the cube with the set of vertices \ ( x _ 1 , x _ 2 , x _...

Let QQ be the cube with the set of vertices {(x1,x2,x3)∈R3:x1,x2,x3∈{0,1}}\left\{\left({x}_{1}, {x}_{2}, {x}_{3}\right) \in {\mathbb{R}}^{3}: {x}_{1}, {x}_{2}, {x}_{3} \in \left\{0, 1\right\}\right\}. Let FF be the set of all twelve lines containing the diagonals of the six faces of the cube QQ. Let SS be the set of all four lines containing the main diagonals of the cube QQ; for instance, the line passing through the vertices (0,0,0)\left(0, 0, 0\right) and (1,1,1)\left(1, 1, 1\right) is in SS. For lines ℓ1andℓ2{ℓ}_{1} and {ℓ}_{2}, let dd\left({ℓ}_{1}, {ℓ}_{2}\right)denotetheshortestdistancebetweenthem.Thenthemaximumvalueofd(ℓ1,ℓ2)denote the shortest distance between them. Then the maximum value ofd\left({ℓ}_{1}, {ℓ}_{2}\right), as ℓ1{ℓ}_{1} varies over Fandℓ2F and {ℓ}_{2} varies over SS, isLet 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\ \ .

Options

  1. A16\dfrac{1}{\sqrt{6}}
  2. B18\dfrac{1}{\sqrt{8}}
  3. C13\dfrac{1}{\sqrt{3}}
  4. D112\dfrac{1}{\sqrt{12}}

Correct answer

A. 16\dfrac{1}{\sqrt{6}}

Step-by-step solution

Plotting the diagram of cube we get,

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\ \ .

Now equation of ODOD line will be,

\overset{\to}{r} = \overset{\to}{0} + \lambda\left(\overset{^}{i} + \overset{^}{j}\right)

And equation of diagonal BEBE will be,

\overset{\to}{r}1 = \overset{^}{j} + \mu\left(\overset{^}{i} - \overset{^}{j} + \overset{^}{k}\right)

Now finding the shortest distance between line OD and BEOD \text{ and } BE we get,

S . D = \left|\dfrac{\left(\overset{^}{j} - 0\right) \cdot \left[\left(\overset{^}{i} + \overset{^}{j}\right) \times \left(\overset{^}{i} - \overset{^}{j} + \overset{^}{k}\right)\right]}{\sqrt{{1}^{2} + {1}^{2} + {0}^{2}} \cdot \sqrt{{1}^{2} + {1}^{2} + {1}^{2}}}\right|

⇒ S . D = \left|\dfrac{\overset{^}{j} \cdot \left(\overset{^}{i} - \overset{^}{j} - 2\overset{^}{k}\right)}{\sqrt{6}}\right| = \dfrac{1}{\sqrt{6}}

Now in other case shortest distance will be zero.

Practice Three Dimensional Geometry on Quantrex Academy →

More from Three Dimensional Geometry

Let P be the plane such that it contains the straight line (x-1)/(2) = (y-3)/(3) =… JEE Advanced 2026Let L be the straight line joining the points P(1, 2, -1) and Q(2, 3, 1) . Let S be the… JEE Advanced 2026Let the foot of perpendicular from the point (λ, 2, 3) on the line (x-4)/(1) = (y-9)/(2)… JEE Main 2026Let a line L_1 pass through the origin and be perpendicular to the lines L_2: r = (3+t) i… JEE Main 2026The shortest distance between the lines (x-4)/(1) = (y-3)/(2) = (z-2)/(-3) and (x+2)/(2)… JEE Main 2026Let the image of the point P(0, -5, 0) in the line (x-1)/(2) = (y)/(1) = (z+1)/(-2) be… JEE Main 2026Let the image of the point P(1, 6, a) in the line L: (x)/(1) = (y-1)/(2) = (z-a+1)/(b)… JEE Main 2026Let a line L be perpendicular to both the lines L_1: (x+1)/(3) = (y+3)/(5) = (z+5)/(7)… JEE Main 2026Let a triangle PQR be such that P and Q lie on the line (x+3)/(8) = (y-4)/(2) = (z+1)/(2)… JEE Main 2026If the distance of the point (a, 2, 5) from the image of the point (1, 2, 7) in the line… JEE Main 2026The square of the distance of the point P(5, 6, 7) from the line (x-2)/(2) = (y-5)/(3) =… JEE Main 2026The square of the distance of the point of intersection of the lines r = ( i + j - k ) +… JEE Main 2026The shortest distance between the lines r =((1)/(3) i +2 j +(8)/(3) k )+λ(2 i -5 j +6 k )… JEE Main 2026If (2α+1, α^2-3α, (α-1)/(2)) is the image of (α, 2α, 1) in the line… JEE Main 2026A line with direction ratios 1, -1, 2 intersects the lines (x)/(2) = (y)/(3) = (z+1)/(3)… JEE Main 2026The square of the distance of the point (-2, -8, 6) from the line (x-1)/(1) = (y-1)/(2) =… JEE Main 2026 Full Three Dimensional Geometry list Three Dimensional Geometry JEE Maths PYQsJEE Mathematics PYQs All JEE Advanced PYQs