QJEEAdv23Ma4f81: JEE Advanced 2023 Mathematics - Class 12 JEEAdv23Ma4f81 Solution: Let be the cube with the set of vertices . Let be the set of…

Question

Verified Solution • 2025–26 Session moderate

Let

Q

be the cube with the set of vertices

\left\{\left(x_1, x_2, x_3\right) \in \mathbb{R}^3: x_1, x_2, x_3 \in\{0,1\}\right\}

. 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

\ell_1

and

\ell_2

, let

d\left(\ell_1, \ell_2\right)

denote the shortest distance between them. Then the maximum value of

d\left(\ell_1, \ell_2\right)

, as

\ell_1

varies over

F

and

\ell_2

varies over

S

, is :

Accepted Answer

\mathrm{DR}^{\prime} \mathrm{S}

of

\mathrm{OG}=1,1,1

DR'S of AF

=-1,1,1

DR'S of CE = 1, 1, -1

\mathrm{DR}^{\prime} \mathrm{S}

of

\mathrm{BD}=1,-1,1

Equation of

\mathrm{OG} \Rightarrow \frac{\mathrm{x}}{1}=\frac{\mathrm{y}}{1}=\frac{\mathrm{z}}{1}

Equation of

\mathrm{AB} \Rightarrow \frac{\mathrm{x}-1}{1}=\frac{\mathrm{y}}{-1}=\frac{\mathrm{z}}{0}

Normal to both the line's

=\left|\begin{array}{ccc}\hat{i} &amp; \hat{j} &amp; \hat{k} \\1 &amp; 1 &amp; 1 \\1 &amp; -1 &amp; 0\end{array}\right|=\hat{i}+\hat{j}-2 \hat{k}

\overrightarrow{\mathrm{OA}}=\hat{\mathrm{i}}

S.D.

=\frac{|\hat{\mathrm{i}} \cdot(\hat{\mathrm{i}}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}})|}{|\hat{\mathrm{i}}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}}|}=\frac{1}{\sqrt{6}}

Step-by-Step Verified Solution