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Question #40610

Difficulty: 8/10 | 🟢 Free

The Question

Let $\lambda$ be the rank of the word "PRIME" in the dictionary formed by permuting its letters in alphabetical order. Consider the two lines in 3D space given by: $L_1: \frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4}$ $L_2: \frac{x - 2}{3} = \frac{y - 4}{4} = \frac{z - 6}{\lambda - 89}$ The shortest distance between $L_1$ and $L_2$ is:

Options

A.
0
B.
$\frac{1}{\sqrt{5}}$
C.
$\frac{1}{\sqrt{33}}$
D.
$\frac{2}{\sqrt{29}}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Rank of a Word|Coplanarity Condition|Shortest distance between skew lines

Why This Question Is Tricky

Common Mistakes

This question fuses combinatorics (Rank of a Word) with 3D Vector Geometry.

The primary trap lies in the scalar triple product $[\vec{a}_2 - \vec{a}_1, \vec{b}_1, \vec{b}_2]$ .

Students who blindly apply the shortest distance formula without checking for coplanarity first will waste time calculating the cross product magnitude.

Furthermore, a single off-by-one error in the dictionary rank calculation (e.g., getting 93 or 95) will lead to a non-zero scalar triple product, perfectly matching the meticulously crafted distractor options (B and C).

This tests both computational precision in permutations and edge-case recognition in vector geometry.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Permutations and Combinations - Class 11
Concept	Rank of a Word
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free