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Question #41872

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all permutations of the letters of the word "PRIME". Let $a_1$ be the number of words in $S$ where the vowels are together. Let $a_2$ be the number of words in $S$ where 'P' is adjacent to 'R', and 'M' is adjacent to 'E'. Let $a_3$ be the number of words in $S$ where 'I' is exactly in the middle position. Let $b_1$ be the number of words in $S$ where 'E' is at the rightmost position. Let $b_3$ be the number of words in $S$ where all consonants are together, and all vowels are together.

Consider two lines $L_1$ and $L_2$ in 3D space. Line $L_1$ is parallel to the vector $\vec{u} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}$ . Line $L_2$ is parallel to the vector $\vec{v} = (a_2 - a_1) \hat{i} + (b_1 - a_1 - a_2) \hat{j} + b_3 \hat{k}$ . If $\theta$ is the angle between the lines $L_1$ and $L_2$ , then $\theta$ is equal to:

Options

A.
$\frac{\pi}{3}$
B.
$\frac{2\pi}{3}$
C.
$\cos^{-1}\left(\frac{1}{6}\right)$
D.
$\frac{\pi}{6}$

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

String Method (Adjacency)|Angle between two lines|Dot Product Projection Sign

Why This Question Is Tricky

Common Mistakes

This question tests the 'Dot Product Projection Sign' trap embedded within a multi-step combinatorics obfuscation.

Students must first correctly apply the String Method to find permutation counts, which serve as vector components.

The critical trap lies in the geometric definition: the angle between two lines is strictly acute (or right), $\theta \in [0, \pi/2]$ .

The constructed vectors $\vec{u}$ and $\vec{v}$ have an obtuse angle between them ( $2\pi/3$ ) because their dot product is negative.

If a student blindly applies the vector angle formula $\cos \theta = (\vec{u} \cdot \vec{v}) / (|\vec{u}||\vec{v}|)$ without taking the absolute value of the dot product (which is required for lines), they will incorrectly select $2\pi/3$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Permutations and Combinations - Class 11
Concept	String Method (Adjacency)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free