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Question #75717

Difficulty: 5/10 | 🟢 Free

The Question

Let $L_1$ be a line passing through the point $P(1, 0, -1)$ with direction vector $\vec{d}_1 = x\hat{i} + |x-4|\hat{j} + \hat{k}$ , where $x \in \mathbb{R}$ . The line $L_1$ is parallel to the plane $\Pi: 2x - y - 2z = 5$ . Let $L_3$ be another line passing through the points $Q(2, 1, 0)$ and $R(3, 3, 2)$ . If $\theta$ is the acute angle between the lines $L_1$ and $L_3$ , then the value of $\cos \theta$ is:

Options

A.
$\frac{8}{9}$
B.
$\frac{4}{\sqrt{41}}$
C.
$\frac{2}{3}$
D.
$\frac{5}{\sqrt{41}}$

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

Unit Vector Normalization|Angle between two lines|Modulus Function Breakdown

Why This Question Is Tricky

Examiner Levers

Extraneous Data

Provides unnecessary variables (Red Herrings) to test conceptual focus.

Difficulty Boost : +3

Common Mistakes

This question tests the student's ability to maintain parameter consistency across piecewise functions (Modulus Function Breakdown) within a 3D geometry context.

The primary trap occurs when solving the orthogonality condition $|x-4| = 2x - 2$ .

Students commonly square both sides to get $x^2 - 8x + 16 = 4x^2 - 8x + 4$ , yielding $x = \pm 2$ .

If they fail to verify the roots against the domain of the modulus or the non-negativity of the RHS, they will incorrectly accept the extraneous root $x = -2$ .

Using $x = -2$ produces an incorrect direction vector $\langle -2, 6, 1 \rangle$ , which perfectly leads to the trap option $\frac{4}{\sqrt{41}}$ .

The points $P$ and the constant $5$ in the plane equation act as extraneous data to distract from the core algebraic constraint.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Vector Algebra - Class 12
Concept	Unit Vector Normalization
Difficulty	5
Cognitive Load	6
Target Solve Time	150 sec
Access	Free