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Question #41892

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{u}$ be a fixed unit vector in $\mathbb{R}^3$ . A solid region $\Omega$ consists of all vectors $\vec{v} \in \mathbb{R}^3$ that satisfy the following two conditions:

$|\vec{v}| \le 3$
$\sqrt{3} (\vec{v} \cdot \vec{u}) > \sqrt{|\vec{v}|^2 - (\vec{v} \cdot \vec{u})^2}$

Find the volume of the region $\Omega$ .

Options

A.
$9\pi$
B.
$18\pi$
C.
$\frac{9\pi}{2}$
D.
$27\pi$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Indicator Function Logic|Dot Product Projection Sign|Modulus Sign Dependency

Why This Question Is Tricky

Common Mistakes

The question tests the student's ability to handle inequalities involving square roots and vector projections.

The common trap is to immediately square both sides of the inequality $\sqrt{3}(\vec{v} \cdot \vec{u}) > \sqrt{|\vec{v}|^2 - (\vec{v} \cdot \vec{u})^2}$ , which algebraically leads to $4\cos^2\theta > 1 \implies |\cos\theta| > 1/2$ .

This incorrectly introduces the region $\theta \in (2\pi/3, \pi]$ where the scalar projection is negative, effectively calculating the volume of a double cone and doubling the answer to $18\pi$ .

Students must recognize that the LHS must be strictly positive before squaring, enforcing the sign dependency of the scalar component.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 11
Concept	Indicator Function Logic
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free