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Question #41019

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{a}, \vec{b}, \vec{c}$ be three non-coplanar vectors such that the volume of the parallelepiped formed by them is $325$ . A point $P$ with position vector $\vec{r} = x\vec{a} + y\vec{b} + z\vec{c}$ is chosen uniformly at random inside this parallelepiped. What is the volume of the region consisting of all such points $P$ for which line segments of lengths $x, y,$ and $z$ can form an acute-angled triangle?

Options

A.
$325 \left(1 - \frac{\pi}{4}\right)$
B.
$325 \left(1 - \frac{\pi}{6}\right)$
C.
$\frac{325}{2}$
D.
$325 \left(1 - \frac{\pi}{3}\right)$

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

Equiprobability Fallacy|Scalar Triple Product Volume Logic|Triangle Inequality Mind Games

Why This Question Is Tricky

Common Mistakes

This question tests the 'Equiprobability Fallacy' by tempting students to analyze the geometric lengths and angles of the vector $\vec{r}$ itself, rather than recognizing that the coefficients $(x,y,z)$ are uniformly distributed in a unit cube and mapped via a constant Jacobian.

Furthermore, the 'Triangle Inequality Mind Games' trap is deployed by asking for an acute triangle condition, which creates a non-convex domain bounded by conical surfaces.

Students often mistakenly assume linear boundaries, attempt to use Heron's formula, or fail to realize the three non-acute regions are mutually exclusive, leading to incorrect integration limits or overlapping volume subtractions.

Quick Facts About This Question

Question Information

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