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Question #41570

Difficulty: 8/10 | 🟢 Free

The Question

Let $S_k$ be the region in $\mathbb{R}^3$ defined by the inequalities: $0 \le 2kx + y + z \le 1$ $0 \le x + 2ky + z \le 1$ $0 \le x + y + 2kz \le 1$ Let $V(k)$ denote the volume of the region $S_k$ . Evaluate the integral: $I = \int_{-2}^{0} \frac{1}{V(k)} \, dk$

Options

A.
22
B.
16
C.
-16
D.
24

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

Region Ambiguity|Scalar Triple Product Volume Logic|Scalar Multiplication Geometric Scaling

Why This Question Is Tricky

Common Mistakes

This question tests the 'Region Ambiguity' and 'Scalar Multiplication Geometric Scaling' traps in 3D coordinate transformations.

Students must recognize that the volume of a parallelepiped defined by linear inequalities requires the absolute value of the Jacobian determinant (Scalar Triple Product).

The lethal trap lies in the integration: failing to account for the absolute value implies the region can have 'negative volume' when the scaling factor $k$ causes the orientation of the basis vectors to invert (at $k < -1$ ).

Integrating without splitting the domain at the root $k=-1$ yields the erroneous result of -16 or 16, punishing algorithmic calculation devoid of geometric intuition.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Integrals - Class 12
Concept	Region Ambiguity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free