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Question #41535

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{a}, \vec{b}, \vec{c}$ be three non-coplanar vectors such that $[\vec{a}, \vec{b}, \vec{c}] = \sqrt{3}$ . A varying vector $\vec{p}(t)$ is defined as $\vec{p}(t) = (\sin t)\vec{a} + (\sin t - \cos t)\vec{b} + (\cos 2t)\vec{c}$ for $t \in [0, 2\pi]$ . Let $V(t)$ be the volume of the parallelepiped formed by the vectors $\vec{a}+\vec{b}$ , $\vec{b}+\vec{c}$ , and $\vec{p}(t)$ . If $S$ is the set of all $t \in [0, 2\pi]$ such that the angle between $\vec{p}(t)$ and $(\vec{a}+\vec{b}) \times (\vec{b}+\vec{c})$ is obtuse, find the value of $\int_S V(t) dt$ .

Options

A.
$\frac{9}{2}$
B.
9
C.
$-\frac{9}{2}$
D.
$3\sqrt{3}$

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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

The Sign-Consistency Principle|Scalar Triple Product Volume Logic|Dot Product Projection Sign|Scalar Multiplication Geometric Scaling

Why This Question Is Tricky

Common Mistakes

This problem tests the 'Sign-Consistency Principle' in vector volume integrals.

Students often mechanically compute the scalar triple product and integrate it directly, ignoring that volume is an absolute value.

The trap lies in the dot product projection condition (obtuse angle), which restricts the domain to where the scalar triple product is strictly negative.

A common mistake is integrating the signed scalar triple product without taking its absolute value, yielding -9/2, or ignoring the domain restriction and integrating over the entire period, yielding 0 or 9.

This enforces a rigorous understanding of geometric orientation and piecewise volume accumulation.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Integrals - Class 12
Concept	The Sign-Consistency Principle
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free