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Question #41474

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{a} = \sqrt{3}\hat{i} - \hat{j}$ and $\vec{b}(x) = (\sin x)\hat{i} + (\cos x)\hat{j} + (x^2 e^x)\hat{k}$ be two vectors. Consider the integral $I = \int_0^\pi \sqrt{|\vec{a}|^2 |\vec{b}(x)|^2 - |\vec{a} \times \vec{b}(x)|^2} \, dx$ The value of $I$ is:

Options

A.
$4$
B.
$2\sqrt{3}$
C.
$2$
D.
$4 - 2\sqrt{3}$

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

The Sign-Consistency Principle|Lagrange Identity|Dot Product Projection Sign

Why This Question Is Tricky

Common Mistakes

This question tests the application of Lagrange's Identity and the Sign-Consistency Principle.

The core trap lies in the simplification of $\sqrt{(\vec{a} \cdot \vec{b})^2}$ .

Students frequently simplify this to $\vec{a} \cdot \vec{b}$ instead of the absolute value $|\vec{a} \cdot \vec{b}|$ .

Because the dot product projection $\sqrt{3}\sin x - \cos x$ changes sign at $x = \pi/6$ within the integration domain $[0, \pi]$ , omitting the absolute value leads to the incorrect answer $2\sqrt{3}$ (the net signed area) rather than the correct total area $4$ .

The $x^2 e^x$ term in the $\hat{k}$ component acts as a distractor to discourage direct expansion of the cross product, forcing reliance on vector identities.

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