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Question #72331

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{u}$ and $\vec{v}$ be two variable vectors in $\mathbb{R}^3$ such that $|\vec{u}| = 2$ and $|\vec{v}| = 3$ . Consider the scalar function $f(\vec{u}, \vec{v}) = |\vec{u} \times \vec{v}|^2 - 3(\vec{u} \cdot \vec{v})^2 + 12\vec{u} \cdot (2\vec{v} + \vec{u} \times \vec{v})$ If $M$ and $m$ are the absolute maximum and absolute minimum values of $f(\vec{u}, \vec{v})$ respectively, find the value of $M - m$ .

Options

A.
324
B.
288
C.
36
D.
144

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

Lagrange Identity|Geometric Interpretation of Cross Product|Maximum/Minimum of Quadratic Expressions

Why This Question Is Tricky

Examiner Levers

Negative Thinking

Uses "NOT", "EXCEPT", or "INCORRECT" to flip the logic gate.

Difficulty Boost : +2

Extraneous Data

Provides unnecessary variables (Red Herrings) to test conceptual focus.

Difficulty Boost : +3

Logical Assertion

Tests the causal relationship between two true statements.

Difficulty Boost : +4

Recursive Logic

The answer to part A is required to solve part B (Step-wise failure test).

Difficulty Boost : +5

Common Mistakes

This question tests the application of the Lagrange Identity and the optimization of a resulting quadratic function under vector constraints.

The primary trap lies in the domain of the dot product $x = \vec{u} \cdot \vec{v} \in [-6, 6]$ .

Students often fail to check the boundary conditions (evaluating only at the critical point $x=3$ and one boundary, or assuming $x \in [0, 6]$ because they incorrectly restrict the angle between vectors to acute angles).

Additionally, the term $12\vec{u} \cdot (\vec{u} \times \vec{v})$ acts as extraneous data to test conceptual focus on scalar triple product properties.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Vector Algebra - Class 12
Concept	Lagrange Identity
Difficulty	8
Cognitive Load	8
Target Solve Time	180 sec
Access	Free