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Question #40945

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{u}(t)$ and $\vec{v}(t)$ be two differentiable vector-valued functions in $\mathbb{R}^3$ . At $t=0$ , the following properties are given: $|\vec{u}(0)| = 2$ , $|\vec{v}(0)| = 3$ , and the angle between $\vec{u}(0)$ and $\vec{v}(0)$ is $\cos^{-1}\left(\frac{2}{3}\right)$ . The rates of change of their magnitudes at $t=0$ are $3$ and $-1$ respectively. Additionally, the rate of change of their scalar product at $t=0$ is $5$ . Evaluate the limit: $\lim_{h \to 0} \frac{|\vec{u}(h) \times \vec{v}(h)|^2 - |\vec{u}(0) \times \vec{v}(0)|^2}{h}$

Options

A.
44
B.
124
C.
22
D.
84

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

First Principle of Derivative|Geometric Interpretation of Cross Product|Lagrange Identity

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to connect the First Principle of derivatives with vector calculus, specifically targeting the Lagrange Identity as a computational trap.

By asking for the derivative of the squared magnitude of a cross product, students are forced to evaluate the scalar product of cross product derivatives.

The common mistake is failing to account for the negative sign in Lagrange's Identity $[(\vec{a}\times\vec{b})\cdot(\vec{c}\times\vec{d}) = (\vec{a}\cdot\vec{c})(\vec{b}\cdot\vec{d}) - (\vec{a}\cdot\vec{d})(\vec{b}\cdot\vec{c})]$ , which leads to incorrect addition of the scalar product derivative term (yielding 124 instead of 44).

The problem elegantly disguises the required dot products as rates of change of magnitudes and scalar products, rewarding algebraic discipline and a deep understanding of geometric vector identities.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Limits and Derivatives - Class 11
Concept	First Principle of Derivative
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free