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Question #41346

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{a}(t)$ and $\vec{b}(t)$ be twice differentiable vector-valued functions such that at $t=1$ : $\vec{a}(1) = \vec{b}(1) = 2\hat{i} + \hat{j} + \hat{k}$ $\vec{a}'(1) = 2\hat{i} + 2\hat{j} + 3\hat{k}, \quad \vec{b}'(1) = \hat{j} + 2\hat{k}$ $\vec{a}''(1) = 2\hat{j} + 6\hat{k}, \quad \vec{b}''(1) = \vec{0}$ Let $\vec{c} = \hat{i} - \hat{j} + 2\hat{k}$ . Evaluate the limit: $\lim_{t \to 1} \frac{|\vec{a}(t) \times \vec{b}(t)|}{|[\vec{a}(t), \vec{b}(t), \vec{c}]|}$ (where $[\vec{x}, \vec{y}, \vec{z}]$ denotes the scalar triple product).

Options

A.
$\frac{\sqrt{5}}{3}$
B.
$\frac{2\sqrt{5}}{3}$
C.
$\frac{\sqrt{5}}{6}$
D.
$\frac{5}{3}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Rate of Change Dependency|Geometric Interpretation of Cross Product|Collinearity vs Coplanarity

Why This Question Is Tricky

Common Mistakes

This question tests the 'Lethal Collision' trap in vector kinematics.

Students often assume that if two moving vectors collide (become identical), the rate of change of their cross product is simply determined by the cross product of their velocities ( $\vec{a}' \times \vec{b}'$ ).

However, the first derivative of the cross product vanishes entirely here because the relative velocity $\vec{a}' - \vec{b}'$ is collinear with the position vector.

This forces the student to recognize the instantaneous collinearity trap and compute the second derivative (involving acceleration and mixed velocity terms) to resolve the $0/0$ indeterminate limit via Taylor expansion or L'Hôpital's rule, distinguishing between the vanishing of the cross product (collinearity) and the scalar triple product (coplanarity).

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Derivatives - Class 12
Concept	Rate of Change Dependency
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free