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Question #75880

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{a}, \vec{b}, \vec{c}$ be mutually orthogonal unit vectors. Let $\vec{r}(x) = (x^3 - 3px^2)\vec{b} + (x^3 - 3px^2)\vec{c} + \vec{a}$ , where $p > 0$ is a constant. Let $f(x) = |\vec{a} \times (\vec{r}(x) \times \vec{a})|$ . Let $x_0 > 0$ be the point of local minimum of the function $f(x)$ . If the vectors $\vec{u} = \vec{a}$ , $\vec{v} = \vec{b}$ , and $\vec{w} = \vec{a} + \vec{b} + \left( \lim_{x \to x_0} \frac{1 - \cos(x - x_0)}{(f(x))^2} - k \right) \vec{c}$ are coplanar, then the value of $k$ is:

Options

A.
$\frac{1}{324 p^4}$
B.
$\frac{1}{162 p^4}$
C.
0
D.
$\frac{1}{81 p^4}$

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

Coplanarity Condition|Vector Triple Product Expansion|L'Hopital's Pitfall|First Derivative Test Trap

Why This Question Is Tricky

Examiner Levers

Variable Manipulation

Algebraic solutions only (No numbers); tests pure symbolic logic.

Difficulty Boost : +4

Common Mistakes

This question tests the 'First Derivative Test Trap' and 'L'Hopital's Pitfall' within vector calculus.

Students often ignore the absolute value generated by the vector magnitude, incorrectly applying the first derivative test to the signed polynomial $h(x) = x^3 - 3px^2$ .

This leads them to falsely identify $x=2p$ as the local minimum, whereas the absolute value reflects this into a local maximum, making the non-differentiable root $x=3p$ the true local minimum.

Furthermore, the limit requires careful expansion; applying L'Hopital's rule prematurely without squaring the VTP magnitude denominator leads to undefined or incorrect limits due to the cusp at $x=3p$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Vector Algebra - Class 12
Concept	Coplanarity Condition
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free