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Question #40678

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{a}$ and $\vec{b}$ be two unit vectors such that the angle between them is $\frac{\pi}{6}$ . A sequence of position vectors is defined for $k \in \mathbb{N}$ as: $\vec{r}_k = \vec{\Omega} + \sum_{n=1}^{\infty} \left( \frac{k}{k+1} \right)^n \vec{a} + \sum_{n=1}^{\infty} \left( \frac{k^2}{k^2+1} \right)^n \vec{b}$ where $\vec{\Omega} = \left( \lim_{x \to 0} \frac{\sin x - x}{x^3} \right) \vec{a} + \left( \int_{0}^{\pi/2} \ln(\tan x) dx \right) \vec{b} + \left( \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n} \right) (\vec{a} \times \vec{b})$ . Find the area of the triangle formed by the points with position vectors $\vec{r}_1, \vec{r}_2,$ and $\vec{r}_3$ .

Options

A.
$\frac{1}{2}$
B.
1
C.
$\frac{\sqrt{3}}{4}$
D.
$\frac{1}{4}$

Check Your Answer

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

Infinite GP Convergence|Geometric Interpretation of Cross Product|Position Vector Origin Dependence

Why This Question Is Tricky

Common Mistakes

This question tests the student's understanding of translational invariance in vector geometry, specifically the 'Position Vector Origin Dependence' trap.

The position vectors $\vec{r}_k$ are defined with a highly complex constant vector $\vec{\Omega}$ and infinite geometric progressions.

A student who relies on brute-force procedural calculation will waste significant time evaluating the limits, integrals, and series in $\vec{\Omega}$ , and then attempt to compute the area using the origin-dependent formula $\frac{1}{2}|\vec{r}_1 \times \vec{r}_2 + \vec{r}_2 \times \vec{r}_3 + \vec{r}_3 \times \vec{r}_1|$ .

The trap is failing to recognize that the area of a triangle is invariant under origin shifts.

By defining a new origin at $\vec{\Omega}$ , the constant vector completely vanishes, and the infinite GPs simplify to basic polynomials $k$ and $k^2$ , reducing a seemingly monstrous problem to a trivial cross product.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sequences and Series - Class 11
Concept	Infinite GP Convergence
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free