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Question #41923

Difficulty: 8/10 | 🟢 Free

The Question

Let $T_k(x) = (x^2+x) \log_2\left( \frac{(k+1)^2}{k(k+2)} \right)$ for $k \in \mathbb{N}$ , and define $S(x) = \sum_{k=1}^{\infty} T_k(x)$ . Consider the following vectors in $\mathbb{R}^3$ : $\vec{v}_1 = [S(x)] \hat{i} + S(x) \hat{j} + \hat{k}$ $\vec{v}_2 = S(x) \hat{i} + [S(x)] \hat{j} + \hat{k}$ $\vec{v}_3 = \hat{i} + \hat{j} + \hat{k}$ where $[ \cdot ]$ denotes the greatest integer function. Find the number of values of $x \in [-2, 2]$ for which the vectors $\vec{v}_1, \vec{v}_2, \vec{v}_3$ are linearly dependent.

Options

A.
5
B.
7
C.
10
D.
Infinitely many

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

Linear Independence vs Dependence|Sum to n terms vs. General Term|Integral Part and Fractional Part

Why This Question Is Tricky

Common Mistakes

The question tests the intersection of infinite series evaluation, linear algebra, and discrete step-functions.

The primary trap lies in the determinant condition for linear dependence: students often ignore the discontinuous nature of the greatest integer function $[S(x)]$ , treating it as a continuous variable $S(x)$ .

This leads to a determinant of zero everywhere, falsely implying 'Infinitely many' solutions.

Furthermore, even if they correctly identify that $S(x)$ must be an integer, a secondary trap awaits: assuming $x$ itself must be an integer, which yields only 5 solutions.

The rigorous path requires solving the quadratic $x^2+x = k$ for $k \in \{0,1,..,6\}$ and carefully filtering the roots within the domain $[-2, 2]$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Vector Algebra - Class 12
Concept	Linear Independence vs Dependence
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free