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Question #40686

Difficulty: 8/10 | 🟢 Free

The Question

Let $\alpha$ and $\beta$ be the roots of the quadratic equation $x^2 - 2kx + (k^2 + k - 2) = 0$ , where $k \in \mathbb{R}$ . An Arithmetic-Geometric Progression is defined by its partial sum $S_n = \sum_{m=1}^{n} (m|\alpha| + |\beta|) r^{m-1}$ , where the common ratio is $r = \frac{|\alpha + \beta|}{|\alpha| + |\beta|}$ . Let $D$ be the set of all values of $k$ for which $\lim_{n \to \infty} S_n$ exists and is finite. Which of the following represents the set $D$ ?

Options

A.
$(-2, 1) \cup (2, \infty)$
B.
$[-2, 1] \cup [2, \infty)$
C.
$(-\infty, -2) \cup (1, 2)$
D.
$(-2, \infty) \setminus \{1, 2\}$

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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

Arithmetic-Geometric Progression (AGP)|Quadratic Roots in Sequences|Triangle Inequality Mind Games|Closed vs Open Intervals

Why This Question Is Tricky

Common Mistakes

This question traps students who mechanically apply the infinite AGP sum formula without verifying the open interval of convergence ( $|r| < 1$ ).

By using the Triangle Inequality, $r$ is naturally bounded by 1.

Students often mistakenly solve for $r \le 1$ , leading them to include the boundaries $k=-2, 1, 2$ where $r=1$ and the series diverges.

Furthermore, it elegantly combines the condition for complex roots and the opposite-sign condition for real roots, forcing a comprehensive analysis of the quadratic's discriminant and root product.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sequences and Series - Class 11
Concept	Arithmetic-Geometric Progression (AGP)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free