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Question #41146

Difficulty: 8/10 | 🟢 Free

The Question

Let $S = \mathbb{R} \setminus \{\alpha\}$ . A binary operation $*$ is defined on $S$ by $x * y = x y - \lambda x - \lambda y + \lambda^2 + \lambda$ where $\lambda = \sum_{n=2}^{\infty} \frac{2n+6}{n^3-n}$ . If the operation $*$ admits a unique identity element in $S$ , what is the value of $\alpha$ ?

Options

A.
2
B.
3
C.
4
D.
6

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

Identity Element Uniqueness|Method of Differences|Denominator Restriction

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of algebraic structures and the method of differences.

The core trap is the 'Denominator Restriction' in telescoping series.

Students often find the difference function $V_n = \frac{2n+2}{n^2-n}$ and blindly evaluate it at $n=1$ , ignoring that the sum explicitly starts at $n=2$ because $V_1$ is undefined (division by zero).

This leads to incorrect values for $\lambda$ .

Furthermore, the problem requires understanding that for a binary operation $x * y = (x-\lambda)(y-\lambda) + \lambda$ , the identity element $e = \lambda + 1$ is only unique if the annihilator element $x = \lambda$ is excluded from the domain, elegantly linking series convergence to group theory axioms.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 12
Concept	Identity Element Uniqueness
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free