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Question #40450

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all real numbers $x$ that satisfy the equation $(x^2 - 4x + 4) \sin\left(\frac{1}{x-2}\right) = \sqrt{x^2 - 4x + 4}$ Find the number of elements in $S$ .

Options

A.
0
B.
1
C.
2
D.
\text{Infinitely many}

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

Extraneous Solution Rejection|Squeeze Theorem (Sandwich Theorem)|Limit Existence at Discontinuity

Why This Question Is Tricky

Common Mistakes

This question tests the candidate's ability to distinguish between limiting behavior and functional domain, a classic JEE Advanced trap.

The Squeeze (Sandwich) Theorem is required to evaluate the limit of the LHS as $x \to 2$ , which perfectly matches the RHS limit of 0.

This simulates a removable discontinuity, tempting students to accept $x=2$ as a valid solution.

However, the original equation is undefined at $x=2$ , necessitating Extraneous Solution Rejection.

Furthermore, algebraic simplification for $x \neq 2$ yields equations of the form $\sin(u)=u$ and $\sin(v)=-v$ for non-zero arguments, which have no solutions.

The trap effectively catches students who 'fill the hole' without verifying the initial domain.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Trigonometric Functions - Class 11
Concept	Extraneous Solution Rejection
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free