Before You Begin

Question #41130

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: \mathbb{R} \to \mathbb{R}$ be defined as $f(x) = \left| \ln \left( \frac{x^2 - x + 1}{x^2 + x + 1} \right) \right|$ . The set of all real values of $k$ for which the equation $f(x) = k$ has exactly $4$ distinct real solutions is:

Options

A.
$(0, \ln 3)$
B.
$(0, \ln 3]$
C.
$(0, 1)$
D.
$(0, \infty)$

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

Bijectivity and Inverse|Logarithmic Differentiation Domain|Modulus Function Breakdown

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle the global domain and branch points of a modulus function applied to a transcendental composite function.

The primary trap is performing logarithmic differentiation without accounting for the modulus, which leads students to miss the artificial critical point at $x=0$ where the argument of the logarithm crosses $1$ .

By ignoring the 'W' or 'M' shape created by the modulus reflection of the negative logarithmic branch, students incorrectly assume the function is monotonic between the natural critical points $x=-1$ and $x=1$ , leading to an incorrect count of solutions for $y=k$ and a failure to properly apply the horizontal line test for bijectivity/injectivity.

Quick Facts About This Question

Question Information

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