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Question #40523

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all integers $x$ that satisfy the inequality: $\lim_{t \to x} \frac{(1 - \cos(2t - 6)) \cdot (|t^2 - 4t - 5| - 9)}{(t-3)^2 (t^2 + 1)} \le 0$ Find the number of elements in $S$ .

Options

A.
8
B.
9
C.
10
D.
7

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

Even Multiplicity Roots|Modulus Inequality Logic|Limit Existence at Discontinuity

Why This Question Is Tricky

Common Mistakes

This question tests the 'Limit Existence at Discontinuity' trap within a modulus inequality.

Students commonly apply the wavy-curve method or sign-chart logic directly to the expression, noting that the denominator $(t-3)^2$ makes the function undefined at $x=3$ .

Consequently, they incorrectly exclude $x=3$ from the solution set, yielding 8 integers.

However, the inequality is defined via a limit.

Because the zero at $x=3$ has an even multiplicity in both the numerator (via the cosine expansion) and the denominator, the limit exists, is negative, and satisfies the inequality, making $x=3$ a valid solution.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Inequalities - Class 11
Concept	Even Multiplicity Roots
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free