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Question #40311

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: \mathbb{R} \to \mathbb{R}$ be defined as: $f(x) = \begin{cases} \frac{\sin(\pi x)}{x-8} + \frac{\ln(1 + (x-10)^2)}{(x-10)^2} + \frac{|x-13|}{x-13}, & x \notin \{8, 10, 13\} \\ 2024, & x \in \{8, 10, 13\} \end{cases}$ Let $S$ be the set of all points $c \in \mathbb{R}$ where $f(x)$ is discontinuous at $x=c$ but $\lim_{x \to c} f(x)$ exists. Let $T$ be the set of all integers $k$ such that the equation $3x^4 - 8x^3 - 6x^2 + 24x = k$ has exactly three distinct real roots. Find the sum of the elements in the set $S \Delta T$ , where $\Delta$ denotes the symmetric difference of two sets.

Options

A.
10
B.
21
C.
23
D.
31

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

Difference vs. Symmetric Difference|Bijectivity and Inverse|Limit Existence at Discontinuity

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to rigorously classify discontinuities and analyze the roots of a polynomial using its derivative.

The primary trap lies in the definition of set $S$ : students often incorrectly assume that any point where the denominator is zero implies the limit does not exist, failing to recognize the removable singularities at $x=8$ and $x=10$ .

Furthermore, the symmetric difference operation $S \Delta T$ acts as a secondary trap, penalizing candidates who merely take the union of the sets.

Finally, finding set $T$ requires careful graphing of a quartic function to identify that exactly three real roots occur only at the local extrema that are not the global minimum.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sets - Class 11
Concept	Difference vs. Symmetric Difference
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free