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Question #51613

Difficulty: 5/10 | 🟢 Free

The Question

Let $S$ be the set of all real numbers $a$ such that the function $f(x) = \frac{(x^2-1)(x-3)^2}{|x-1|(x+1)(x-3)(x-5)}$ is discontinuous at $x=a$ , but $\lim_{x \to a} f(x)$ exists. Let $g: [0, 3] \to \mathbb{R}$ be defined by $g(x) = x^3 - 3x^2 + 2$ . Let $T$ be the set of all integers $k$ such that the equation $g(x) = k$ has more than one distinct real solution. Find the sum of the elements in the set $S \Delta T$ , where $S \Delta T$ denotes the symmetric difference of sets $S$ and $T$ .

Options

A.
4
B.
5
C.
6
D.
7

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 distinguish between removable and non-removable discontinuities, and their understanding of symmetric difference.

The primary trap lies in evaluating the limit at $x=1$ and $x=-1$ for $f(x)$ ; students often incorrectly assume that the presence of an absolute value $|x-1|$ implies the limit does not exist at $x=-1$ , or they fail to check the left and right-hand limits at $x=1$ .

Additionally, when finding set $T$ , students might incorrectly include $k=-2$ (which only has one solution at the local minimum) or exclude $k=2$ (which has two solutions at the boundary and local maximum), leading to an incorrect symmetric difference computation.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sets - Class 11
Concept	Difference vs. Symmetric Difference
Difficulty	5
Cognitive Load	6
Target Solve Time	140 sec
Access	Free