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Question #40366

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the domain of the function $f(x) = \ln \left( \frac{3\{x\} - 1}{[x]^2 - 7[x] + 10} \right) + \arccos\left(\frac{x-4}{4}\right)$ , where $[x]$ denotes the greatest integer less than or equal to $x$ , and $\{x\}$ denotes the fractional part of $x$ . If $S$ is expressed as the union of disjoint intervals, then the sum of the lengths of these intervals is:

Options

A.
$\frac{8}{3}$
B.
$\frac{10}{3}$
C.
$\frac{11}{3}$
D.
4

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

Set Complement Boundaries|Domain Restrictions|Greatest Integer Function Traps

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle domain restrictions involving the fractional part and greatest integer functions simultaneously.

The primary trap lies in the rational inequality inside the logarithm.

Students often correctly identify the positive/positive case but completely overlook the negative/negative case (Case 2), forgetting that the fractional part {x} can be less than 1/3 while the quadratic in [x] is negative.

Furthermore, the piecewise nature of {x} requires intersecting the conditions on [x] with the periodic intervals of {x}, rather than treating them as simple continuous inequalities.

Failing to account for the negative/negative case leads directly to the trap option A.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sets - Class 11
Concept	Set Complement Boundaries
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free