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Question #40539

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all $x \in [0, 3]$ satisfying the inequality $\int_0^x ||t-1| - 1| \, dt \ge \frac{x}{2}$ Find the set $S$ .

Options

A.
{0} \cup [1, 2] \cup {3}
B.
[1, 2] \cup {3}
C.
{0} \cup [1, 2]
D.
[1, 3]

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

Modulus Inequality Logic|Inequalities in Definite Integrals|Modulus Function Breakdown

Why This Question Is Tricky

Common Mistakes

This question tests the 'Modulus Function Breakdown' trap within definite integrals.

Students must partition the integral based on the critical points of the nested modulus ( $t=1$ and $t=2$ ).

The core trap lies in solving the resulting quadratic inequalities within strictly bounded sub-intervals.

Failing to properly intersect the solution of each inequality with its respective domain $[0, 1]$ , $(1, 2]$ , and $(2, 3]$ causes students to miss the isolated boundary points $x=0$ and $x=3$ , or incorrectly include the interval $(2, 3)$ , leading to incomplete or extraneous solution sets.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Inequalities - Class 11
Concept	Modulus Inequality Logic
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free