Before You Begin

Question #68612

Difficulty: 5/10 | 🟢 Free

The Question

Let $S$ be the set of all continuously differentiable functions $f: [0, \pi] \to \mathbb{R}$ satisfying $(f'(x))^2 = \sin^2(2x)$ and $f(0)=0$ . Let $R$ be the region bounded by all the curves in $S$ . Let $S'$ be the subset of $S$ consisting of functions that do NOT satisfy the condition $\int_0^{\pi/2} f(x) \cos(2x) \, dx \ge 0$ , and let $R'$ be the region bounded by the curves in $S'$ . If $A$ and $A'$ are the areas of regions $R$ and $R'$ respectively, what is the value of $A - A'$ ?

Options

A.
$\frac{\pi}{2}$
B.
$\pi$
C.
$\frac{3\pi}{2}$
D.
$2\pi$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Geometric Interpretation|Inequalities in Definite Integrals|Extraneous Solution Rejection|The Sign-Consistency Principle

Why This Question Is Tricky

Examiner Levers

Negative Thinking

Uses "NOT", "EXCEPT", or "INCORRECT" to flip the logic gate.

Difficulty Boost : +2

Common Mistakes

This question tests the student's ability to handle absolute value ODEs and extraneous solutions combined with integral inequalities (Sign-Consistency Principle).

The ODE $(f'(x))^2 = \sin^2(2x)$ implies $f'(x) = \pm \sin(2x)$ .

Because $f'(x)$ must be continuous, it can only change signs at the roots of $\sin(2x)$ , specifically at $x = \pi/2$ .

This creates 4 distinct valid branches for $f(x)$ .

Students often fail to identify all 4 branches, assuming $f'(x)$ must be uniformly $\sin(2x)$ or $-\sin(2x)$ , which would yield an incorrect total area $A$ .

Furthermore, the negative phrasing 'do NOT satisfy' forces students to correctly evaluate the integral inequality to isolate the specific subset $S'$ .

Failing to apply the sign-consistency check or misinterpreting the geometric boundaries of the piecewise functions leads to incorrect area calculations.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Differential Equations - Class 12
Concept	Geometric Interpretation
Difficulty	5
Cognitive Load	6
Target Solve Time	140 sec
Access	Free