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Question #41320

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all $x \in (0, 4\pi)$ for which there exists at least one real number $y$ such that the point $(x, y)$ lies on the curve $2\cos(y) + \cos(2x) = \sqrt{2}$ and the tangent to the curve at $(x, y)$ is parallel to the $x$ -axis. The sum of all elements in $S$ is

Options

A.
$14\pi$
B.
$6\pi$
C.
$8\pi$
D.
$10\pi$

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

Implicit Differentiation Implicit Domain|Rolle's Theorem Anchors|Extraneous Solution Rejection

Why This Question Is Tricky

Common Mistakes

This question tests the 'Extraneous Solution Rejection' trap within the context of implicit differentiation.

Students often mechanically differentiate $F(x,y)=0$ to find $y' = -F_x/F_y$ and solve $F_x = 0$ to find horizontal tangents.

The critical trap is failing to check if the original implicit relation $F(x,y)=0$ actually yields a real value for $y$ at these critical $x$ -coordinates.

By setting up a trigonometric implicit function where half of the derivative's roots fall outside the implicit domain (yielding $\cos(y) > 1$ ), the problem punishes rote application of calculus without verifying the topological existence of the point on the curve.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Continuity and Differentiability - Class 12
Concept	Implicit Differentiation Implicit Domain
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free