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Question #41487

Difficulty: 8/10 | 🟢 Free

The Question

Consider the curves $C_1$ and $C_2$ defined parametrically by $C_1: x = t + \sin t, y = \cos t$ and $C_2: x = t + \sin t, y = \cos 2t$ . Let $P$ be the point of intersection of $C_1$ and $C_2$ on the $y$ -axis, and $Q$ be their next point of intersection with $x > 0$ . The area of the region bounded by $C_1$ and $C_2$ between $P$ and $Q$ is:

Options

A.
$\frac{\pi}{3} + \frac{3\sqrt{3}}{8}$
B.
$\frac{\pi}{2}$
C.
$\frac{2\pi}{3} + \frac{3\sqrt{3}}{4}$
D.
$\frac{\pi}{3} + \frac{\sqrt{3}}{4}$

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

Intersection Point Accuracy|Parametric Curve Area|Algebraic-Trigonometric Graphical Mapping|General Solution Principal Values

Why This Question Is Tricky

Common Mistakes

The problem tests the student's ability to find the area between parametric curves while navigating the trap of general trigonometric solutions.

A common mistake is assuming parametric curves can intersect at different parameter values, but here $x(t)$ is strictly increasing, guaranteeing intersections only occur at identical $t$ values.

Furthermore, students often misidentify the intersection bounds by incorrectly solving the trigonometric equation, leading to erroneous limits like $t=\pi$ .

The integration also requires careful application of trigonometric product-to-sum formulas, acting as a secondary accuracy filter.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Integrals - Class 12
Concept	Intersection Point Accuracy
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free