Before You Begin

Question #41447

Difficulty: 8/10 | 🟢 Free

The Question

The family of curves $C_1$ satisfies the differential equation $\frac{dy}{dx} = \frac{\cos x}{y \sin x - \sin^4 x \cos^8 x}$ . Let $C_2$ be the orthogonal trajectory to $C_1$ that passes through the point $(\pi, 0)$ . If the area of the region bounded by the curve $C_2$ , the x-axis, and the line $x = \frac{3\pi}{2}$ is $A$ , then $A$ equals:

Options

A.
$\frac{3\pi}{512} - \frac{8}{693}$
B.
$\frac{3\pi}{512} + \frac{8}{693}$
C.
$\frac{3\pi}{256} - \frac{8}{693}$
D.
$\frac{3\pi}{256} + \frac{8}{693}$

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

Wallis' Formula|Orthogonal Trajectories|Integrating Factor (IF) Sign Sensitivity

Why This Question Is Tricky

Common Mistakes

The problem tests the integration of orthogonal trajectories leading to a linear ODE, combined with a rigorous application of Wallis' formula on a shifted interval.

The primary trap lies in the evaluation of the area integral using integration by parts.

Students frequently ignore the phase shift when applying Wallis' formula to $\int_{\pi}^{3\pi/2} \sin^5 x \cos^6 x dx$ .

Because the power of sine is odd, the substitution $x = \pi + w$ introduces a negative sign, making the integral $-8/693$ .

Blindly applying the standard Wallis formula without adjusting for the quadrant yields $+8/693$ , leading directly to the incorrect Option B.

Additionally, the absolute value in the Integrating Factor $e^{\int \tan x dx} = |\sec x|$ tests rigorous ODE solving, though the boundary condition absorbs the sign.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Wallis' Formula
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free