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Question #71647

Difficulty: 8/10 | 🟢 Free

The Question

Let $k$ be the area of the region bounded by the curve $y = \frac{x \sin x}{1+\cos^2 x}$ , the x-axis, and the lines $x=0, x=\pi$ . A variable plane $P(\theta)$ is defined by the equation: $(1-\cos 2\theta)x + (1+\cos 2\theta)y + 2z = 4 - \frac{2k \sec^2 \theta \tan^2 \theta}{1 + e^{\tan^3 \theta}}$ The line $L: x-1 = y-1 = \frac{1-z}{2} = t$ intersects the plane $P(\theta)$ at a point $Q(\theta)$ corresponding to the parameter $t = t(\theta)$ . Find the value of the integral $\int_{-\pi/4}^{\pi/4} t(\theta) d\theta$ .

Options

A.
$\frac{\pi^2}{12}$
B.
$\frac{\pi^2}{6}$
C.
$\frac{\pi^2}{24}$
D.
$\frac{\pi^2}{8}$

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

Line-Plane intersection logic|Variable plane intercepts|Substitution (Change of Variable)|Even and Odd Function Symmetry

Why This Question Is Tricky

Examiner Levers

Recursive Logic

The answer to part A is required to solve part B (Step-wise failure test).

Difficulty Boost : +5

Common Mistakes

This question tests the student's ability to decouple geometric intersection logic from advanced definite integration properties.

The trap lies in the variable plane equation, which initially appears as a complex 3D geometry problem but collapses into a single parameter $t(\theta)$ upon line substitution.

The integration of $t(\theta)$ acts as a Jacobian-style gatekeeper, requiring a non-linear substitution ( $u = \tan \theta$ ) followed by the application of the even/odd symmetry property $\int_{-a}^{a} \frac{E(x)}{1+a^{O(x)}} dx = \int_{0}^{a} E(x) dx$ .

A step-wise failure test is embedded by making the plane's constant term dependent on a prior integral $k$ , forcing recursive accuracy.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Three-Dimensional Geometry - Class 12
Concept	Line-Plane intersection logic
Difficulty	8
Cognitive Load	7
Target Solve Time	170 sec
Access	Free