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Question #41412

Difficulty: 8/10 | 🟢 Free

The Question

Let $P(\theta) = (\sin\theta, \cos\theta, \sin 2\theta)$ be a moving point in 3D space and $\Pi$ be the plane $x + y - z = 1$ . If $d(\theta)$ denotes the perpendicular distance from $P(\theta)$ to the plane $\Pi$ , evaluate the integral $I = \int_{-\pi/2}^{0} \sqrt{3} \, d(\theta) (\cos\theta - \sin\theta) d\theta$

Options

A.
1
B.
$\frac{2}{3}$
C.
$\frac{5}{6}$
D.
$\frac{1}{2}$

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

Substitution (Change of Variable), Distance from point to plane, Line-Plane intersection logic, Projection vs. Coordinate

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of substitution techniques with the geometric definition of distance.

The core trap lies in the absolute value inherent to the point-to-plane distance formula.

When the substitution $t = \sin\theta + \cos\theta$ is applied, the integrand simplifies to $|t - t^2|$ .

Students often blindly drop the absolute value, assuming the expression maintains a constant sign, which leads to integrating $t - t^2$ over $[-1, 1]$ and obtaining $-2/3$ (or $2/3$ ).

Recognizing that the path of the point intersects the plane at $t=0$ (where the distance is zero) requires partitioning the integral to correctly evaluate the piecewise function, rewarding geometric intuition and rigorous algebraic execution.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Substitution (Change of Variable)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free