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Question #41579

Difficulty: 8/10 | 🟢 Free

The Question

Let $P_1$ and $P_2$ be the planes $x+y+z=1$ and $x-2y=2$ respectively. Let $\Pi$ be a plane passing through the line of intersection of $P_1$ and $P_2$ such that the perpendicular distance from the origin to $\Pi$ is maximized. Consider the vector fields $\vec{u} = x\hat{i} + y\hat{j} + \hat{k}$ , $\vec{v} = y\hat{i} - x\hat{j} + \hat{k}$ , and $\vec{w} = \hat{i} - \hat{j} + \hat{k}$ . Let $S$ be the area of the portion of the plane $\Pi$ that satisfies the inequality $[\vec{u} \;\; \vec{v} \;\; \vec{w}] \ge -3$ . The value of $S$ is:

Options

A.
$4\pi \sqrt{6}$
B.
$4\pi \sqrt{3}$
C.
$3\pi \sqrt{6}$
D.
$4\pi$

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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

Compound Function Boundaries|Determinant as Geometric Area/Volume|Equation of plane through intersection|Parametric form pitfalls

Why This Question Is Tricky

Common Mistakes

This question tests the geometric optimization of planes and the evaluation of surface integrals over domains defined by vector products.

Trap 1: Students often use tedious calculus to maximize the distance of the plane from the origin, risking algebraic errors, instead of using the geometric insight that the normal must be parallel to the position vector of the closest point on the intersection line.

Trap 2: The scalar triple product boundary requires careful expansion and completing the square; a common error is misidentifying the radius as $\sqrt{3}$ instead of 2.

Trap 3 (Parametric Pitfall): When calculating the final surface area, students frequently forget the secant factor (Jacobian of the surface) $\sqrt{1+z_x^2+z_y^2} = \sqrt{6}$ , incorrectly answering $4\pi$ (the area of the projected base).

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Integrals - Class 12
Concept	Compound Function Boundaries
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free