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Question #40794

Difficulty: 8/10 | 🟢 Free

The Question

Let $P$ be the parabola $x^2 - 2x + 4y - 11 = 0$ and $AB$ be its chord of contact drawn from the point $M(4, 1)$ . Let $S$ be the region bounded by the parabola $P$ and the chord $AB$ . Evaluate the integral $E = 540 \iint_S \left[ \frac{x}{2} \right] dy \, dx$ where $[ \cdot ]$ denotes the greatest integer function.

Options

A.
270
B.
180
C.
360
D.
25

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

Parabola Shifted Vertex|Chord of Contact Logic|Greatest Integer Function (GIF)|Concavity Analysis

Why This Question Is Tricky

Common Mistakes

This question tests the generalized application of the chord of contact ( $T=0$ ) for a shifted parabola, penalizing students who incorrectly apply origin-vertex formulas.

The core trap lies in the double integral over the bounded region $S$ with a Greatest Integer Function (GIF) integrand.

Students must correctly analyze the concavity of the parabola to establish the upper and lower bounds ( $y_{upper}$ vs $y_{lower}$ ) to avoid negative area artifacts.

The GIF forces a split-case integration at $x=4$ , acting as a step-function boundary.

Failing to split the integral or ignoring the $1/4$ scaling factor from the parabola's equation leads directly to the secondary trap options (180 or 360).

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Conic Sections - Class 11
Concept	Parabola Shifted Vertex
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free