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Question #41486

Difficulty: 8/10 | 🟢 Free

The Question

Let $x_1$ and $x_2$ ( $x_1 < x_2$ ) be the abscissae of the intersection points of the parabola $P: y = 3x^2 - \frac{3}{2}$ and the hyperbola $H: 4y^2 - 4x^2 = 5$ that lie in the region $y < 0$ . Evaluate the integral $I = \int_{0}^{1} \left| x^2 - x_2^2 \right| \, dx - 5x_2^5$

Options

A.
$\frac{61}{243}$
B.
$\frac{49}{243}$
C.
$\frac{22}{81}$
D.
$\frac{2}{9}$

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

Intersection Point Accuracy|Intersection of Conics|Modulus Inequality Logic|Limit Existence at Discontinuity

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to accurately determine the intersection of conics and use those geometric properties as critical bounds in calculus.

The primary trap lies in the modulus function inside the definite integral.

If a student fails to recognize that the intersection point $x_2 = 1/3$ acts as the non-differentiable turning point of the modulus within the integration limits $[0, 1]$ , they will incorrectly evaluate the integral without splitting it (yielding the trap answer 49/243).

The secondary scalar anchor '61/243' is constructed to prevent trial-and-error guessing and strictly validates the piecewise integration.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Integrals - Class 12
Concept	Intersection Point Accuracy
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free