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Question #41421

Difficulty: 8/10 | 🟢 Free

The Question

Let $H$ be the rectangular hyperbola $x^2 - y^2 = 2$ . A curve $C$ is obtained by rotating $H$ about the origin by an angle of $\pi/4$ clockwise. Let $P(x, y)$ be a point on $C$ in the first quadrant with $x \ge 1$ , and let $D(x)$ be the distance from $P$ to the origin. Define the integral: $I(\lambda) = \int_{1}^{\infty} \frac{x^2 - 1}{x^2 D(x)^2 + (\lambda - 2)x^2} \, dx$ Using the method of differentials, the approximate value of $I(4.48)$ is:

Options

A.
0.48
B.
0.46
C.
0.52
D.
0.44

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

Partial Fractions Decomposition|Locus Problems via Transformation|Rectangular Hyperbola|Approximation via Differentials

Why This Question Is Tricky

Common Mistakes

This question exploits a common trap where students attempt direct Partial Fractions Decomposition on the non-linearized rational function $\frac{x^2-1}{x^4 + 2.48x^2 + 1}$ .

Because the roots of the denominator are complex and cumbersome, direct integration leads to an arctangent expression that is extremely difficult to evaluate without a calculator.

The problem specifically tests the candidate's ability to recognize that the integral defines a parameterized function $I(\lambda)$ , and that 'Approximation via Differentials' (Taylor expansion to first order) around a clean base point ( $\lambda=4$ ) bypasses the intractable partial fractions entirely.

The initial locus transformation serves as a rigorous barrier to ensure only candidates who can synthesize coordinate geometry with advanced calculus reach the integration phase.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Partial Fractions Decomposition
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free