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Question #40495

Difficulty: 8/10 | 🟢 Free

The Question

Let $P(x) = x^3 - 5x^2 + 9x - 5$ . Let $\alpha$ be the real root and $\beta \pm i\gamma$ (where $\beta, \gamma \in \mathbb{R}$ and $\gamma > 0$ ) be the complex roots of the equation $P(x) = 0$ .

Let $I = \int_{\alpha}^{\beta} \frac{2x-4}{|x^2-4x+1|} \, dx$ and $L = \lim_{n \to \infty} \sum_{k=1}^{n} \frac{2k - 2n}{|k^2 - 2kn - 2n^2|}$

Find the value of $e^{I + L}$ .

Options

A.
$\frac{4}{9}$
B.
$\frac{9}{4}$
C.
1
D.
$\frac{2}{3}$

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

Conjugate Root Theorem|Integral as the Limit of a Sum|Logarithmic Integration Modulus

Why This Question Is Tricky

Common Mistakes

This question tests the Conjugate Root Theorem to identify integration bounds and the evaluation of an Integral as the Limit of a Sum.

The core trap is the 'Logarithmic Integration Modulus'.

Students frequently ignore the absolute value in the denominator or the domain of the argument in $\ln|f(x)|$ .

Because the quadratics $x^2-4x+1$ and $x^2-2x-2$ are strictly negative on their respective integration intervals, dropping the modulus incorrectly flips the sign of the integrals, leading to the trap answer $9/4$ instead of the correct $4/9$ .

Furthermore, it rewards advanced algebraic substitution by hiding a symmetry where $L$ is simply a shifted version of $I$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Complex Numbers and Quadratic Equations - Class 11
Concept	Conjugate Root Theorem
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free